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2. Relative Generalized Hamming Weights of One-Point Algebraic Geometric Codes<xref ref-type="fn" rid="fn1">1</xref><fn id="fn1"><label>1</label><p>The paper is registered to the ORCID of Olav Geil. For more details please visit ...
- Author
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Geil, Olav, Martin, Stefano, Matsumoto, Ryutaroh, Ruano, Diego, and Luo, Yuan
- Subjects
- *
HAMMING codes , *CODING theory , *ALGEBRAIC geometric codes , *LINEAR codes , *CRYPTOGRAPHY - Abstract
Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes. In this paper, we elaborate on the implication of these parameters and devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes, our bound is often tight. Furthermore, for these codes, the relative generalized Hamming weights are often much larger than the corresponding generalized Hamming weights. [ABSTRACT FROM AUTHOR]
- Published
- 2014
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3. New Constructions of Optimal Cyclic (r, δ) Locally Repairable Codes From Their Zeros.
- Author
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Qiu, Jing, Zheng, Dabin, and Fu, Fang-Wei
- Subjects
CYCLIC codes ,REED-Solomon codes ,PAPER arts ,GENERALIZATION - Abstract
An $(r, \delta)$ -locally repairable code ($(r, \delta)$ -LRC for short) was introduced by Prakash et al. for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of $r$ -LRCs produced by Gopalan et al.. An $(r, \delta)$ -LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. generalized the construction of cyclic $r$ -LRCs proposed by Tamo et al. , and constructed several classes of optimal $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of , , this paper first characterizes $(r, \delta)$ -locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic $(r, \delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively from the product of two sets of zeros. Our constructions include all optimal cyclic $(r,\delta)$ -LRCs proposed in , , and our method seems more convenient to obtain optimal cyclic $(r, \delta)$ -LRCs with flexible parameters. Moreover, many optimal cyclic $(r,\delta)$ -LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$ , respectively with $(r+\delta -1)\nmid n$ can be obtained from our method. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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4. Weighted Block-wise Burst Error Correcting Codes.
- Author
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Sethi, Amita and Sharma, Arpana
- Abstract
In this paper, we construct new types of codes, called Weighted Block-wise Burst Error Correcting Codes (Weighted BBEC Codes), which are an improvement on the previously known block wise burst error correcting codes in the sense of their weight, i.e., (n = n
1 + n2 , k) linear codes that correct all the bursts of length b1 (fixed) with the weight w1 (w1 ≤ b1 ) or more (less) in the first sub-block of length n1 and all the bursts of length b2 (fixed) with the weight w2 or more (less) in the next sub-block of length n2 . [ABSTRACT FROM AUTHOR]- Published
- 2024
5. Optimal Quaternary Hermitian LCD Codes.
- Author
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Lu, Liangdong, Li, Ruihu, and Ren, Yuezhen
- Subjects
LINEAR codes ,CRYPTOSYSTEMS - Abstract
Linear complementary dual (LCD) codes, which are a class of linear codes introduced by Massey, have been extensively studied in the literature recently. It has been shown that LCD codes play a role in measures to counter passive and active side-channel analyses on embedded cryptosystems. In this paper, tables are presented of good quaternary Hermitian LCD codes and they are used in the construction of puncturing, shortening and combination codes. The results of this, including three tables of the best-known quaternary Hermitian LCD codes of any length n ≤ 25 with corresponding dimension k, are presented. In addition, many of these quaternary Hermitian LCD codes given in this paper are optimal and saturate the lower or upper bound of Grassl's code table, and some of them are nearly optimal. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
6. Linear feedback coding scheme for multiple-access fading channels with degraded message sets.
- Author
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Liao, Yuan and Wang, Xiaofang
- Subjects
LINEAR codes ,LINEAR network coding ,CHANNEL coding ,PHYSICAL layer security ,WIRELESS communications ,COMPUTER simulation - Abstract
Channel coding technology plays an important role in wireless communication systems, and it serves as a crucial mechanism to reduce interference during the transmission process. As the fifth-generation (5G) and sixth-generation (6G) wireless communication systems rapidly advance, requirements of the users on the quality and security of wireless service are increasing. To solve these problems, it calls for us to explore the new channel coding technologies. In this paper, a linear feedback coding scheme for fading multiple-access channels with degraded message sets (FMAC-DMS) is proposed. In this scheme, the transmitting beamforming and channel splitting are used to transform the channel with complex signals into scalar equivalent sub-channels. Then, the extended Schalkwijk-Kailath coding scheme (SK) is further applied to each sub-channel. The channel capacity, finite blocklength (FBL) sum-rate and FBL secrecy achievable sum-rate of the FMAC-DMS in single-input single-output (SISO) and multi-input single-output (MISO) cases are derived. Finally, we show that the proposed scheme not only provides a FBL coding solution but also guarantees physical layer security(PLS). The numerical and simulation results show the effectiveness of the proposed scheme as a channel coding solution. The study of this paper provides a new method to construct a practical FBL scheme for the FMAC-DMS. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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7. On a family of linear MRD codes with parameters [8×8,16,7]q.
