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BCH codes are a subclass of cyclic codes with many interesting properties. In many cases BCH codes are the best linear codes. For example, among all binary cyclic codes of odd length n with n≤125 the best cyclic code is always a BCH code except for two special cases. Reed-Solomon codes are also BCH codes and are widely used in communication devices and consumer electronics. Binary BCH codes were discovered by Hocquenghem and independently by Bose and Ray-Chaudhuri. They were extended to BCH codes over finite fields by Gorenstein and Zierler in 1961. In fact, BCH codes were introduced by Bush in 1952 in the context of orthogonal arrays. In the past sixty years, a lot of progress on the study of BCH codes has been made. The objective of this survey is to present the state-of-the-art of BCH cyclic codes and their applications. © 2024 Elsevier B.V.

Most of the approaches published in the literature to construct S-boxes via Cellular Automata (CA) work by either iterating a finite CA for several time steps, or by a one-shot application of the global rule. The main characteristic that brings together these works is that they employ a single CA rule to define the vectorial Boolean function of the S-box. In this work, we explore a different direction for the design of S-boxes that leverages on Orthogonal CA (OCA), i.e. pairs of CA rules giving rise to orthogonal Latin squares. The motivation stands on the facts that an OCA pair already defines a bijective transformation, and moreover the orthogonality property of the resulting Latin squares ensures a minimum amount of diffusion. We exhaustively enumerate all S-boxes generated by OCA pairs of diameter 4 <= d <= 6, and measure their nonlinearity. Interestingly, we observe that for d = 4 and d = 5 all S-boxes are linear, despite the underlying CA local rules being nonlinear. The smallest nonlinear S-boxes emerges for d = 6, but their nonlinearity is still too low to be used in practice. Nonetheless, we unearth an interesting structure of linear OCA S-boxes, proving that their Linear Components Space is itself the image of a linear CA, or equivalently a polynomial code. We finally classify all linear OCA S-boxes in terms of their generator polynomials.

Constacyclic codes contain cyclic codes as a subclass and have nice algebraic structures. Constacyclic codes have theoretical importance, as they are connected to a number of areas of mathematics and outperform cyclic codes in several aspects. Negacyclic codes are a subclass of constacyclic codes and are distance-optimal in many cases. However, compared with the extensive study of cyclic codes, negacyclic codes are much less studied. In this paper, several families of ternary negacyclic codes and their duals are constructed and analysed. These families of negacyclic codes and their duals contain distance-optimal codes and have very good parameters in general. The duals of three families of ternary negacyclic codes presented in this paper are distance-optimal. IEEE

Cyclic codes are important in theory, as they are closely related to a number of areas of mathematics. Cyclic codes are also important in practice, as they have efficient encoding and decoding algorithms. An infinite family of cyclic codes over GF(q) is said to have linearly-best-known parameters if for any [n,k,d] code C in this family, there is no known [n, k, d′] linear code over GF(q) such that d′ > d. An infinite family of cyclic codes over GF(q) is said to have cyclicly-best-known parameters if for any [n, k, d] code C in this family, there is no known [n, k, d′] cyclic code over GF(q) such that d′ > d. It is very rare to see an infinite family of binary cyclic codes with cyclicly-best-known parameters whose duals codes have also cyclicly-best-known parameters. The objective of this paper is to study such family of binary cyclic codes of length 2m – 1 and dimension 2m – 1 – m(m – 1)/2, denoted by C (2,m,2), and their dual codes C⊥ (2,m,2). The weight distribution of C⊥ (2,m,2) is settled and the parameters of C(2,m,2) are investigated in this paper. A larger family of binary cyclic codes C(2,m,r) and their duals are also constructed and studied in this paper, where 0 ≤ r ≤ m – 1. IEEE

There are many infinite families of 2-designs and 3-designs supported by linear codes in the literature. Recently, several infinite families of 4-designs from linear codes were discovered. But no infinite family of linear codes supporting an infinite family of nontrivial simple 5-designs is reported in the literature. It is not easy to construct 5-designs from sporadic codes. Only a small number of sporadic 5-designs supported by linear codes are available in the literature. In this paper, we construct eight 5-designs from some lifted codes of some known linear codes. © 2024 Elsevier B.V.

