466 results
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2. Remarks on the paper: 'On pseudo BCH-algebras'
- Author
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Andrzej Walendziak
- Subjects
Algebra ,Mathematics Subject Classification ,Mathematical sciences ,Binary relation ,Applied Mathematics ,Well-defined ,BCH code ,Computer Science::Information Theory ,Mathematics - Abstract
In [Y.B. Jun, S.S. Ahn, On pseudo BCH-algebras, Applied Mathematical Sciences, 39 (2015), 1931-1939], the authors introduced pseudo BCH-algebras and defined in such algebras a binary relation , but it is not well defined. Mathematics Subject Classification: 06F35, 03G25
- Published
- 2015
3. On Bose distance of a class of BCH codes with two types of designed distances
- Author
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Gan, Chunyu, Li, Chengju, Qian, Haifeng, and Shi, Xueying
- Published
- 2024
- Full Text
- View/download PDF
4. Mutually disjoint Steiner systems from BCH codes
- Author
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Yan, Qianqian and Zhou, Junling
- Published
- 2024
- Full Text
- View/download PDF
5. On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and Their Consequences.
- Author
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Liu, Qi, Ding, Cunsheng, Mesnager, Sihem, Tang, Chunming, and Tonchev, Vladimir D.
- Subjects
AUTOMORPHISM groups ,ALGEBRAIC coding theory ,REPRESENTATIONS of groups (Algebra) ,DATA transmission systems ,QUANTUM information science ,GROUP theory ,LINEAR codes ,CYCLIC codes - Abstract
The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
6. Infinite Families of Linear Codes Supporting More t -Designs.
- Author
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Yan, Qianqian and Zhou, Junling
- Subjects
AUTOMORPHISM groups ,CYBERNETICS ,AUTOMORPHISMS ,LINEAR codes - Abstract
Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes $\mathcal {C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^{m}$ and established that the codewords of the minimum (or the second minimum) weight in these codes support 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$ -designs with $t=3,4$. In particular, we prove that codewords of weight 7 in $\mathcal {C}_{(q,q+1,4,1)}$ support 4-designs for odd $m \geqslant 5$ and they support 3-designs for even $m \geqslant 4$ , which provide infinite classes of simple $t$ -designs with new parameters. Another significant class of $t$ -designs we produce in this paper has complementary designs with parameters 4- $(2^{2s+1}+ 1,5,5)$ ; these designs have the smallest index among all the known simple 4- $(q+1,5,\lambda)$ designs derived from codes for prime powers $q$ ; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL $(2,2^{2s+1})$ as the automorphism group constructed by Alltop in 1969. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
7. Two Families of Optimal Linear Codes and Their Subfield Codes.
- Author
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Heng, Ziling, Wang, Qiuyan, and Ding, Cunsheng
- Subjects
LINEAR codes ,CYCLIC codes ,FAMILIES - Abstract
In this paper, a family of $[{q}^{2}-1, 4, {q}^{2}-{q}-2]$ cyclic codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2}-1,{q}^{2}-5,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of cyclic codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Some of the subfield codes are optimal and some have the best known parameters. It is shown that the subfield codes are equivalent to a family of primitive BCH codes and thus the parameters of the BCH codes are solved. The duals of the subfield codes are also optimal with respect to the sphere-packing bound. A family of $[{q}^{2}, 4, {q}^{2}-{q}-1]$ linear codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2},{q}^{2}-4,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of linear codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Five infinite families of 2-designs are also constructed with three families of linear codes of this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
8. The q -Ary Antiprimitive BCH Codes.
- Author
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Zhu, Hongwei, Shi, Minjia, Wang, Xiaoqiang, and Helleseth, Tor
- Subjects
CYCLIC codes ,LINEAR codes ,DECODING algorithms ,LIQUID crystal displays - Abstract
It is well-known that cyclic codes have efficient encoding and decoding algorithms. In recent years, antiprimitive BCH codes have attracted a lot of attention. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. Some lower bounds on the minimum distance of antiprimitive BCH codes are given. The BCH codes presented in this paper have good parameters in general, containing many optimal linear codes. In particular, two open problems about the minimum distance of BCH codes of this type are partially solved in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
9. On Hulls of Some Primitive BCH Codes and Self-Orthogonal Codes.
- Author
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Gan, Chunyu, Li, Chengju, Mesnager, Sihem, and Qian, Haifeng
- Subjects
ORTHOGONALIZATION ,FINITE fields ,LINEAR codes ,LIQUID crystal displays - Abstract
Self-orthogonal codes are an important type of linear codes due to their wide applications in communication and cryptography. The Euclidean (or Hermitian) hull of a linear code is defined to be the intersection of the code and its Euclidean (or Hermitian) dual. It is clear that the hull is self-orthogonal. The main goal of this paper is to obtain self-orthogonal codes by investigating the hulls. Let $\mathcal {C}_{(r,r^{m}-1,\delta,b)}$ be the primitive BCH code over $\mathbb {F}_{r}$ of length $r^{m}-1$ with designed distance $\delta $ , where $\mathbb {F}_{r}$ is the finite field of order $r$. In this paper, we will present Euclidean (or Hermitian) self-orthogonal codes and determine their parameters by investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes. Several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances. Furthermore, some Hermitian self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code $\mathcal {C}_{(r,r^{2}-1,\delta,1)}$ and its hull in both Hermitian and Euclidean cases for $2 \le \delta \le r^{2}-1$. We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
10. Binary [ n , (n + 1)/2] Cyclic Codes With Good Minimum Distances.
- Author
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Tang, Chunming and Ding, Cunsheng
- Subjects
CYCLIC codes ,REED-Muller codes ,BINARY codes ,LINEAR codes - Abstract
The binary quadratic-residue codes and the punctured Reed-Muller codes ${\mathcal {R}}_{2}((m-1)/2, m))$ are two families of binary cyclic codes with parameters $[n, (n+1)/2, d \geq \sqrt {n}]$. These two families of binary cyclic codes are interesting partly due to the fact that their minimum distances have a square-root bound. The objective of this paper is to construct two families of binary cyclic codes of length $2^{m}-1$ and dimension near $2^{m-1}$ with good minimum distances. When $m \geq 3$ is odd, the codes become a family of duadic codes with parameters $[2^{m}-1, 2^{m-1}, d]$ , where $d \geq 2^{(m-1)/2}+1$ if $m \equiv 3 \pmod {4}$ and $d \geq 2^{(m-1)/2}+3$ if $m \equiv 1 \pmod {4}$. The two families of binary cyclic codes contain some optimal binary cyclic codes. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
11. The Subfield Codes and Subfield Subcodes of a Family of MDS Codes.
- Author
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Tang, Chunming, Wang, Qi, and Ding, Cunsheng
- Subjects
CYCLIC codes ,LIQUID crystal displays ,LINEAR codes - Abstract
Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of $[{2^{m}+1, 2u-1, 2^{m}-2u+3}]$ MDS codes for $1 \leq u \leq 2^{m-1}$ , which are cyclic, reversible and BCH codes over ${\mathrm {GF}}(2^{m})$. The objective of this paper is to study the quaternary subfield subcodes and quaternary subfield codes of a subfamily of the MDS codes for even $m$. A family of quaternary cyclic codes is obtained. These quaternary codes are distance-optimal in some cases and very good in general. Furthermore, two infinite families of 3-designs from these quaternary codes and their duals are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
12. The dual-containing primitive BCH codes with the maximum designed distance and their applications to quantum codes
- Author
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Shi, Xueying, Yue, Qin, and Wu, Yansheng
- Published
- 2019
- Full Text
- View/download PDF
13. Parameters and characterizations of hulls of some projective narrow-sense BCH codes
- Author
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Huang, Yuwen, Li, Chengju, Wang, Qi, and Du, Zongrun