- Author
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Timpanella, Marco and Zini, Giovanni
- Subjects
ALGEBRAIC geometry ,ALGEBRAIC varieties ,FINITE fields ,FINITE geometries ,PROJECTIVE spaces ,LINEAR codes ,FAMILIES - Abstract
In this paper we consider a family F of 2n-dimensional F q -linear rank metric codes in F q n × n arising from polynomials of the form x q s + δ x q n 2 + s ∈ F q n [ x ] . The family F was introduced by Csajbók et al. (JAMA 548:203–220) as a potential source for maximum rank distance (MRD) codes. Indeed, they showed that F contains MRD codes for n = 8 , and other subsequent partial results have been provided in the literature towards the classification of MRD codes in F for any n. In particular, the classification has been reached when n is smaller than 8, and also for n greater than 8 provided that s is small enough with respect to n. In this paper we deal with the open case n = 8 , providing a classification for any large enough odd prime power q. The techniques are from algebraic geometry over finite fields, since our strategy requires the analysis of certain 3-dimensional F q -rational algebraic varieties in a 7-dimensional projective space. We also show that the MRD codes in F are not equivalent to any other MRD codes known so far. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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8. New classes of NMDS codes with dimension 3.
- Author
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Fan, Cuiling, Wang, An, and Xu, Li
- Subjects
LINEAR codes ,ELLIPTIC curves ,ALGEBRAIC codes ,INTEGERS - Abstract
The singleton defect of an [n, k, d] linear code C is defined as s (C) = n - k + 1 - d . Codes with s (C) = s (C ⊥) = 1 are called near maximum distance separable (NMDS) codes. It is known that an [ n , 3 , n - 3 ] NMDS code is equivalent to an (n, 3)-arc in PG(2, q). In this paper, by adding some suitable projective points into some known (q + 5 , 3) -arcs in PG(2, q), we obtain two families of [ q + 7 , 3 , q + 4 ] NMDS codes for even prime power q and a family of [ q + 6 , 3 , q + 3 ] NMDS codes for odd prime power q. In addition, when q = 2 m and m is odd, by adding m suitable projective points into the maximum arcs in PG(2, q), we obtain a family of [ q + m + 2 , 3 , q + m - 1 ] NMDS codes over F q , from which we further induce a family of NMDS codes with parameters [ q t + m + 2 , 3 , q t + m - 1 ] over the extension field F q t for any odd integer t. All the resulting NMDS codes in this paper are shown to be linearly inequivalent to the NMDS codes constructed from elliptic curves, and their weight distributions are completely determined. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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9. The algebraic structure of additive codes over 28.
- Author
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Cheng, Xiangdong
- Subjects
FINITE fields ,LINEAR operators ,ADDITIVES ,LINEAR codes - Abstract
In this paper, we investigate the algebraic structure of 2 r 8 s -additive codes, where r and s are nonnegative integers, 2 (respectively, 8 ) denotes the finite field of order 2 (respectively, 8). We first give the generator polynomials of additive cyclic codes over 8 and then the generator polynomials of additive cyclic codes over 2 8 is also given. In addition, we introduce a linear map W : 2 8 → 2 , and study its properties. What's more, the dual of additive cyclic codes over 2 8 are investigated as well. And we get that the duals of any additive cyclic codes over 2 8 are also additive cyclic codes. Finally, separable 2 8 -additive cyclic codes are investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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10. On Building Self-Complementary Codes and Their Application in Information Hiding.
- Author
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Kosolapov, Y. V., Pevnev, F. S., and Yagubyants, M. V.
- Abstract
Line codes are widely used to protect data transmission and storage systems against errors, keep various cryptographic algorithms and protocols working stably, and protect hidden information from errors in a stegocontainer. One of the code classes applied in a number of the listed areas is linear self-complementary codes over a binary field. These codes contain a vector of all ones and their weight numerator is a symmetric polynomial. In applied problems, self-complementary [n, k] codes are often required to have the maximal possible code distance d(k, n) at given length n and size k. The values of d(k, n) are already known for n < 13. The task formulated in this paper for self-complementary codes with length n = 13, 14, 15 is to find lower estimates of d(k, n) and values proper of d(k, n). The development of an efficient method for obtaining a lower estimate close to d(k, n) is an urgent task, because finding values proper of d(k, n) is generally a difficult task. The paper proposes four methods for finding lower estimates. These methods are based on cyclic codes, residual codes, the (u|u + v) structure, and the tensor product of codes. The methods are used together for the considered lengths to efficiently obtain lower estimates which either coincide with found values of d(k, n) or differ from them by one. The paper proposes a sequence of checks, which in some cases helps prove the absence of a self-complementary [n, k] code with code distance d. The final part of the work proposes an information hiding structure based on self-complementary codes. This structure is resistant to interference in the stegocontainer. The calculations show that the new structure is more efficient when compared with the known counterparts. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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11. Optimal Ferrers diagram rank-metric codes from MRD codes.
- Author
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Liu, Shuangqing
- Subjects
LINEAR network coding ,ALGEBRAIC coding theory ,LINEAR codes - Abstract
Subspace codes, motivated by their extensive application in random network coding, have become one of central topics in algebraic coding theory during the last 10 years. Due to the significant application in subspace codes, Ferrers diagram rank-metric (FDRM) codes also have drawn a lot of attention. In this paper, we introduce two new constructions based on subcodes of MRD codes. The first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes. Apply the first construction to solve the optimality of [ F , 4 ] q -FDRM codes, where F = [ 2 , 2 , 4 , 4 , ... , 2 l , 2 l ] , which was raised in Etzion et al. (IEEE Trans Inf Theory 62:1616–1630, 2016). By the restricted Gabidulin codes and improving the way to select the subcodes, the second construction is presented, which unifies and generalizes all known constructions based on subcodes of Gabidulin codes. By the second construction, we can give new families of optimal FDRM codes, whose numbers of codewords are unequal to q v 0 . This paper also shows new families of FDRM codes whose optimality cannot be obtained by the constructions based on subcodes of F q m -linear MRD codes. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
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12. Steiner systems S(2,4,2m) supported by a family of extended cyclic codes.