The classical way of extending an [n,k,d] linear code C is to add an overall parity-check coordinate to each codeword of the linear code C. This extended code, denoted by C‾(−1) and called the standardly extended code of C, is a linear code with parameters [n+1,k,d¯], where d¯=d or d¯=d+1. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper. © 2024 Elsevier Inc.

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. Inspired by the recent work on binary cyclic codes published in Tang and Ding (2022) [26] and the works in Liu et al. (2023) [15] and Liu et al. (2023) [16], the objectives of this paper are the construction and analysis of four infinite families of ternary cyclic codes with length n=3m−1 for odd m and dimension k∈{n/2,(n+2)/2} whose minimum distances have a square-root-like lower bound. Their duals have parameters [n,k⊥,d⊥], where k⊥∈{n/2,(n−2)/2} and d⊥ also has a square-root-like lower bound. These families of codes and their duals contain distance-optimal cyclic codes. © 2023 Elsevier Inc.

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. Inspired by the recent work on binary cyclic codes published in Tang and Ding (2022) [14], the objectives of this paper are the construction and analyses of five infinite families of binary cyclic codes with parameters [n, k] and (n - 6)/3 < k < 2(n +6)/3. Three of the five families of binary cyclic codes and their duals have a very good lower bound on their minimum distances and contain distance-optimal codes. The other two families of binary cyclic codes are composed of binary duadic codes with a square-root-like lower bound on their minimum distances. As a by-product, two infinite families of self-dual binary codes with a square-root-like lower bound on their minimum distances are obtained. & COPY; 2023 Elsevier Inc. All rights reserved.

Interplay between coding theory and combinatorial t -designs has been a hot topic for many years for combinatorialists and coding theorists. Some infinite families of cyclic codes supporting infinite families of 3-designs have been constructed in the past 50 years. However, no infinite family of negacyclic codes supporting an infinite family of 3-designs has been reported in the literature. This is the main motivation of this paper. Let q=pm , where p is an odd prime and m≥2 is an integer. The objective of this paper is to present an infinite family of cyclic codes over GF(q) supporting an infinite family of 3-designs and two infinite families of negacyclic codes over GF(q2) supporting two infinite families of 3-designs. The parameters and the weight distributions of these codes are determined. The subfield subcodes of these negacyclic codes over GF(q) are studied. Three infinite families of almost MDS codes are also presented. A constacyclic code over GF(4) supporting a 4-design and seven open problems are also presented in this paper.

Locally recoverable codes (LRCs) were proposed for the recovery of data in distributed and cloud storage systems about nine years ago. A lot of progress on the study of LRCs has been made by now. However, there is a lack of general theory on the minimum locality of linear codes. In addition, the minimum locality of many known families of linear codes has not been studied in the literature. Motivated by these two facts, this paper develops some general theory about the minimum locality of linear codes, and investigates the minimum locality of a number of families of linear codes, such as q-ary Hamming codes, q-ary Simplex codes, generalized Reed-Muller codes, ovoid codes, maximum arc codes, the extended hyperoval codes, and near MDS codes. Many classes of both distance-optimal and dimension-optimal LRCs are presented in this paper. To this end, the concepts of linear locality and minimum linear locality are specified. The minimum linear locality of many families of linear codes are settled with the general theory developed in this paper.

Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of [2m+1,2u-1,2m-2u+3] MDS codes for 1≤u≤2m-1, which are cyclic, reversible and BCH codes over GF(2m). 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. IEEE

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 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 PGL(2, pm)-invariant codes over GF(ph) is completed. As an application of this result, the p-ranks of all incidence structures invariant under the projective general linear group PGL(2, pm) 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. IEEE

Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of [2m+1,2u-1,2m-2u+3] MDS codes for 1≤u≤2m-1, which are cyclic, reversible and BCH codes over GF(2m). 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. IEEE

The binary quadratic-residue codes and the punctured Reed-Muller codes R2((m−1)/2,m)) are two families of binary cyclic codes with parameters [n,(n+1)/2,d≥√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 2m−1 and dimension near 2m−1 with good minimum distances. When m≥3 is odd, the codes become a family of duadic codes with parameters [2m−1,2m−1,d] , where d≥2(m−1)/2+1 if m≡3(mod4) and d≥2(m−1)/2+3 if m≡1(mod4) . The two families of binary cyclic codes contain some optimal binary cyclic codes. IEEE

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 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 PGL(2, pm)-invariant codes over GF(ph) is completed. As an application of this result, the p-ranks of all incidence structures invariant under the projective general linear group PGL(2, pm) 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. IEEE

Let q= 2 m. The projective general linear group PGL (2 , q) acts as a 3-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over GF (2 h) that are invariant under PGL (2 , q) are trivial codes: the repetition code, the whole space GF(2h)2m+1, and their dual codes. As an application of this result, the 2-ranks of the (0,1)-incidence matrices of all 3 - (q+ 1 , k, λ) designs that are invariant under PGL (2 , q) are determined. The second objective is to present two infinite families of cyclic codes over GF (2 m) such that the set of the supports of all codewords of any fixed nonzero weight is invariant under PGL (2 , q) , therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters [ q+ 1 , q- 3 , 4 ] q, where q= 2 m, and m≥ 4 is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3-(q+ 1 , 4 , 2) design. A code from the second family has parameters [ q+ 1 , 4 , q- 4 ] q, q= 2 m, m≥ 4 even, and the minimum weight codewords support a 3-(q+ 1 , q- 4 , (q- 4) (q- 5) (q- 6) / 60) design, whose complementary 3-(q+ 1 , 5 , 1) design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over GF (q) that can support a 3-(q+ 1 , q- 4 , (q- 4) (q- 5) (q- 6) / 60) design is proved, and it is shown that the designs supported by the codewords of minimum weight in the codes from the second family of codes meet this bound. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

A special function is a function either of special form or with a special property. Special functions have interesting applications in coding theory and combinatorial t-designs. The main objective of this paper is to survey t-designs constructed from special functions, including quadratic functions, almost perfect nonlinear functions, almost bent functions, bent functions, bent vectorial functions, and planar functions. These combinatorial designs are not constructed directly from such functions, but come from linear codes which are constructed with such functions. As a byproduct, this paper also surveys linear codes from certain special functions.

In this paper, the 3-rank of the incidence matrices of 2-designs supported by the minimum weight codewords in a family of ternary linear codes considered in Ding and Li (Discret Math 340(10):2415–2431, 2017) are computed. A lower bound on the minimum distance of the ternary codes spanned by the incidence matrices of these designs is derived, and it is proved that the codes are subcodes of the 4th order generalized Reed–Muller codes. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

A special function is a function either of special form or with a special property. Special functions have interesting applications in coding theory and combinatorial t-designs. The main objective of this paper is to survey t-designs constructed from special functions, including quadratic functions, almost perfect nonlinear functions, almost bent functions, bent functions, bent vectorial functions, and planar functions. These combinatorial designs are not constructed directly from such functions, but come from linear codes which are constructed with such functions. As a byproduct, this paper also surveys linear codes from certain special functions.