- Published
- 2022
- Full Text
- View/download PDF
14. The Dual Codes of Several Classes of BCH Codes.
- Author
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Gong, Binkai, Ding, Cunsheng, and Li, Chengju
- Subjects
LINEAR codes ,HAMMING distance ,TELECOMMUNICATION systems ,CYCLIC codes - Abstract
As a special subclass of cyclic codes, BCH codes have wide applications in communication and storage systems. A BCH code of length $n$ over $\mathbb {F}_{q}$ is always relative to an $n$ -th primitive root of unity $\beta $ in an extension field of $\mathbb {F}_{q}$ , and is called a dually-BCH code if its dual is also a BCH code relative to the same $\beta $. The question as to whether a BCH code is a dually-BCH code is in general very hard to answer. In this paper, an answer to this question for primitive narrow-sense BCH codes and projective narrow-sense ternary BCH codes is given. Sufficient and necessary conditions in terms of the designed distances $\delta $ will be presented to ensure that these BCH codes are dually-BCH codes. In addition, the parameters of the primitive narrow-sense BCH codes and their dual codes are investigated. Some lower bounds on minimum distances of the dual codes of primitive and projective narrow-sense BCH codes are developed. Especially for binary primitive narrow-sense BCH codes, the new bounds on the minimum distances of the dual codes improve the classical Sidel’nikov bound, and are also better than the Carlitz and Uchiyama bound for large designed distances $\delta $. The question as to what subclasses of cyclic codes are BCH codes is also answered to some extent. As a byproduct, the parameters of some subclasses of cyclic codes are also investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
15. A 5.4 \muW Soft-Decision BCH Decoder for Wireless Body Area Networks.
- Author
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Yang, Chia-Hsiang, Huang, Ting-Ying, Li, Mao-Ruei, and Ueng, Yeong-Luh
- Subjects
BODY area networks ,IEEE 802 standard ,INTEGRATED circuits ,BIT error rate ,ERROR-correcting codes ,DECODERS (Electronics) - Abstract
This paper presents an IEEE 802.15.6 compliant soft-decision BCH decoder for energy-constrained wireless body area networks. The proposed soft-decision decoder (SDD) provides a 1 dB coding gain compared to the hard-decision decoder (HDD). The improvement in BER performance can translate into power savings at the transmitter. The energy dissipation and area of the soft-decision BCH decoder is minimized by jointly considering the algorithm, architecture, and circuit parameters. An early termination strategy is proposed to reduce the number of redundant test patterns. Probabilistic sorting is proposed to determine the test patterns, and its hardware complexity is only 54.7% of the conventional sorting method. The HDD kernel is implemented by adopting the Peterson rule, reducing the area by 44.2%. A pass-transistor logic based Chien search circuit consumes 33.3% less energy compared to the standard-cell based implementation. The chip is designed to operate at the minimum energy point of 0.29 V, yielding an energy reduction of 94% compared to a direct-mapped SDD at SNR=5\ dB. Fabricated in 90 nm CMOS, the chip dissipates 5.4 \muW at 500 kHz, achieving a throughput of 6.38 Mbps. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
16. BCH Codes for the Rosenbloom–Tsfasman Metric.
- Author
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Zhou, Wei, Lin, Shu, and Abdel-Ghaffar, Khaled A. S.
- Subjects
BCH codes ,CYCLIC codes ,HASSE diagrams ,GRAPHIC methods for partially ordered sets ,HAMMING codes - Abstract
The Rosenbloom–Tsfasman metric has attracted the attention of many researchers as a generalization of the Hamming metric that is relevant to practical problems. Codes for this metric were considered. In particular, Reed–Solomon codes were generalized to be compatible with this metric. In this paper, a generalization of BCH codes for the Rosenbloom–Tsfasman metric is proposed. This generalization is based on considering BCH codes as subfield subcodes of Reed–Solomon codes. By characterizing these subfield subcodes, an explicit construction of BCH codes for the Rosenbloom–Tsfasman metric is provided. Two important properties of Reed–Solomon codes and BCH codes for the Rosenbloom–Tsfasman metric are studied and compared with those for the Hamming metric. These properties are cyclic structure and duality. The approach is based on Galois-Fourier transforms associated with Hasse derivatives. [ABSTRACT FROM PUBLISHER]
- Published
- 2016
- Full Text
- View/download PDF
17. A class of narrow-sense BCH codes over Fq of length qm-12.
- Author
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Ling, Xin, Mesnager, Sihem, Qi, Yanfeng, and Tang, Chunming
- Subjects
LINEAR codes ,QUADRATIC forms ,DECODING algorithms ,CIPHERS - Abstract
BCH codes with efficient encoding and decoding algorithms have many applications in communications, cryptography and combinatorial design. This paper studies a class of linear codes of length q m - 1 2 over F q with special trace representation, where q is an odd prime power. With the help of the inner distributions of some subsets of association schemes of quadratic forms, we determine the weight enumerators of these codes. Determining some cyclotomic coset leaders δ i of cyclotomic cosets modulo q m - 1 2 , we prove that narrow-sense BCH codes of length q m - 1 2 with designed distance δ i = q m - q m - 1 2 - 1 - q ⌊ m - 3 2 ⌋ + i - 1 2 have the corresponding trace representation, and have the minimal distance d = δ i and the Bose distance d B = δ i , where 1 ≤ i ≤ ⌊ m + 11 6 ⌋ . [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
18. The covering radii of a class of binary cyclic codes and some BCH codes
- Author
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Kavut, Selçuk and Tutdere, Seher
- Published
- 2019
- Full Text
- View/download PDF
19. Reliable State Estimation of an Unmanned Aerial Vehicle Over a Distributed Wireless IoT Network.
- Author
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Noor-A-Rahim, Md., Khyam, M. O., Ali, G. G. Md. Nawaz, Liu, Zilong, Pesch, Dirk, and Chong, Peter H. J.