- Author
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Wang, Qi
- Subjects
CYCLIC codes ,EXTENDED families ,STEINER systems ,LINEAR codes - Abstract
In [C. Ding, An infinite family of Steiner systems $ S(2, 4, 2^m) $ from cyclic codes, J. Combin. Des. 26 (2018), no.3, 126–144], Ding constructed a family of Steiner systems $ S(2, 4, 2^m) $ for all $ m \equiv 2 \pmod{4} \ge 6 $ from a family of extended cyclic codes. The objective of this paper is to present a family of Steiner systems $ S(2, 4, 2^m) $ for all $ m \equiv 0 \pmod{4} \ge 4 $ supported by this family of extended cyclic codes. The main result of this paper complements the previous work of Ding, and the results in the two papers will show that there exists a binary extended cyclic code that can support a Steiner system $ S(2, 4, 2^m) $ for all even $ m \geq 4 $. Furthermore, this paper also determines the parameters of other $ 2 $-designs supported by this family of extended cyclic codes. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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13. Weight Enumerators and Cardinalities for Number-Theoretic Codes.
- Author
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Nozaki, Takayuki
- Subjects
HAMMING weight ,RINGS of integers ,BINARY codes ,LINEAR codes ,ERROR-correcting codes - Abstract
The number-theoretic code is a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts’ codes, correcting single insertion or deletion. Moreover, we show that the formula deduces the MacWilliams identity for the linear codes over the ring of integers modulo $r$. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
14. Reversible codes in the Rosenbloom-Tsfasman metric.
- Author
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Gopinadh, Bodigiri Sai and Marka, Venkatrajam
- Subjects
LINEAR codes ,BINARY codes ,TELECOMMUNICATION systems ,DATA warehousing ,CRYPTOGRAPHY - Abstract
Reversible codes have a range of wide applications in cryptography, data storage, and communication systems. In this paper, we investigated reversible codes under the Rosenbloom-Tsfasman metric (RT-metric). First, some properties of reversible codes in the RT-metric were described. An essential condition for a reversible code to be a maximum distance separable code (MDS code, in short) in the RT-metric was established. A necessary condition for a binary self-dual code to be reversible was proven and the same was generalized for q-ary self-dual reversible codes. Several constructions for reversible RT-metric codes were provided in terms of their generator matrices. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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15. Construction of quantum codes from multivariate polynomial rings.
- Author
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Yu, Cong, Zhu, Shixin, and Tian, Fuyin
- Subjects
POLYNOMIAL rings ,ERROR-correcting codes ,LINEAR codes ,QUANTUM rings - Abstract
In this paper, we use multivariate polynomial rings to construct quantum error-correcting codes (QECCs) via Hermitian construction. We establish a relation between linear codes and ideals of multivariate polynomial rings. We give a necessary and suffcient condition for a multivariate polynomial to generate a Hermitian dual-containing code. By comparing with the literatures in recent years, we construct 31 new QECCs over F q , where q = 3 , 4 , 5 , 7 . Some of them reach quantum singleton bound and some of them exceed quantum GV bound. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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16. Codes arising from directed strongly regular graphs with μ=1.
- Author
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Huilgol, Medha Itagi and D'Souza, Grace Divya
- Subjects
DIRECTED graphs ,REGULAR graphs ,LINEAR codes ,FINITE fields ,ERROR-correcting codes ,RESEARCH personnel - Abstract
The rank of adjacency matrix plays an important role in construction of linear codes from a directed strongly regular graph using different techniques, namely, code orthogonality, adjacency matrix determinant and adjacency matrix spectrum. The problem of computing the dimensions of such codes is an intriguing one. Several conjectures to determine the rank of adjacency matrix of a DSRG Γ over a finite field, keep researchers working in this area. To address the same to an extent, we have considered the problem of finding the rank over a finite field of the adjacency matrix of a DSRG Γ (v , k , t , λ , μ) with μ = 1 , including some mixed Moore graphs and corresponding codes arising from them, in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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17. Construction of lattices over the real sub-field of ℚ(ς8) for block fading (wiretap) coding.
- Author
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Singh, Ankur, Kumar, Pratyush, and Shukla, Ankur
- Subjects
- *
BINARY codes , *WIRETAPPING , *REAL numbers , *BLOCK codes , *LINEAR codes , *JACOBI forms , *LATTICE theory - Abstract
In this paper, we consider the challenge of ensuring secure communication in block-fading wiretap channels using encoding techniques. We investigate the coding for block-fading wiretap channels using stacked lattice codes constructed over completely real number fields, which is a well-established technique. In this paper, we consider degree-four complex multiplication field 풦 = ℚ(ς8) over ℚ. We employ binary codes to generate a lattice over ℚ(ς8), which is subsequently used to form an integral lattice. The resulting integral lattice can be effectively applied to enhance the security of communication within block-fading wiretap channels. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
18. Expanded low-rank parity-check codes and their application to cryptography.
- Author
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Franck Rivel, Kamwa Djomou, Emmanuel, Fouotsa, and Calvin, Tadmon
- Subjects
CRYPTOGRAPHY ,CRYPTOSYSTEMS ,LINEAR codes ,PUBLIC key cryptography ,DECODING algorithms - Abstract
In this paper, we define expanded LRPC codes from LRPC codes with a decoding algorithm using the one of the underlying LRPC codes. Next, we propose to use these codes for cryptography by deriving two cryptosystems in a McEliece setting. in order to reduce the key sizes, we use a generator matrix in systematic form for the first scheme, and an m − order quasi-cyclic LRPC code that we define for the second scheme. The obtained code has a very poor structure and is more likely to be a random linear code. Next, we give some security parameters and compare the key sizes of our public key with the key sizes of some cryptosystems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
19. MDS codes with l-Galois hulls of arbitrary dimensions.
- Author
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Qian, Liqin, Cao, Xiwang, Wu, Xia, and Lu, Wei
- Subjects
REED-Solomon codes ,LINEAR codes ,FINITE fields ,PROJECTIVE planes ,INTERSECTION graph theory - Abstract
The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The objective of this paper is to construct some MDS codes with l-Galois hulls of arbitrary dimensions by using the generalized Reed–Solomon codes over finite fields with regard to l-Galois inner product. We give a general construction theorem and some construction ideas of MDS with l-Galois hulls of arbitrary dimensions. Our approach provides a general framework that effectively unifies similar known techniques for constructing MDS codes. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. Linear complexity of two classes of quaternary sequences based on sign alternation transformation.