In this paper, the 3-rank of the incidence matrices of 2-designs supported by the minimum weight codewords in a family of ternary linear codes considered in Ding and Li (Discret Math 340(10):2415–2431, 2017) are computed. A lower bound on the minimum distance of the ternary codes spanned by the incidence matrices of these designs is derived, and it is proved that the codes are subcodes of the 4th order generalized Reed–Muller codes. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Given a {q} -ary frequency hopping sequence set of length {n} and size {M} with Hamming correlation {H}, one can obtain a {q} -ary (nonlinear) cyclic code of length {n} and size nM with Hamming distance n-H. Thus, every upper bound on the size of a code from coding theory gives an upper bound on the size of a frequency hopping sequence set. Indeed, all upper bounds from coding theory have been converted to upper bounds on frequency hopping sequence sets [1]. On the other hand, a lower bound from coding theory does not automatically produce a lower bound for frequency hopping sequence sets. In particular, the most important lower bound, the Gilbert-Varshamov bound in coding theory, has not been transformed to a valid lower bound on frequency hopping sequence sets. The purpose of this paper is to transform the Gilbert-Varshamov bound from coding theory to frequency hopping sequence sets by establishing a connection between a special family of cyclic codes (which are called hopping cyclic codes in this paper) and frequency hopping sequence sets. We provide two proofs of the Gilbert-Varshamov bound. One is based on a probabilistic method that requires advanced tool-martingale. This proof covers the whole rate region. Another proof is purely elementary but only covers part of the rate region.

Given a {q} -ary frequency hopping sequence set of length {n} and size {M} with Hamming correlation {H}, one can obtain a {q} -ary (nonlinear) cyclic code of length {n} and size nM with Hamming distance n-H. Thus, every upper bound on the size of a code from coding theory gives an upper bound on the size of a frequency hopping sequence set. Indeed, all upper bounds from coding theory have been converted to upper bounds on frequency hopping sequence sets [1]. On the other hand, a lower bound from coding theory does not automatically produce a lower bound for frequency hopping sequence sets. In particular, the most important lower bound, the Gilbert-Varshamov bound in coding theory, has not been transformed to a valid lower bound on frequency hopping sequence sets. The purpose of this paper is to transform the Gilbert-Varshamov bound from coding theory to frequency hopping sequence sets by establishing a connection between a special family of cyclic codes (which are called hopping cyclic codes in this paper) and frequency hopping sequence sets. We provide two proofs of the Gilbert-Varshamov bound. One is based on a probabilistic method that requires advanced tool-martingale. This proof covers the whole rate region. Another proof is purely elementary but only covers part of the rate region.

A special function is a function either of special form or with a special property. Special functions have interesting applications in coding theory and combinatorial t-designs. The main objective of this paper is to survey t-designs constructed from special functions, including quadratic functions, almost perfect nonlinear functions, almost bent functions, bent functions, bent vectorial functions, and planar functions. These combinatorial designs are not constructed directly from such functions, but come from linear codes which are constructed with such functions. As a byproduct, this paper also surveys linear codes from certain special functions.

In this paper, the 3-rank of the incidence matrices of 2-designs supported by the minimum weight codewords in a family of ternary linear codes considered in Ding and Li (Discret Math 340(10):2415–2431, 2017) are computed. A lower bound on the minimum distance of the ternary codes spanned by the incidence matrices of these designs is derived, and it is proved that the codes are subcodes of the 4th order generalized Reed–Muller codes. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

It is proved that for every d≥ 2 such that d- 1 divides q- 1 , where q is a power of 2, there exists a Denniston maximal arc A of degree d in PG (2 , q) , being invariant under a cyclic linear group that fixes one point of A and acts regularly on the set of the remaining points of A. Two alternative proofs are given, one geometric proof based on Abatangelo–Larato’s characterization of Denniston arcs, and a second coding-theoretical proof based on cyclotomy and the link between maximal arcs and two-weight codes. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