- Subjects
TELECOMMUNICATION systems ,WIRELESS Internet ,WIRELESS communications ,INTERNET of things ,WIRELESS sensor networks ,REMOTELY piloted vehicles - Abstract
Unmanned aerial vehicles (UAVs) have attracted a lot of attention due to their enormous potentiality in civil and military applications over the past years. In order to allow accurate control action of UAV, a robust and real-time state estimation technique is required. In this paper, we propose a Kalman filter based UAV state estimation technique when the communication takes place over wireless links in an Internet of Things (IoT) network. We consider that a set of sensors observes the state of the UAV and transmits the observation to a control center (central server) over a distributed wireless IoT network. To deal with the communication impairments due to wireless communication links between the UAV's sensors and the IoT system components, e.g., IoT gateways, a Bose–Chaudhuri–Hocquenghem coded communication system is presented. Based on the received signals at the IoT gateways, a global state estimation technique is proposed. Performance of the proposed communication and estimation scheme is demonstrated through numerical results for different conditions. From the comparison with a conventional estimation scheme, it is observed that the proposed scheme significantly outperforms the conventional scheme in terms of state estimation and error performance. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
20. A class of negacyclic BCH codes and its application to quantum codes
- Author
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Zhu, Shixin, Sun, Zhonghua, and Li, Ping
- Published
- 2018
- Full Text
- View/download PDF
21. Optimization of a Systolic Array BCH encoder with Tree-Type Structure.
- Author
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Lim, Duk-Gyu, Shakya, Sharad, and Lee, Je-Hoon
- Subjects
SYSTOLIC array circuits ,ERROR-correcting codes ,PARALLEL processing ,ALGORITHMS ,COMPARATIVE studies ,FIELD programmable gate arrays - Abstract
BCH code is one of the most widely used error correcting code for the detection and correction of random errors in the modern digital communication systems. The conventional BCH encoder that is operated in bit-serial manner cannot adequate with the recent high speed appliances. Therefore, parallel encoding algorithms are always a necessity. In this paper, we introduced a new systolic array type BCH parallel encoder. To study the area and speed, several parallel factors of the systolic array encoder is compared. Furthermore, to prove the efficiency of the proposed algorithm using tree-type structure, the throughput and the area overhead was compared with its counterparts also. The proposed BCH encoder has a great flexibility in parallelization and the speed was increased by 40% than the original one. The results were implemented on synthesis and simulation on FPGA using VHDL. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
22. Decoding of QOSTBC Concatenates BCH Code Based on Parallel Interference Cancellation.
- Author
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Yan Zheng-Hang, Yang Yu-Hang, Ma Maode, and Lu Yilong
- Subjects
MAXIMUM likelihood statistics ,ALGORITHMS ,SIGNAL-to-noise ratio ,ANTENNAS (Electronics) ,FORWARD error correction - Abstract
It has been shown that quasi orthogonal space time block code (QOSTBC) can achieve high transmission rate with partial diversity. In this paper, a QOSTBC concatenating Bose-Chaudhuri- Hocquenghem (BCH) code structure is presented. At the receiver, pairwise detection and error correction are first implemented. The decoded data are regrouped. Parallel interference cancellation (PIC) and dual orthogonal space time block code (OSTBC) decoding are deployed to the regrouped data. The pure concatenated scheme is shown to have higher diversity order and better error performance at high signal-to-noise ratio (SNR) scenario than both QOSTBC and OSTBC schemes. The PIC and dual OSTBC decoding algorithm can further obtain approximate 1.2 dB gains than the pure concatenated scheme at 10
-6 bit error probability. [ABSTRACT FROM AUTHOR]- Published
- 2011
23. Coding Schemes for Noiseless and Noisy Asynchronous CDMA Systems.
- Author
-
Ying-Wah Wu and Shih-Chun Chang
- Subjects
ASYNCHRONOUS transfer mode ,WIRELESS communications ,CODE division multiple access ,SPREAD spectrum communications ,INFORMATION theory - Abstract
Novel coding schemes for noiseless and noisy asynchronous code division multiple access (A-CDMA) systems are presented in this paper. The schemes use Wu-Chang spreading code, block interleaver, and synchronizable channel codes to support A-CDMA communications with random delays. For the noiseless case, each active user generates one of M messages, M = ⌊(2
T-D - 1)/(D + 1)⌋, and a (T - D)-stage MLSR encoder encodes the message into a codeword of length T. Given a maximum random delay of D chips, the receiver can decode messages of all active users without ambiguity. For the noisy case, extended Bose-Caldwell cyclic codes are used to encode messages. The extended cyclic codes provide error correction and delay recovery effectively. By using a BCH code of length n as the cyclic code, the extended code can correct up to τ ≤ ⌊n/4⌋ errors and recover a maximum random delay of n - 1 chips. [ABSTRACT FROM AUTHOR]- Published
- 2010
- Full Text
- View/download PDF
24. Cancelable Fingerprint Cryptosystem Using Multiple Spiral Curves and Fuzzy Commitment Scheme.
- Author
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Sandhya, Mulagala and Prasad, Munaga V. N. K.
- Subjects
BIOMETRIC identification ,HUMAN fingerprints ,FUZZY algorithms ,CURVILINEAR motion ,PARAMETRIC equations ,ANTHROPOMETRY ,FUZZY sets - Abstract
The increased use of biometric-based authentication systems in a variety of applications has made biometric template protection an important issue. Unlike conventional systems, biometric cannot be revoked or changed. This made template protection a critical issue to be considered in the recent years. This paper proposes a cancelable fingerprint cryptosystem using multiple spiral curves and fuzzy commitment scheme. The method is built by combining cancelable biometrics and biometric cryptosystems. First, we compute transformed minutiae features using multiple spiral curves. Further, these transformed features are encrypted using fuzzy commitment scheme. Hence, a secure template is obtained. Experimental results and analysis prove the credibility of proposed method with recently presented methods of fingerprint template protection. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
25. A class of narrow-sense BCH codes over Fqqm-12 of length Fqqm-12
- Author
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Ling, Xin, Mesnager, Sihem, Qi, Yanfeng, and Tang, Chunming
- Published
- 2020
- Full Text
- View/download PDF
26. New Quantum BCH Codes of Length n=2(q4−1).
- Author
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Wang, Jun Li, Li, Rui Hu, Ma, Yue Na, and Liu, Yang
- Subjects
CIPHERS ,NANOELECTRONICS ,DIMENSIONS - Abstract
In this paper, narrow-sense and non-narrow-sense BCH codes of length n =2(q
4 −1) that contain their Hermitian duals over Fq2 are studied, where q is an odd prime power. Theory of cyclotomic coset is applied to confirm the maximum designed distances and dimensions of these BCH codes. Finally, some new quantum codes are obtained from such BCH codes, many of our quantum codes are much better than those from narrow-sense BCH codes in [12]. [ABSTRACT FROM AUTHOR]- Published
- 2019
- Full Text
- View/download PDF
27. Some binary BCH codes with length n = 2m + 1.
- Author
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Liu, Yang, Li, Ruihu, Fu, Qiang, Lu, Liangdong, and Rao, Yi
- Subjects
- *
BCH codes , *BINARY codes , *CYCLOTOMIC fields , *CYCLIC codes , *FINITE fields - Abstract
Abstract Under research for nearly sixty years, Bose–Chaudhuri–Hocquenghem (BCH) codes have played increasingly important roles in many applications such as communication, data storage and information security. However, the dimension and minimum distance of BCH codes have been seldom solved by now because of their intractable characteristics. The objective of this paper is to study the dimensions of some binary BCH codes with length n = 2 m + 1. Many new techniques are employed to investigate the coset leaders modulo n. For m = 2 t + 1 , 4 t + 2 , 8 t + 4 and m ≥ 10 , the first five largest coset leaders modulo n are determined, and the dimensions of some BCH codes of length n with designed distance δ > 2 ⌈ m 2 ⌉ are presented. These new skills and results may be helpful to study other types of cyclic codes over finite fields. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
28. The q-Ary Antiprimitive BCH Codes
- Author
-
Hongwei Zhu, Tor Helleseth, Xiaoqiang Wang, and Minjia Shi
- Subjects
Discrete mathematics ,Computer science ,Minimum distance ,Data_CODINGANDINFORMATIONTHEORY ,Library and Information Sciences ,Type (model theory) ,Computer Science Applications ,Finite field ,Computer Science::Discrete Mathematics ,Encoding (memory) ,Decoding methods ,BCH code ,Computer Science::Information Theory ,Information Systems - Abstract
It is well-known that cyclic codes have efficient encoding and decoding algorithms. In recent years, antiprimitive BCH codes have attracted a lot of attention. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. Some lower bounds on the minimum distance of antiprimitive BCH codes are given. The BCH codes presented in this paper have good parameters in general, containing many optimal linear codes. In particular, two open problems about the minimum distance of BCH codes of this type are partially solved in this paper.