- Author
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Zhao, Lu, Pei, Yongzhen, Cao, Tianqing, and Du, Jiao
- Subjects
FINITE fields ,LINEAR codes - Abstract
Two classes of optimal quaternary sequences have been constructed by applying the sign alternation transform and Gray mapping to Legendre sequence pair, twin-prime sequence pair and GMW sequence pair. In this paper, a new method for investigating the linear complexity over finite field is proposed, and the exact values of the linear complexity over finite field F 4 and Galois ring Z 4 of the quaternary sequences are determined. The results show that their linear complexity are quite good. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
21. Perfect mixed codes from generalized Reed–Muller codes.
- Author
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Romanov, Alexander M.
- Subjects
REED-Muller codes ,PRODUCT coding ,LINEAR codes - Abstract
In this paper, we propose a new method for constructing 1-perfect mixed codes in the Cartesian product F n × F q n , where F n and F q are finite fields of orders n = q m and q. We consider generalized Reed-Muller codes of length n = q m and order (q - 1) m - 2 . Codes whose parameters are the same as the parameters of generalized Reed-Muller codes are called Reed-Muller-like codes. The construction we propose is based on partitions of distance-2 MDS codes into Reed-Muller-like codes of order (q - 1) m - 2 . We construct a set of q q cn nonequivalent 1-perfect mixed codes in the Cartesian product F n × F q n , where the constant c satisfies c < 1 , n = q m and m is a sufficiently large positive integer. We also prove that each 1-perfect mixed code in the Cartesian product F n × F q n corresponds to a certain partition of a distance-2 MDS code into Reed-Muller-like codes of order (q - 1) m - 2 . [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
22. On the parameters of extended primitive cyclic codes and the related designs.
- Author
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Yan, Haode and Yin, Yanan
- Subjects
CYCLIC codes ,HAMMING weight ,HAMMING codes ,LINEAR codes ,EXTENDED families ,SHIFT registers - Abstract
Very recently, Heng et al. studied a family of extended primitive cyclic codes. It was shown that the supports of all codewords with any fixed nonzero Hamming weight in this code support a 2-design. In this paper, we study this family of extended primitive cyclic codes in more details. The weight distribution is determined and the parameters of the related 2-designs are also given. Moreover, we prove that the minimum weight codewords in this code support a 3-design when p = 2 , which gives an affirmative answer to Heng's conjecture. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
23. Infinite families of minimal binary codes via Krawtchouk polynomials.
- Author
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Du, Xiaoni, Rodríguez, René, and Wu, Hao
- Subjects
BINARY codes ,LINEAR codes ,BOOLEAN functions ,POLYNOMIALS ,COMBINATORICS ,DATA warehousing ,QUANTUM cryptography - Abstract
Linear codes play a crucial role in various fields of engineering and mathematics, including data storage, communication, cryptography, and combinatorics. Minimal linear codes, a subset of linear codes, are particularly essential for designing effective secret sharing schemes. In this paper, we introduce several classes of minimal binary linear codes by carefully selecting appropriate Boolean functions. These functions belong to a renowned class of Boolean functions, namely, the general Maiorana–McFarland class. We employ a method first proposed by Ding et al. (IEEE Trans Inf Theory 64(10):6536–6545, 2018) to construct minimal codes violating the Ashikhmin–Barg bound (wide minimal codes) by using Krawtchouk polynomials. The lengths, dimensions, and weight distributions of the obtained codes are determined using the Walsh spectrum distribution of the chosen Boolean functions. Our findings demonstrate that a vast majority of the newly constructed codes are wide minimal. Furthermore, our proposed codes exhibit a significantly larger minimum distance, in some cases, compared to some existing similar constructions. Finally, we address this method, based on Krawtchouk polynomials, more generally, and highlight certain generic properties related to it. These general results offer insights into the scope of this approach. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
24. Optimized decoder for low-density parity check codes based on genetic algorithms.
- Author
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El Ouakili, Hajar, El Ghzaoui, Mohammed, and El Alami, Rachid
- Subjects
DECODING algorithms ,GENETIC algorithms ,CODING theory ,BLOCK codes ,LINEAR codes ,GENETIC code ,LOW density parity check codes ,ERROR-correcting codes - Abstract
Low-density parity check (LDPC) codes, are a family of error-correcting codes, their performances close to the Shannon limit make them very attractive solutions for digital communication systems. There are several algorithms for decoding LDPC codes that show great diversity in terms of performance related to error correction. Also, very recently, many research papers involved the genetic algorithm (GA) in coding theory, in particular, in the decoding linear block codes case, which has heavily contributed to reducing the bit error rate (BER). In this paper, an efficient method based on the GA is proposed and it is used to improve the power of correction in terms of BER and the frame error rate (FER) of LDPC codes. Subsequently, the proposed algorithm can independently decide the most suitable moment to stop the decoding process, moreover, it does not require channel information (CSI) making it adaptable for all types of channels with different noise or intensity. The simulations show that the proposed algorithm is more efficient in terms of BER compared to other LDPC code decoders. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
