Search

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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 = ⌊(2T-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]

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]

In this paper, narrow-sense and non-narrow-sense BCH codes of length n =2(q4−1) 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]

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]

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]

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]

A novel scheme is presented for encoding and iterative soft-decision decoding of cyclic codes of prime lengths. The encoding of a cyclic code of a prime length is performed on a collection of codewords which are mapped through Galois Fourier transform into a codeword in a low-density parity-check code with a binary parity-check matrix for transmission. Using this matrix, binary iterative soft-decision decoding algorithm is applied to jointly decode a collection of codewords from the cyclic code. The joint-decoding allows for information sharing among the received vectors corresponding to the codewords in the collection during the iterative decoding process. For decoding Reed-Solomon and BCH codes of prime lengths, the proposed decoding scheme not only requires much lower decoding complexity than other soft-decision decoding algorithms for these codes, but also yields superior performance. The proposed decoding scheme can also achieve a joint-decoding gain over the maximum likelihood decoding of individual codewords. The decoding scheme is also applied to quadratic residue codes. [ABSTRACT FROM AUTHOR]

In this paper, we give the dimension and the minimum distance of two subclasses of narrow-sense primitive BCH codes over F q with designed distance δ = a q m − 1 − 1 (resp. δ = a q m − 1 q − 1) for all 1 ≤ a ≤ q − 1 , where q is a prime power and m > 1 is a positive integer. As a consequence, we obtain an affirmative answer to two conjectures proposed by C. Ding in 2015. Furthermore, using the previous part, we extend some results of Yue and Hu [16] , and we give the dimension and, in some cases, the Bose distance for a large designed distance in the range a q m − 1 q − 1 , a q m − 1 q − 1 + T for 0 ≤ a ≤ q − 2 , where T = q m + 1 2 − 1 if m is odd, and T = 2 q m 2 − 1 if m is even. [ABSTRACT FROM AUTHOR]

In the paper we investigate the structure of i -components of two classes of codes: the Kerdock codes and the duals of the linear uniformly packed codes with parameters of the primitive double-error-correcting BCH code. We prove that for any admissible length the punctured Kerdock code consists of two i -components and the duals of the linear uniformly packed codes with parameters of the primitive double-error-correcting BCH code is an i -component for any i. We give an alternative proof for the fact presented by De Caen and van Dam in 1999 that the restrictions of the Hamming scheme to the doubly shortened Kerdock codes are association schemes. [ABSTRACT FROM AUTHOR]

In magnetic recording, a standard code architecture consists of an outer Reed-Solomon code in concatenation with an inner parity code. The inner parity code is used to detect and correct common error events. Generally, a parity code with short block length performs better, as multiple error events within one block and, consequently, miscorrection are less likely. In this paper, we study an inner code that offers the same system performance as a parity code with very short block length, even as short as the symbol length (in bits) of the outer Reed-Solomon code, but with higher code rate. This code is a tensor-product code, with a Bose-Chauduri-Hocquenghem (BCH) code and a short parity code as constituent codes. The decoder for this code is not much more complex than the optimal decoder of the baseline parity-coded channel; in fact, the only additional steps are Viterbi detection matched to the channel and decoding of the BCH code. [ABSTRACT FROM AUTHOR]

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. The objective of this paper is to investigate the parameters of some q -ary cyclic codes of length n = q 2 m − 1 q + 1. The dimensions of these codes are settled and the lower bounds on their minimum distances are presented by employing the Hartmann-Tzeng bound or BCH bound. Moreover, when q = 2 , 3 , we give the largest coset leaders δ 1 modulo n and investigate the parameters of the BCH codes C (q , n , δ 1 , 1). [ABSTRACT FROM AUTHOR]

An explicit construction of an infinite family of cyclic codes is presented which, over GF(4) (resp., GF(8)), have approximately 8/9 (resp., 48/49) the redundancy of BCH codes of the same minimum distance and length. As such, the new codes are the best codes currently known in a regime where the minimum distance is fixed and the code length goes to infinity. [ABSTRACT FROM PUBLISHER]