- Published
- 2022
29. Efficient Sub-Codeword Key Equation Solver for Generalized Integrated Interleaved BCH Decoder
- Author
-
Xinmiao Zhang and Zhenshan Xie
- Subjects
Iterative and incremental development ,Computer science ,Code word ,Key equation solver ,Data_CODINGANDINFORMATIONTHEORY ,Electrical and Electronic Engineering ,Solver ,Arithmetic ,Error detection and correction ,Throughput (business) ,Decoding methods ,BCH code - Abstract
Generalized integrated interleaved (GII) error-correcting codes nest sub-codewords to form codewords of more powerful codes. They can achieve hyper-speed decoding with good error correction capability. For GII codes built on BCH codes, the first decoding stage is to decode individual BCH sub-words. This stage largely determines the throughput and dominates the area of the overall decoder. Unlike that in traditional BCH decoding, longer polynomials need to be kept in the key-equation solver (KES) step of the first stage in order to continue the KES step in the second-stage nested decoding of GII codes. To take advantage of the very fast storage class memories (SCMs), GII codes with 3-error-correcting BCH sub-codewords can be utilized. This paper proposes a low-complexity and high-speed design for the KES of 3-error-correcting BCH sub-word decoding. Formulas are developed to compute the KES results directly instead of utilizing the traditional iterative process. More importantly, through analyzing the properties of the involved variables, the coefficients are scaled and reformulated to substantially reduced the complexity. Detailed hardware implementation architectures are also developed in this paper. Our design achieves three times throughput with 20.2% smaller area than the best prior design for a code over GF(210).
- Published
- 2022
30. On Hulls of Some Primitive BCH Codes and Self-Orthogonal Codes
- Author
-
Chunyu Gan, Sihem Mesnager, Chengju Li, and Haifeng Qian
- Subjects
Dimension (graph theory) ,Order (ring theory) ,Library and Information Sciences ,Type (model theory) ,Hermitian matrix ,Upper and lower bounds ,Linear code ,Computer Science Applications ,Combinatorics ,Finite field ,Mathematics::Metric Geometry ,BCH code ,Information Systems ,Mathematics - Abstract
Self-orthogonal codes are an important type of linear codes due to their wide applications in communication and cryptography. The Euclidean (or Hermitian) hull of a linear code is defined to be the intersection of the code and its Euclidean (or Hermitian) dual. It is clear that the hull is self-orthogonal. The main goal of this paper is to obtain self-orthogonal codes by investigating the hulls. Let $\mathcal {C}_{(r,r^{m}-1,\delta,b)}$ be the primitive BCH code over $\mathbb {F}_{r}$ of length $r^{m}-1$ with designed distance $\delta $ , where $\mathbb {F}_{r}$ is the finite field of order $r$ . In this paper, we will present Euclidean (or Hermitian) self-orthogonal codes and determine their parameters by investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes. Several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances. Furthermore, some Hermitian self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code $\mathcal {C}_{(r,r^{2}-1,\delta,1)}$ and its hull in both Hermitian and Euclidean cases for $2 \le \delta \le r^{2}-1$ . We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension.
- Published
- 2021
31. On the dimension and minimum distance of BCH codes over GF( q).
- Author
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Dianwu, Yue and Zhengming, Hu
- Abstract
In this paper, only narrow-sense primitive BCH codes over GF( q) are considered. A formula, that can be used in many cases, is first presented for computing the dimension of BCH codes. It improves the result given by MacWilliams and Sloane in 1977. A new method for finding the dimension of all types of BCH codes is proposed. In second part, it is proved that the BCH bound is the leader of some cyclotomic coset, and we guess that the minimum distance for any BCH code is also the leader of some cyclotomic coset. [ABSTRACT FROM AUTHOR]
- Published
- 1996
- Full Text
- View/download PDF
32. Reduced-Complexity Key Equation Solvers for Generalized Integrated Interleaved BCH Decoders
- Author
-
Zhenshan Xie and Xinmiao Zhang
- Subjects
Block code ,Computer science ,020208 electrical & electronic engineering ,Clock rate ,020206 networking & telecommunications ,Data_CODINGANDINFORMATIONTHEORY ,02 engineering and technology ,Linear code ,0202 electrical engineering, electronic engineering, information engineering ,Code (cryptography) ,Electrical and Electronic Engineering ,Arithmetic ,Error detection and correction ,BCH code ,Decoding methods ,Block (data storage) - Abstract
Generalized integrated interleaved (GII) codes nest linear block codewords to generate codewords belonging to stronger linear block codes. They can achieve very high throughput with excellent error-correcting capability. GII codes can be based on either Reed-Solomon (RS) or BCH codes. GII-BCH codes are the most promising candidate for error correction in next-generation terabit/s Flash and storage class memories, and optical communications. On the other hand, GII decoder implementation faces many challenges. In particular, the key equation solver (KES) in the nested decoding process causes not only clock frequency bottleneck but also large area. Although techniques have been developed recently to address these issues for GII-RS codes, they cannot be directly extended for binary GII-BCH codes if every other iteration of the nested KES is skipped to reduce the latency. Two instead of one higher-order syndromes need to be incorporated into each nested KES iteration and this makes the implementation much more challenging. In this paper, algorithmic reformulations and architectural modifications are developed to eliminate the clock frequency bottleneck and reduce the area of GII-BCH nested KES. Additionally, the number of processing elements is substantially reduced by analyzing the minimum number of coefficients to keep for the involved polynomials without undesirable degradation on the error-correcting performance. The critical path of the proposed scaled nested BCH KES architecture with reduced processing elements is only one multiplier and three adders. For an example GII-BCH code over ${GF}(2^{12})$ that has a ${t}_{v}=58$ -error-correcting code as the most powerful code generated by the nesting, our design also further reduces the area by 33%.
- Published
- 2020
33. An SRAM-Based PUF With a Capacitive Digital Preselection for a 1E-9 Key Error Probability
- Author
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Yizhak Shifman, Avi Miller, Yoav Weizman, Joseph Shor, and Osnat Keren
- Subjects
Computer science ,business.industry ,Concatenated error correction code ,020208 electrical & electronic engineering ,Physical unclonable function ,Word error rate ,02 engineering and technology ,Code rate ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Node (circuits) ,Static random-access memory ,Electrical and Electronic Engineering ,Error detection and correction ,business ,BCH code ,Computer hardware - Abstract
In this paper, a novel architecture of an SRAM-based Physical Unclonable Function (PUF) whose unstable cells are identified and masked by a “capacitive tilt” preselection test is proposed. During the preselection test, either one of the cross-coupled inverter nodes is loaded with a digitally controlled capacitance, which can tilt the PUF towards a “1” or a “0”. Any cell which maintains its value during this test is deemed stable. Each capacitor bank is shared between four PUF cells, such that the additional area required is only 20%. This test does not require precision analog voltages, as did previous tests, and can thus be implemented by the user during key enrollment. In a test-chip implementation in TSMC 65nm node, the preselection test identified and disqualified all the cells that were measured as unstable in 18 PUF arrays, of 800 bits each. An overall scheme for PUF stabilization is proposed, composed of two error-reductions steps, comprising Inner and Outer error control blocks. The outer error control is typically a BCH code, which guarantees a required key error rate. The inner error control can either be the proposed tilt test, a BCH code, or another error correction scheme. To maximize the ratio of reliable key bits to physical PUF cells, i.e., the overall PUF code rate, several solutions were analyzed. It is shown that in order to achieve a key error rate of 1E-9, the most efficient construction is a tilt test followed by BCH[184,128,15], with a PUF code rate of 0.49. In addition, the tilt test is more efficient in terms of power, area and execution time than the other inner error correction mechanisms. Therefore, this construction is not only efficient by its overall code rate, but also by its area and its power, and further, it enables a more lightweight BCH implementation for the outer code.