25. Text-independent speaker identification system using discrete wavelet transform with linear prediction coding.
- Author
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Alrusaini, Othman and Daqrouq, Khaled
- Subjects
LINEAR codes ,ADDITIVE white Gaussian noise ,SYSTEM identification ,GAUSSIAN mixture models ,DISCRETE systems ,FEATURE extraction ,VIDEO coding ,DISCRETE wavelet transforms ,WAVELET transforms - Abstract
One of the key problems of the modern day is the presentation of an identity verification system that can perform sufficient accuracy in identity verification, is resilient to assaults and noises, and can be recorded in the simplest possible method. In this study, a new speaker feature extraction which based on discrete wavelet transform (DWT) and linear prediction coding (LPC) algorithm (WLPCA) are investigated. This paper's primary objective is to evidence the performance of the new method for speaker identification by a Gaussian mixture model (GMM). The proposed method improves the recognition rate over the Mel-frequency cepstral coefficient (MFCC). Experimental evaluation of the process performance is performed on two speech databases; our recorded database and the publicly available TIMIT database. We show that the speech features derived by the newly proposed method are more suitable for GMM (91.53%), in terms of the time-consumed, by requiring less Gaussian mixtures than MFCC (85.77%). For testing the presented method in a noisy environment, Additive white Gaussian noise (AWGN) was added to the TIMIT database, where a slight improvement in the performance of the presented method (60.02%) over the MFCC (59.89%) was observed. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
26. Some new constructions of optimal linear codes and alphabet-optimal (r,δ)-locally repairable codes
- Author
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Qiu, Jing and Fu, Fang-Wei
- Published
- 2024
- Full Text
- View/download PDF
27. LCD codes from equitable partitions of association schemes.
- Author
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Švob, Andrea
- Subjects
LINEAR codes ,LIQUID crystal displays - Abstract
Linear codes with complementary duals (shortly named LCD codes) are linear codes whose intersection with their duals are trivial. In this paper, we give a method of constructing these type of linear codes from equitable partitions of association schemes. The LCD codes constructed in this paper are of length 2n and dimension n and have the property of being formally self-dual. To illustrate the method we construct LCD codes from some distance-regular graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
28. A note on (3,1) and (3,2) separating systems and bound for (3,1) separating system.
- Author
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Rega, B. and Durairajan, C.
- Subjects
DIGITAL printing ,LINEAR codes - Abstract
Separating codes have been studied due to their applications to digital finger printing, the state assignments, automata theory and to construct hash functions. In this paper, we study the necessary and sufficient conditions for a code to be a (3 , 1) and (3 , 2) -separating systems for q-ary level and also satisfy its intersecting properties. We also construct a bound for (3 , 1) separating system. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
29. On the algebraic structure of quasi-group codes.
- Author
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Borello, Martino and Willems, Wolfgang
- Subjects
LINEAR codes ,PERMUTATION groups ,AUTOMORPHISMS ,AUTOMORPHISM groups ,PERMUTATIONS - Abstract
In this paper, an intrinsic description of some families of linear codes with symmetries is given, showing that they can be described more generally as quasi-group codes, that is, as linear codes allowing a group of permutation automorphisms which acts freely on the set of coordinates. An algebraic description, including the concatenated structure, of such codes is presented. This allows to construct quasi-group codes from codes over rings, and vice versa. The last part of the paper is dedicated to the investigation of self-duality of quasi-group codes. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
30. Double circulant complementary dual codes over F4.
- Author
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Shoaib, Hatoon
- Subjects
LINEAR codes ,LOGICAL prediction - Abstract
Linear codes with complementary-duals (LCD codes) are linear codes that trivially intersect with their dual (Massey, 1992). In this paper, we study double circulant codes (DC codes), which are a special class of quasi-cyclic codes, over F4 that are LCD. The main techniques used are as follows: Chinese reminder theory (CRT) decomposition in the line of (Ling et al. 2001), explicit enumeration, and asymptotics. In particular, we show that the class of codes considered here is asymptotically good. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
31. Several families of irreducible constacyclic and cyclic codes.
- Author
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Sun, Zhonghua, Wang, Xiaoqiang, and Ding, Cunsheng
- Subjects
FINITE fields ,LINEAR codes ,CYCLIC codes - Abstract
In this paper, several families of irreducible constacyclic codes over finite fields and their duals are studied. The weight distributions of these irreducible constacyclic codes and the parameters of their duals are settled. Several families of irreducible constacyclic codes with a few weights and several families of optimal constacyclic codes are constructed. As by-products, a family of [ 2 n , (n - 1) / 2 , d ≥ 2 (n + 1) ] irreducible cyclic codes over GF (q) and a family of [ (q - 1) n , (n - 1) / 2 , d ≥ (q - 1) (n + 1) ] irreducible cyclic codes over GF (q) are presented, where n is a prime such that ord 2 n (q) = (n - 1) / 2 and ord (q - 1) n (q) = (n - 1) / 2 , respectively. The results in this paper complement earlier works on irreducible constacyclic and cyclic codes over finite fields. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
32. On the Combinatorics of Locally Repairable Codes via Matroid Theory.
- Author
-
Westerback, Thomas, Freij-Hollanti, Ragnar, Ernvall, Toni, and Hollanti, Camilla
- Subjects
COMBINATORICS ,COMBINATORIAL probabilities ,COMBINATORIAL group theory ,MATROIDS ,LINEAR dependence (Mathematics) - Abstract
This paper provides a link between matroid theory and locally repairable codes (LRCs) that are either linear or more generally almost affine. Using this link, new results on both LRCs and matroid theory are derived. The parameters $(n,k,d,r,\delta )$ of LRCs are generalized to matroids, and the matroid analog of the generalized singleton bound by Gopalan et al. for linear LRCs is given for matroids. It is shown that the given bound is not tight for certain classes of parameters, implying a nonexistence result for the corresponding locally repairable almost affine codes that are coined perfect in this paper. Constructions of classes of matroids with a large span of the parameters $(n,k,d,r,\delta )$ and the corresponding local repair sets are given. Using these matroid constructions, new LRCs are constructed with prescribed parameters. The existence results on linear LRCs and the nonexistence results on almost affine LRCs given in this paper strengthen the nonexistence and existence results on perfect linear LRCs given by Song et al. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
33. The geometry of (tmodq)-arcs
- Author
-
Kurz, Sascha, Landjev, Ivan, Pavese, Francesco, and Rousseva, Assia
- Published
- 2023
- Full Text
- View/download PDF
34. Constructions of MDS, Near MDS and Almost MDS Codes From Cyclic Subgroups of F* q 2.
- Author
-
Heng, Ziling, Li, Chengju, and Wang, Xinran
- Subjects
CYCLIC codes ,LINEAR codes ,CYCLIC loads - Abstract
Linear codes achieving or nearly achieving the Singleton bound are interesting in both theory and practice. The objective of this paper is to construct several infinite families of MDS, near MDS and almost MDS codes from some special cyclic subgroups of ${\mathbb {F}}_{q^{2}}^{*}$. To this end, the augmentation and extension techniques are used. The codes in this paper have flexible parameters and their lengths could be large. The minimum linear locality of the codes constructed in this paper is also studied. Some infinite families of optimal linearly locally recoverable codes are obtained. Besides, some codes in this paper are proved to be proper for error detection. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
35. Fundamental Limits of Distributed Linear Encoding.
- Author
-
Khooshemehr, Nastaran Abadi and Maddah-Ali, Mohammad Ali
- Subjects
CODING theory ,FINITE fields ,ENCODING ,LINEAR systems ,CHANNEL coding ,LINEAR codes - Abstract
In general coding theory, we often assume that error is observed in transferring or storing encoded symbols, while the process of encoding itself is error-free. Motivated by recent applications of coding theory, in this paper, we consider the case where the process of encoding is distributed and prone to error. We introduce the problem of distributed encoding, comprised of a set of $K \in \mathbb {N}$ isolated source nodes and $N \in \mathbb {N}$ encoding nodes. Each source node has one symbol from a finite field, which is sent to each of the encoding nodes. Each encoding node stores an encoded symbol from the same field, as a function of the received symbols. However, some of the source nodes are controlled by the adversary and may send different symbols to different encoding nodes. Depending on the number of the adversarial nodes, denoted by $\beta \in \mathbb {N}$ , and the cardinality of the set of symbols that each one generates, denoted by $v \in \mathbb {N}$ , the process of decoding from the encoded symbols could be impossible. Assume that a decoder connects to an arbitrary subset of $t \in \mathbb {N}$ encoding nodes and wants to decode the symbols of the honest nodes correctly, without necessarily identifying the sets of honest and adversarial nodes. An important characteristic of a distributed encoding system is $t^{*} \in \mathbb {N}$ , the minimum of such $t$ , which is a function of $K$ , $N$ , $\beta $ , and $v$. In this paper, we study the distributed linear encoding system, i.e. one in which the encoding nodes use linear coding. We show that $t^{*}_{\textrm {Linear}}=K+2\beta (v-1)$ , if $N\ge K+2\beta (v-1)$ , and $t^{*}_{\textrm {Linear}}=N$ , if $N\le K+2\beta (v-1)$. In order to achieve $t^{*}_{\textrm {Linear}}$ , we use random linear coding and show that in any feasible solution that the decoder finds, the messages of the honest nodes are decoded correctly. In order to prove the converse of the fundamental limit, we show that when the adversary behaves in a particular way, it can always confuse the decoder between two feasible solutions that differ in the message of at least one honest node. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
36. Self-orthogonal and quantum codes over chain rings.
- Author
-
Bajalan, Maryam, Moeini, Mina, and Yildiz, Bahattin
- Subjects
GRAY codes ,LINEAR codes - Abstract
In this paper, we investigate the Gray images of codes over chain rings, leading to the derivation of infinite families of self-orthogonal linear codes over the residue field F
q . We determine the parameters of optimal self-orthogonal and divisible linear codes. Additionally, we study the Gray images of quasi-twisted codes, resulting in some self-orthogonal Griesmer quasi-cyclic codes. Finally, we employ the CSS construction to derive some quantum codes based on self-orthogonal linear codes. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
37. On the construction of constacyclically permutable codes from constacyclic codes.
- Author
-
Guanghui Zhang and Shuhua Liang
- Subjects
ALGEBRAIC codes ,FINITE fields ,INTEGERS ,GENERALIZATION ,LINEAR codes - Abstract
In this paper, we propose a way to partition any constacyclic code over a finite field in its equivalence classes according to the algebraic structure of the code. Such a method gives the generalization of cyclically permutable codes (CPCs), which are called constacyclically permutable codes (CCPCs), and it is useful to derive a CCPC from a given constacyclic code. Moreover, we present an enumerative formula for the code size of such a CCPC, with all of the terms being positive integers, and we provide an algebraic method to produce such a CCPC. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