- Published
- 2020
34. Two Families of Optimal Linear Codes and Their Subfield Codes
- Author
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Cunsheng Ding, Qiuyan Wang, and Ziling Heng
- Subjects
Combinatorics ,0202 electrical engineering, electronic engineering, information engineering ,020206 networking & telecommunications ,Dual polyhedron ,02 engineering and technology ,Library and Information Sciences ,Prime power ,BCH code ,Griesmer bound ,Computer Science Applications ,Information Systems - Abstract
In this paper, a family of $[{q}^{2}-1, 4, {q}^{2}-{q}-2]$ cyclic codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2}-1,{q}^{2}-5,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of cyclic codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$ . Some of the subfield codes are optimal and some have the best known parameters. It is shown that the subfield codes are equivalent to a family of primitive BCH codes and thus the parameters of the BCH codes are solved. The duals of the subfield codes are also optimal with respect to the sphere-packing bound. A family of $[{q}^{2}, 4, {q}^{2}-{q}-1]$ linear codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2},{q}^{2}-4,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of linear codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$ . Five infinite families of 2-designs are also constructed with three families of linear codes of this paper.
- Published
- 2020
35. Dimensions of three types of BCH codes over [formula omitted].
- Author
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Liu, Hao, Ding, Cunsheng, and Li, Chengju
- Subjects
- *
BCH codes , *FINITE fields , *MATHEMATICAL formulas , *INFORMATION storage & retrieval systems , *COMMUNICATION - Abstract
BCH codes have been studied for over fifty years and widely employed in consumer devices, communication systems, and data storage systems. However, the dimension of BCH codes is settled only for a very small number of cases. In this paper, we study the dimensions of BCH codes over finite fields with three types of lengths n , namely n = q m − 1 , n = ( q m − 1 ) ∕ ( q − 1 ) and n = q m + 1 . For narrow-sense primitive BCH codes with designed distance δ , we investigate their dimensions for δ in the range 1 ≤ δ ≤ q ⌈ m 2 ⌉ + 1 . For non-narrow sense primitive BCH codes, we provide two general formulas on their dimensions and give the dimensions explicitly in some cases. Furthermore, we settle the minimum distances of some primitive BCH codes. We also explore the dimensions of the BCH codes of lengths n = ( q m − 1 ) ∕ ( q − 1 ) and n = q m + 1 over finite fields. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
36. Low-Latency Unfolded-KES Architecture for Emerging Storage Class Memories
- Author
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Youngjoo Lee, Jeongwon Choe, and Seungsik Moon
- Subjects
Recovery cycle ,Computer science ,020208 electrical & electronic engineering ,02 engineering and technology ,Parallel computing ,Energy consumption ,Solver ,0202 electrical engineering, electronic engineering, information engineering ,Electrical and Electronic Engineering ,Architecture ,Latency (engineering) ,Storage class memory ,Decoding methods ,BCH code - Abstract
This paper presents an advanced key-equation solver (KES) algorithm that can reduce the computing latency of BCH decoding for the high-speed storage class memory (SCM). Adopting the unfolding algorithm as a factor of two, compared to the conventional iterative KES scheme, the proposed work potentially halves the number of processing cycles for KES module, which is normally dominates the overall BCH decoding latency. In contrast that the straight-forward unfolding method increases the critical delay, we accelerate the major computing path that is activated at the most of SCM lifetime, preserving the critical delay of the proposed KES module as similar to that of the original one. When the minor cases are detected, the recovery processing is added at the end of the corresponding iteration. In order to reduce the additional energy consumption due to the unfolded architecture, we carefully deactivate the internal modules during the accelerated processing, which only necessitate for the recovery cycle. Implementation results show that the proposed KES architecture greatly reduces the decoding latency of arbitrary BCH decoder, leading to the high-speed and reliable emerging storages.
- Published
- 2020
37. A Machine Learning Approach to Phase Reference Estimation With Noise
- Author
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Ning Xie, Alex X. Liu, and Le Ou-Yang
- Subjects
business.industry ,Computer science ,Machine learning ,computer.software_genre ,Phase-locked loop ,Phase noise ,Turbo code ,Demodulation ,Artificial intelligence ,Electrical and Electronic Engineering ,business ,computer ,BCH code ,Decoding methods ,Phase-shift keying - Abstract
This paper concerns the problem of phase reference estimation with noise, introduced by the imperfect phase-locked loop (PLL) circuit, or the imperfect channel estimation, or both. Prior solutions for suppressing phase noise focus on improving the accuracy of phase reference estimation. The accuracy of phase reference estimation is not high enough due to the following two limitations. First, since the PLL circuit works in radio-frequency (RF), a PLL circuit with high accuracy leads to high cost and high complexity, which makes the deployment difficult. Second, as data rates increase and wireless channels become more complex, the receiver is more difficult to obtain an ideal channel estimation and the negative effect of phase noise becomes more apparent. In this paper, we propose a machine learning approach to mitigate the negative effect of phase noise by using clustering algorithms. The key intuition of our approach is that the clustering algorithm can adaptively trace the shifted constellation point due to the phase noise. Our approach is adaptive because it can adaptively find each received symbol belongs to its original constellation point if the phase noise is not too large, e.g. , no larger than $0.25 \pi $ . While the shifted distance is not too large, we can map the received symbols into the correct constellation point to mitigate the negative effect of phase noise. Instead of directly using conventional clustering algorithms into the proposed machine learning approach, we propose a new weighted ensemble clustering algorithm to further improve the performance of our approach. In comparison with prior approaches based on RF circuits, our approach has comparable reception performance but with low complexity and low cost. Our experimental results show that, for a QPSK system, our approach improves the demodulation performance and the decoding performance about 10 dB, 8 dB under BCH codes, and 3 dB under Turbo codes, respectively. Even the demodulation performance of our approach without channel coding is better than the decoding performance of the system with channel coding about 5 dB under BCH codes.
- Published
- 2020
38. Generator Polynomial Model-Based Eye Diagram Estimation Method for Bose-Chaudhuri-Hocquenghem (BCH) Code and Reed-Solomon (RS) Code
- Author
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Joungho Kim and Junyong Park
- Subjects
Polynomial code ,Computer science ,Code word ,020206 networking & telecommunications ,Data_CODINGANDINFORMATIONTHEORY ,02 engineering and technology ,Condensed Matter Physics ,Atomic and Molecular Physics, and Optics ,Bathtub curve ,Reed–Solomon error correction ,0202 electrical engineering, electronic engineering, information engineering ,Code (cryptography) ,Electrical and Electronic Engineering ,Encoder ,Algorithm ,BCH code ,Block (data storage) - Abstract
This paper proposes a generator polynomial model-based eye diagram estimation method for the error-correction codes (ECCs), which are the Bose–Chaudhuri–Hocquenghem (BCH) code and the Reed–Solomon (RS) code. The ECCs compensate for the channel loss at the bit level, so most of the systems include the ECC block. The BCH and RS codes are the generator polynomial-based ECCs; this paper proposes the generator polynomial model-based method. The model provides the accurate probability density function of the code words for the BCH and RS code. Thus, the proposed method is capable of achieving the signal integrity analysis for the system level. For verification, we design the BCH and RS encoders. The estimated eye diagrams are nearly the same as that of the transient simulation for the BCH and RS codes. Furthermore, the voltage bathtub curves are the same up to 10−8. For a smaller bit-error rate value, the proposed method only provides the bathtub curve. This paper successfully verifies the eye diagram estimation method for the BCH and RS codes.
- Published
- 2020
39. An approach to PAPR Reduction in OFDM using Goppa codes
- Author
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Sharmila Sengupta and B. K. Lande
- Subjects
Orthogonal frequency-division multiplexing ,Computer science ,Modulation index ,Data_CODINGANDINFORMATIONTHEORY ,02 engineering and technology ,Multiplexing ,Scrambling ,Binary Golay code ,0202 electrical engineering, electronic engineering, information engineering ,Electronic engineering ,Forward error correction ,General Environmental Science ,Channel code ,Transmitter ,020206 networking & telecommunications ,Spectral efficiency ,Modulation ,Goppa code ,Bit error rate ,General Earth and Planetary Sciences ,020201 artificial intelligence & image processing ,Hamming code ,BCH code ,Multipath propagation ,Communication channel ,Diversity scheme - Abstract
Orthogonal frequency division multiplexing (OFDM) is a widely used modulation and multiplexing technology applied in many telecommunications standards due to its spectral efficiency and robustness against multipath fading. The multicarrier aspect of OFDM signal is characterized by high peak-to-average power ratio (PAPR), which renders the power amplifier (PA) inefficient and causes distortion in the transmitted signal. Several techniques to overcome and reduce PAPR such as signal distortion, signal scrambling, coding etc. are proposed and implemented on multipath fading channels. Coding methods does not affect system complexity as forward error correction is an inherent block in any digital communication system. The frequency diversity benefits of OFDM is not utilized well unless channel coding is used. Several codes like Hamming, Golay, BCH, RS, Reed Muller etc. have been investigated so far. Goppa codes were needed to be explored in OFDM systems. In this paper, its ability to reduce the peak power is studied for conventional OFDM signal. These codes are used in computer and telecommunication based applications owing to its useful properties required for cryptography. The PAPR reduction of Goppa coded OFDM (GOFDM) is found to be satisfactory for higher order constellation mapping of binary information. A look up table can also be created to select Goppa code words on the basis of a desired threshold of PAPR in dB. A mathematical analysis of the relationship of input data and its PAPR is also executed and certain conclusions are drawn as shown in the lemmas. But coding methods solely cannot achieve substantial PAPR reduction and therefore most of the literature on PAPR reduction methods have combined coding with other techniques. In this paper, G-OFDM is extended to a hybrid method of constant amplitude modulated (CAM)-G-OFDM to completely mitigate the PAPR problem. But it is achieved at the cost of transmitter and receiver complexity and performance parameters like bit error rate (BER), dependency on modulation index etc. which can be taken care of by making some amendments in the system design.
- Published
- 2020
40. Efficient image tampering localization using semi-fragile watermarking and error control codes
- Author
-
Pascal Lefèvre, Caroline Fontaine, Jiwu Huang, Philippe Gaborit, Philippe Carré, College of Information Engineering [Shenzhen], Shenzhen Univerisity [Shenzhen], Synthèse et analyse d'images (XLIM-ASALI), XLIM (XLIM), Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS)-Université de Limoges (UNILIM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Méthodes Formelles (LMF), Institut National de Recherche en Informatique et en Automatique (Inria)-CentraleSupélec-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Ecole Normale Supérieure Paris-Saclay (ENS Paris Saclay), and Mathématiques & Sécurité de l'information (XLIM-MATHIS)
- Subjects
Block code ,Image quality ,Computer science ,02 engineering and technology ,Image (mathematics) ,[INFO.INFO-CR]Computer Science [cs]/Cryptography and Security [cs.CR] ,020401 chemical engineering ,[INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing ,Computer Science::Multimedia ,0202 electrical engineering, electronic engineering, information engineering ,Code (cryptography) ,0204 chemical engineering ,Electrical and Electronic Engineering ,Digital watermarking ,ComputingMilieux_MISCELLANEOUS ,[INFO.INFO-MM]Computer Science [cs]/Multimedia [cs.MM] ,020206 networking & telecommunications ,Control and Systems Engineering ,[INFO.INFO-IT]Computer Science [cs]/Information Theory [cs.IT] ,Signal Processing ,Computer Vision and Pattern Recognition ,Error detection and correction ,Algorithm ,Software ,Decoding methods ,BCH code - Abstract
In this paper, we propose an image tampering localization algorithm using semi-fragile watermarking and Error-Locating codes in the DWT domain. By introducing different families of codes, we show the benefit in terms of image tampering localization and complexity of using control code error localization as an authentication function. Indeed, we first experimentally show that error localization block codes is as precise as using classical error correcting codes (Reed-Solomon and BCH codes) to locate image tampering. However, their corresponding decoding algorithms complexity is at least quadratic which make them impractical for some real time applications. To solve this problem, we introduce error-control codes called Error-Locating codes where error localization is reduced to a single syndrome computation performed with low number of binary operations (detailed later in the paper). We provide comparisons of image quality and tampering localization performances using error-detection, error-localization and error-correction approaches with different error control codes.
- Published
- 2022
41. ОСОБЛИВОСТІ АЕРОДИНАМІЧНОГО РОЗРАХУНКУ ВЕНТИЛЯЦІЙНИХ СИСТЕМ У КОМП’ЮТЕРНІЙ ПРОГРАМІ MAGICAD
- Author
-
S. B. Protsenko, S. S. Perekhodko, and O. S. Novytska
- Subjects
Pipeline transport ,Flow separation ,Computer science ,Computation ,Mode (statistics) ,Mechanical engineering ,General Medicine ,Aerodynamics ,BCH code - Abstract
One of the most popular computer programs for designing interior engineering systems for buildings, including ventilation systems, is the MagiCAD program of the Finnish company MagiCAD Group. MagiCAD allows easily and quickly create three-dimensional models of ventilation systems, selecting area sections of air pipelines sections, to perform aerodynamic and acoustic calculations and balancing of air duct networks, making detailed specifications and etc. Despite the detailed documentation on the program, the MagiCAD mechanism remains hidden from the user. The method of pressure losses definition in MagiCAD is significantly different from the national standards, especially for definition of pressure losses of components. This difference should be taken into account when doing aerodynamic computations designing ventilation systems. The paper considers the mathematical dependencies used by the MagiCAD program for aerodynamic computations of air pipelines. The comparison of these dependencies with the used for national design practice is also provided in the paper. The comparison of computation results is made for specific values of friction pressure losses, pressure losses of components such as bends of round and rectangular sections of different angles, diffusers and confusers, tee fittings, etc. Despite the significant differences in the methods, the computed values of the pressure losses are quite close, the significant differences in the results of the calculations were observed only for confusers. Obviously, significant differences can be expected in the results of the calculation of the distribution nodes operating in the pressure (flow separation) mode, since its computation according to BCH 353-86 is similar to the confusers’ computation, but this issue needs further research.