38. On double cyclic codes over Z2 + uZ2.
- Author
-
Aydogdu, Ismail
- Subjects
CYCLIC codes ,LINEAR codes ,BINARY codes ,LINEAR operators - Abstract
In this paper, we introduced double cyclic codes over R
r × Rs , where R = Z2 + uZ2 = {0, 1, u, 1 + u} is the ring with four elements and u² = 0. We first determined the generator polynomials of R-double cyclic codes for odd integers r and s, then gave the generators of duals of free double cyclic codes over Rr × Rs . By defining a linear Gray map, we looked at the binary images of R-double cyclic codes and gave several examples of optimal parameter binary linear codes obtained from R-double cyclic codes. Moreover, we studied self-dual R-double cyclic codes and presented an example of a self-dual R-double cyclic code. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
39. New entanglement-assisted quantum error-correcting codes from negacyclic codes.
- Author
-
Chen, Xiaojing, Lu, Xingbo, Zhu, Shixin, Jiang, Wan, and Wang, Xindi
- Subjects
ERROR-correcting codes ,LINEAR codes ,INTEGERS ,GENERALIZATION - Abstract
Entanglement-assisted quantum error-correcting (EAQEC) codes are a generalization of quantum error-correcting (QEC) codes, which can be constructed from arbitrary classical linear codes by relaxing the dual-containing condition and by using preshared entangled states between the sender and the receiver. In this paper, we investigate EAQEC codes of length n = 2 (q 2 + 1) a , where q is an odd prime power, a = m 2 + 1 and m is an odd integer. The resulting EAQEC codes are entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes when the minimum distance d ≤ n + 2 2 . [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. A class of three-weight linear codes over finite fields of odd characteristic.
- Author
-
Duan, Bingbing, Han, Guangguo, and Qi, Yanfeng
- Subjects
LINEAR codes ,DATA warehousing ,FINITE fields ,GAUSSIAN sums - Abstract
Applied in communication, data storage system, secret sharing schemes, authentication codes and association schemes, linear codes attract much attention. In this paper, a class of three-weight linear codes is obtained by the defining sets over finite fields of odd characteristic. The parameters and weight distributions of linear codes are determined by the additive characters, multiplicative characters and Gauss sums. Further, most of linear codes obtained are minimal, which can be used to construct secret sharing schemes. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Construction of quasi self-dual codes over a commutative non-unital ring of order 4.
- Author
-
Kim, Jon-Lark and Roe, Young Gun
- Subjects
COMMUTATIVE rings ,TWO-dimensional bar codes ,CODE generators ,LINEAR codes ,NONCOMMUTATIVE algebras ,TORSION - Abstract
Let I be the commutative non-unital ring of order 4 defined by generators and relations. I = a , b ∣ 2 a = 2 b = 0 , a 2 = b , a b = 0. Alahmadi et al. have classified QSD codes, Type IV codes (QSD codes with even weights) and quasi-Type IV codes (QSD codes with even torsion code) over I up to lengths n = 6 , and suggested two building-up methods for constructing QSD codes. In this paper, we construct more QSD codes, Type IV codes and quasi-Type IV codes for lengths n = 7 and 8, and describe five new variants of the two building-up construction methods. We find that when n = 8 there is at least one QSD code with minimun distance 4, which attains the highest minimum distance so far, and we give a generator matrix for the code. We also describe some QSD codes, Type IV codes and quasi-Type IV codes with new weight distributions. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. Skew cyclic codes over Fq[u1, u2, ..., ur]/?ui³ - ui, uiuj - ujui?i, jr=1.
- Author
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Rai, Pradeep, Singh, Bhupendra, and Gupta, Ashok Ji
- Abstract
Our paper delves into exploring skew cyclic codes over a generalized class of rings denoted by T = T
r . We define Tr = Fq [u1 , u2 , ..., ur ]/(ui 3 - ui , ui uj -uj ui )i,j r =1, q = pm and p is some odd prime. Our study introduces a Gray map for the ring T and explores its properties. Using a decomposition theorem, we analyze the structural features of skew cyclic codes over T . Additionally, we offer a formula to find the count of skew cyclic codes of length n over the ring T under specific conditions. Further, we derive a criterion to get Linear Complementary Dual (LCD) codes over T from skew cyclic codes. Moreover, we present a technique for deriving quantum codes from a particular class of skew cyclic codes over T which contain their dual. [ABSTRACT FROM AUTHOR]- Published
- 2024
43. On Linear Codes over Finite Singleton Local Rings.
- Author
-
Alabiad, Sami, Alhomaidhi, Alhanouf Ali, and Alsarori, Nawal A.
- Subjects
LOCAL rings (Algebra) ,LINEAR codes ,BINARY codes ,CODING theory ,TWO-dimensional bar codes ,ISOMORPHISM (Mathematics) - Abstract
The study of linear codes over local rings, particularly non-chain rings, imposes difficulties that differ from those encountered in codes over chain rings, and this stems from the fact that local non-chain rings are not principal ideal rings. In this paper, we present and successfully establish a new approach for linear codes of any finite length over local rings that are not necessarily chains. The main focus of this study is to produce generating characters, MacWilliams identities and generator matrices for codes over singleton local Frobenius rings of order 32. To do so, we first start by characterizing all singleton local rings of order 32 up to isomorphism. These rings happen to have strong connections to linear binary codes and Z 4 codes, which play a significant role in coding theory. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
44. Hulls of linear codes from simplex codes.
- Author
-
Xu, Guangkui, Luo, Gaojun, Cao, Xiwang, and Xu, Heqian
- Subjects
AUTOMORPHISM groups ,LINEAR codes ,VECTOR spaces ,PROJECTIVE spaces - Abstract
The hull of a linear code plays an important role in determining the complexity of algorithms for checking permutation equivalence of two linear codes and computing the automorphism group of a linear code. Regarding the quantum error correction, linear codes with determined hull are used to construct quantum codes. In this paper, we focus on the hull of Simplex codes and punctured Simplex codes. We firstly study the properties of the matrix produced by the column vectors of a projective space and determine the Euclidean and Hermitian hull of punctured Simplex codes completely. Secondly, we investigate the Euclidean and Hermitian hull of several classes of linear codes from Simplex codes. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. On the equivalence of Zps-linear generalized Hadamard codes.