- Published
- 2019
42. Energy-Efficient Symmetric BC-BCH Decoder Architecture for Mobile Storages
- Author
-
Daesung Kim, Jeongseok Ha, In-Cheol Park, Youngjoo Lee, Seokha Hwang, Seungsik Moon, and Jaehwan Jung
- Subjects
Chien search ,Computer science ,business.industry ,020208 electrical & electronic engineering ,NAND gate ,Initialization ,Data_CODINGANDINFORMATIONTHEORY ,02 engineering and technology ,Hardware and Architecture ,Encoding (memory) ,0202 electrical engineering, electronic engineering, information engineering ,Code (cryptography) ,Electrical and Electronic Engineering ,Error detection and correction ,business ,BCH code ,Decoding methods ,Computer hardware - Abstract
Recently, symmetric block-wise concatenated-BCH (SBC-BCH) codes are proposed as strong error-correcting codes (ECCs) based on hard-decision channel outputs, which is especially suited for storages using NAND flash memories. Targeting energy-efficient NAND flash memory applications, this paper presents an energy-optimized decoder architecture which includes an iterative decoder for a SBC-BCH code as a main decoder and a low-complexity auxiliary decoder for a block-wise single parity-check (BSPC) code. The auxiliary decoder is opportunistically in action to break the dominant error bound associated with the SBC-BCH code, which allows one to lower the uncorrectable bit-error-rate (UBER) to 10−15 in an energy efficient way. This work presents several design-level optimizations for further enhancing the energy-efficiency of the iterative SBC-BCH decoder. More precisely, the new initialization scheme is proposed for ensuring the energy-efficient seamless decoding scenario. The syndrome tracking is applied to eliminate the previous syndrome calculation and the reordered Chien search further enhances the energy-efficiency as well as the decoding throughput. Targeting a 0.9-rate 4KB SBC-BCH code for commercialized storages using NAND flash memories, a prototype decoder consisting of both the iterative main and auxiliary decoders is designed in a 65-nm CMOS process. By applying the proposed optimizations, the prototype decoder achieves an energy-efficiency of 3.43 pJ/b while providing a decoding throughput of 13.2 Gb/s, which is superior to the previous state-of-the-art decoders for mobile storages.
- Published
- 2019
43. On the Use of Soft-Decision Error-Correction Codes in nand Flash Memory
- Author
-
Guiqiang Dong, Ningde Xie, and Tong Zhang
- Subjects
Decision error ,Computer science ,Nand flash memory ,business.industry ,Computer engineering ,Computer data storage ,Electronic engineering ,Electrical and Electronic Engineering ,Low-density parity-check code ,Latency (engineering) ,business ,Scale down ,BCH code ,Decoding methods - Abstract
As technology continues to scale down, NAND Flash memory has been increasingly relying on error-correction codes (ECCs) to ensure the overall data storage integrity. Although advanced ECCs such as low-density parity-check (LDPC) codes can provide significantly stronger error-correction capability over BCH codes being used in current practice, their decoding requires soft-decision log-likelihood ratio (LLR) information. This results in two critical issues. First, accurate calculation of LLR demands fine-grained memory-cell sensing, which nevertheless tends to incur implementation overhead and access latency penalty. Hence, it is critical to minimize the fine-grained memory sensing precision. Second, accurate calculation of LLR also demands the availability of a memory-cell threshold-voltage distribution model. As the major source for memory-cell threshold-voltage distribution distortion, cell-to-cell interference must be carefully incorporated into the model. However, these two critical issues have not been ever addressed in the open literature. This paper attempts to address these open issues. We derive mathematical formulations to approximately model the threshold-voltage distribution of memory cells in the presence of cell-to-cell interference, based on which the calculation of LLRs is mathematically formulated. This paper also proposes a nonuniform memory sensing strategy to reduce the memory sensing precision and, thus, sensing latency while still maintaining good error-correction performance. In addition, we investigate these design issues under the scenario when we can also sense interfering cells and hence explicitly estimate cell-to-cell interference strength. We carry out extensive computer simulations to demonstrate the effectiveness and involved tradeoffs, assuming the use of LDPC codes in 2-bits/cell NAND Flash memory.
- Published
- 2011
44. A Class of Narrow-Sense BCH Codes
- Author
-
Zhonghua Sun, Shixin Zhu, and Xiaoshan Kai
- Subjects
FOS: Computer and information sciences ,Physics ,Class (set theory) ,Information Theory (cs.IT) ,Computer Science - Information Theory ,Dimension (graph theory) ,020206 networking & telecommunications ,02 engineering and technology ,Sense (electronics) ,Library and Information Sciences ,Lambda ,Computer Science Applications ,Combinatorics ,Weight distribution ,0202 electrical engineering, electronic engineering, information engineering ,BCH code ,Griesmer bound ,Information Systems - Abstract
BCH codes are an important class of cyclic codes which have applications in satellite communications, DVDs, disk drives, and two-dimensional bar codes. Although BCH codes have been widely studied, their parameters are known for only a few special classes. Recently, Ding et al. made some new progress in BCH codes. However, we still have very limited knowledge on the dimension of BCH codes, not to mention the weight distribution of BCH codes. In this paper, we generalize the results on BCH codes from several previous papers. 1) The dimension of narrow-sense BCH codes of length $((q^{m}-1)/{\lambda })$ with designed distance $2\leq \delta \leq (({q^{\lceil (m+1)/2 \rceil }-1})/(\lambda)+1)$ is settled, where $\lambda $ is any factor of $(q-1)$ . 2) The weight distributions of two classes of narrow-sense BCH codes of length $(({q^{m}-1})/2)$ with designed distance $\delta =(({(q-1)q^{m-1}-q^{\lfloor (m-1)/2\rfloor }-1})/2)$ and $\delta =(({(q-1)q^{m-1}-q^{\lfloor (m+1)/2\rfloor }-1})/2)$ are determined. 3) The weight distribution of a class of BCH codes of length $((q^{m}-1)/({q-1}))$ is determined. In particular, a subclass of this class of BCH codes is optimal with respect to the Griesmer bound. Some optimal linear codes obtained from this class of BCH codes are characterized.