- Author
-
Bhunia, Dipak K., Fernández-Córdoba, Cristina, Vela, Carlos, and Villanueva, Mercè
- Subjects
HADAMARD codes ,LINEAR codes - Abstract
Linear codes of length n over Z p s , p prime, called Z p s -additive codes, can be seen as subgroups of Z p s n . A Z p s -linear generalized Hadamard (GH) code is a GH code over Z p which is the image of a Z p s -additive code under a generalized Gray map. It is known that the dimension of the kernel allows to classify these codes partially and to establish some lower and upper bounds on the number of such codes. Indeed, in this paper, for p ≥ 3 prime, we establish that some Z p s -linear GH codes of length p t having the same dimension of the kernel are equivalent to each other, once t is fixed. This allows us to improve the known upper bounds. Moreover, up to t = 10 if p = 3 or t = 8 if p = 5 , this new upper bound coincides with a known lower bound based on the rank and dimension of the kernel. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. Fq2-double cyclic codes with respect to the Hermitian inner product.
- Author
-
Aydogdu, Ismail, Abualrub, Taher, and Samei, Karim
- Subjects
CYCLIC codes ,LINEAR codes ,FINITE fields - Abstract
In this paper, we introduce F q 2 -double cyclic codes of length n = r + s , where F q 2 is the Galois field of q 2 elements, q is a power of a prime integer p and r, s are positive integers. We determine the generator polynomials for any F q 2 -double cyclic code. For any F q 2 -double cyclic code C , we will define the Euclidean dual code C ⊥ based on the Euclidean inner product and the Hermitian dual code C ⊥ H based on the Hermitian inner product. We will construct a relationship between C ⊥ and C ⊥ H and then find the generator polynomials for the Hermitian dual code C ⊥ H. As an application of our work, we will present examples of optimal parameter linear codes over the finite field F 4 and also examples of optimal quantum codes that were derived from F 4 -double cyclic codes using the Hermitian inner product. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
47. Theoretical Approaches of Interval-Valued Fuzzy Code and Fuzzy Soft Code.
- Author
-
Woldie, Masresha Wassie, Mebrat, Jejaw Demamu, and Taye, Mihret Alamneh
- Subjects
FUZZY sets ,LINEAR algebraic groups ,LINEAR codes ,ALGORITHMS ,DIAGNOSIS - Abstract
In this study, we attempted to demonstrate the interval-valued fuzzy code by extending the concept of an intervalvalued fuzzy set. Further, we discussed the operations of the interval-valued fuzzy code. The interval-valued fuzzy soft code is introduced, and various related properties are investigated in this paper. Finally, we show that the operations of interval-valued fuzzy soft code are discussed. Through this paper, we use the set of integers modulo 2, that is Z2= {0, 1}. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. Some variations of Tanner's construction for short length QC‐LDPC codes.
- Author
-
Kim, Wonjun, Cho, Hyunwoo, Song, Hong‐Yeop, and Song, Min Kyu
- Abstract
This paper proposes a modification to Tanner's work for constructing girth‐8 quasi‐cyclic low‐density parity‐check codes. The main contribution of this paper is to use an arithmetic sequence at the leftmost column for the exponent matrix so that the lifting size is not necessarily restricted to the prime numbers. Two theorems on the lifting sizes that achieve girth at least 8 using this approach is also provided. This construction exhibits better frame error rate results to the modified 5G new radio (NR) low‐density parity‐check codes for lengths around 500. Also, this construction achieves better frame error rate performance results than the recently proposed one using the Golomb rulers at around frame error rate of 10−6. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
49. Almost Optimal Construction of Functional Batch Codes Using Extended Simplex Codes.
- Author
-
Yohananov, Lev and Yaakobi, Eitan
- Subjects
LINEAR codes ,INFORMATION retrieval ,LOGICAL prediction - Abstract
A functional $k$ -batch code of dimension $s$ consists of $n$ servers storing linear combinations of $s$ linearly independent information bits. Any multiset request of size $k$ of linear combinations (or requests) of the information bits can be recovered by $k$ disjoint subsets of the servers. The goal under this paradigm is to find the minimum number of servers for given values of $s$ and $k$. A recent conjecture states that for any $k=2^{s-1}$ requests the optimal solution requires $2^{s}-1$ servers. This conjecture is verified for $s \leqslant 5$ but previous work could only show that codes with $n=2^{s}-1$ servers can support a solution for $k=2^{s-2} + 2^{s-4} + \left \lfloor{ \frac { 2^{s/2}}{\sqrt {24}} }\right \rfloor $ requests. This paper reduces this gap and shows the existence of codes for $k=\lfloor \frac {5}{6}2^{s-1} \rfloor - s$ requests with the same number of servers. Another construction in the paper provides a code with $n=2^{s+1}-2$ servers and $k=2^{s}$ requests, which is an optimal result. These constructions are mainly based on extended Simplex codes and equivalently provide constructions for parallel Random I/O (RIO) codes. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
50. Some subfield codes from MDS codes.
- Author
-
Xiang, Can and Luo, Jinquan
- Subjects
BINARY codes ,LINEAR codes ,FINITE fields - Abstract
Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, a class of binary subfield codes is constructed from a special family of MDS codes, and their parameters are explicitly determined. The parameters of their dual codes are also studied. Some of the codes presented in this paper are optimal or almost optimal. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
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