- Published
- 2019
45. Advanced Bit Flip Concatenates BCH Code Demonstrates 0.93% Correctable BER and Faster Decoding on (36 864, 32 768) Emerging Memories
- Author
-
Sheyang Ning
- Subjects
010302 applied physics ,Computer science ,020208 electrical & electronic engineering ,Byte ,Data_CODINGANDINFORMATIONTHEORY ,02 engineering and technology ,01 natural sciences ,Memory array ,Decoding latency ,0103 physical sciences ,0202 electrical engineering, electronic engineering, information engineering ,Bit error rate ,Electrical and Electronic Engineering ,Arithmetic ,Low-density parity-check code ,Error detection and correction ,Decoding methods ,BCH code - Abstract
Bose–Chaudhuri–Hocquenghem (BCH) and low-density-parity-check (LDPC) are two popular error correcting codes for non-volatile memories. However, the BCH has limited error correction ability, while the LDPC requires multiple sensing operations per read. In this paper, an advanced bit flip (ABF) scheme is proposed to obtain high-correctable raw bit error rate (BER), fast decoding, and one sensing per read, simultaneously. During write, first, the ABF uses data mapping to correct major program errors. Then, the BCH is used to correct the remaining errors. The performance of concatenated ABF + BCH is calculated by using 108 writes data from a nano-random access memory array. On 32 768 user data bits, ABF + BCH reduces 54% parity size and 17% decoding latency compared with only using BCH. Furthermore, ABF + BCH performances are calculated and analyzed statistically on a hypothetical memory array with binominal error distribution. By using fixed 512 bytes parity, ABF + BCH obtains 0.93% correctable raw BER when set + retention BER is 0.1%. Moreover, ABF + BCH shows less decoding latency than only using BCH.
- Published
- 2018
46. BCH codes for 5G wireless communication systems over multipath fading channel
- Author
-
Lukman Audah and Ghasan Ali Hussain
- Subjects
Control and Optimization ,Computer Networks and Communications ,Orthogonal frequency-division multiplexing ,Computer science ,02 engineering and technology ,Data_CODINGANDINFORMATIONTHEORY ,PAPR ,0202 electrical engineering, electronic engineering, information engineering ,Bandwidth (computing) ,Electronic engineering ,Waveform ,BCH ,Electrical and Electronic Engineering ,OOBE ,Channel code ,Physical layer ,020206 networking & telecommunications ,Transmission (telecommunications) ,Hardware and Architecture ,f-OFDM ,Signal Processing ,020201 artificial intelligence & image processing ,BCH code ,Multipath propagation ,5G ,Information Systems ,Communication channel - Abstract
Due to its large peak to average power ratio (PAPR) and high out of band emission (OOBE), OFDM doesn't meet the requirements of 5G services. Additionally, it supports only one type of waveform parameters in entire bandwidth. In contrast, f-OFDM is dividing the system's bandwidth into a number of subbands to support several waveform parameters based on various service scenarios. So, Filtered-OFDM (f-OFDM) is considered as a modern enabler of the flexible waveform to overcome the OFDM drawbacks while remaining its advantages as well as, to encounter the new challenges that faced 5G. Nonetheless, there is a trade-off among OOBE, PAPR and SNR performance. Meanwhile, channel coding technology is one of the most important issue in physical layer which is playing an essential role in order to achieve the reliability and latency. So, BCH code has been suggested in this paper for f-OFDM system to achieve the reliability of transmission information and thus improving BER performance over multipath fading channel. Whilst, BCH-LTE system is introduced as a baseline in this paper that using for comparison purpose with proposed system. Simulation results showed that the proposed BCH-f-OFDM system was significantly better than BCH-LTE system in terms of decreasing OOBE and achieving improving in BER performance. Although, PAPR levels was stilling high in proposed system due to the trade-off among OOBE, PAPR and SNR performance. However, the proposed system is considered a promising candidate to meet the requirements of 5G services because of its ability to solve two important issues in between three trade-offs'.
- Published
- 2020
47. Hermitian LCD codes from cyclic codes
- Author
-
Chengju Li
- Subjects
Discrete mathematics ,business.industry ,Applied Mathematics ,020206 networking & telecommunications ,Cryptography ,0102 computer and information sciences ,02 engineering and technology ,Construct (python library) ,Type (model theory) ,01 natural sciences ,Hermitian matrix ,Computer Science::Other ,Computer Science Applications ,Finite field ,010201 computation theory & mathematics ,Encoding (memory) ,0202 electrical engineering, electronic engineering, information engineering ,business ,Decoding methods ,BCH code ,Mathematics - Abstract
Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. It was proved that asymptotically good Hermitian LCD codes exist. The objective of this paper is to construct some cyclic Hermitian LCD codes over finite fields and analyse their parameters. The dimensions of these codes are settled and the lower bounds on their minimum distances are presented. Most Hermitian LCD codes presented in this paper are not BCH codes.
- Published
- 2017
48. Narrow-Sense BCH Codes Over $ {\mathrm {GF}}(q)$ With Length $n=\frac {q^{m}-1}{q-1}$
- Author
-
Maosheng Xiong, Shuxing Li, Gennian Ge, and Cunsheng Ding
- Subjects
Discrete mathematics ,Minimum distance ,Dimension (graph theory) ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,Sense (electronics) ,Library and Information Sciences ,01 natural sciences ,Electronic mail ,Computer Science Applications ,010201 computation theory & mathematics ,Weight distribution ,0202 electrical engineering, electronic engineering, information engineering ,Ternary operation ,BCH code ,Decoding methods ,Information Systems ,Mathematics - Abstract
Cyclic codes are widely employed in communication systems, storage devices, and consumer electronics, as they have efficient encoding and decoding algorithms. BCH codes, as a special subclass of cyclic codes, are in most cases among the best cyclic codes. A subclass of good BCH codes are the narrow-sense BCH codes over $ {\mathrm {GF}}(q)$ with length $n=(q^{m}-1)/(q-1)$ . Little is known about this class of BCH codes when $q>2$ . The objective of this paper is to study some of the codes within this class. In particular, the dimension, the minimum distance, and the weight distribution of some ternary BCH codes with length $n=(3^{m}-1)/2$ are determined in this paper. A class of ternary BCH codes meeting the Griesmer bound is identified. An application of some of the BCH codes in secret sharing is also investigated.
- Published
- 2017
49. Improving bit error-rate based on adaptive Bose-Chaudhuri Hocquenghem concatenated with convolutional codes
- Author
-
Hatem Zakaria, Ahmed Samy, and Ashraf Y. Hassan
- Subjects
Decoder ,Control and Optimization ,Computer Networks and Communications ,Computer science ,Viterbi ,Concatenation ,Bose-Chaudhuri Hocquenghem ,Data_CODINGANDINFORMATIONTHEORY ,Encoder ,Hardware and Architecture ,Convolutional code ,Signal Processing ,Bit error rate ,Bandwidth (computing) ,Convolutional ,Electrical and Electronic Engineering ,Error detection and correction ,Algorithm ,BCH code ,Decoding methods ,Information Systems ,Parity bit ,Communication channel - Abstract
Several algorithms have been proposed to avoid the error floor region, such as the concatenation codes that requires high computational demands in addition to high complexity. This paper proposes a technique based on using cascaded BCH and convolutional codes that leads to better error correction performance. Moreover, an adaptive method based on sensing the channel's noise to determine the number of the parity bits that will be added to the used BCH that reduces the consumed bandwidth and the transmitted parity bits is presented. A further enhancement is fulfilled by using parallel processing branches, resulting in reducing the consumed time and speed up the performance. The results show that the proposed code presents a better performance. A high reduction in the number of cycles that will be used in the encoding and decoding compared with the classical method and finally a flexible parity bits method based on the signal-to-noise ratio of the channel that reduced the parity bits which leads to reduce the consumed bandwidth. The MATLAB simulation and the field programmable gate array (FPGA) implementation will be provided in this paper to validate the proposed concept.
- Published
- 2021
50. Construction and decoding of BCH codes over chain of commutative rings
- Author
-
Shah, Tariq, Qamar, Attiq, and de Andrade, Antonio Aparecido
- Published
- 2012
- Full Text
- View/download PDF
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