Coding theory and t-designs have close connections and interesting interplay. In this paper, we first introduce a class of ternary linear codes and study their parameters. We then focus on their three-weight subcodes with a special weight distribution. We determine the weight distributions of some shortened codes and punctured codes of these three-weight subcodes. These shortened and punctured codes contain some codes that have the same parameters as the best ternary linear codes known in the database maintained by Markus Grassl at http://www.codetables.de/. These three-weight subcodes with a special weight distribution do not satisfy the conditions of the Assmus–Mattson theorem and do not admit 2-transitive or 2-homogeneous automorphism groups in general. By employing the theory of projective geometries and projective generalized Reed–Muller codes, we prove that they still hold 2-designs. We also determine the parameters of these 2-designs. This paper mainly confirms some recent conjectures of Ding and Li regarding Steiner systems and 2-designs from a special type of ternary projective codes. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

Coding theory and t-designs have close connections and interesting interplay. In this paper, we first introduce a class of ternary linear codes and study their parameters. We then focus on their three-weight subcodes with a special weight distribution. We determine the weight distributions of some shortened codes and punctured codes of these three-weight subcodes. These shortened and punctured codes contain some codes that have the same parameters as the best ternary linear codes known in the database maintained by Markus Grassl at http://www.codetables.de/. These three-weight subcodes with a special weight distribution do not satisfy the conditions of the Assmus–Mattson theorem and do not admit 2-transitive or 2-homogeneous automorphism groups in general. By employing the theory of projective geometries and projective generalized Reed–Muller codes, we prove that they still hold 2-designs. We also determine the parameters of these 2-designs. This paper mainly confirms some recent conjectures of Ding and Li regarding Steiner systems and 2-designs from a special type of ternary projective codes. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

It is proved that for every d≥ 2 such that d- 1 divides q- 1 , where q is a power of 2, there exists a Denniston maximal arc A of degree d in PG (2 , q) , being invariant under a cyclic linear group that fixes one point of A and acts regularly on the set of the remaining points of A. Two alternative proofs are given, one geometric proof based on Abatangelo–Larato’s characterization of Denniston arcs, and a second coding-theoretical proof based on cyclotomy and the link between maximal arcs and two-weight codes. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

Coding theory and t-designs have close connections and interesting interplay. In this paper, we first introduce a class of ternary linear codes and study their parameters. We then focus on their three-weight subcodes with a special weight distribution. We determine the weight distributions of some shortened codes and punctured codes of these three-weight subcodes. These shortened and punctured codes contain some codes that have the same parameters as the best ternary linear codes known in the database maintained by Markus Grassl at http://www.codetables.de/. These three-weight subcodes with a special weight distribution do not satisfy the conditions of the Assmus–Mattson theorem and do not admit 2-transitive or 2-homogeneous automorphism groups in general. By employing the theory of projective geometries and projective generalized Reed–Muller codes, we prove that they still hold 2-designs. We also determine the parameters of these 2-designs. This paper mainly confirms some recent conjectures of Ding and Li regarding Steiner systems and 2-designs from a special type of ternary projective codes. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

It is proved that for every d≥ 2 such that d- 1 divides q- 1 , where q is a power of 2, there exists a Denniston maximal arc A of degree d in PG (2 , q) , being invariant under a cyclic linear group that fixes one point of A and acts regularly on the set of the remaining points of A. Two alternative proofs are given, one geometric proof based on Abatangelo–Larato’s characterization of Denniston arcs, and a second coding-theoretical proof based on cyclotomy and the link between maximal arcs and two-weight codes. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

The punctured binary Reed–Muller code is cyclic and was generalised into the punctured generalised Reed–Muller code over GF(q) in the literature. The first objective of this paper is to present another generalisation of the punctured binary Reed–Muller code and the binary Reed–Muller code, and analyse these codes. The second objective of this paper is to consider two applications of the new codes in constructing LCD codes and 2-designs. The major motivation of constructing and studying the new codes and their extended codes is the construction of 2-designs, which is an interesting topic in combinatorics. It is remarkable that the family of newly generalised cyclic codes contains a subclass of optimal ternary codes with parameters [3m−1,3m−1−2m,4] for all m≥2. Their extended codes have parameters [3m,3m−1−2m,5] for all m≥2, and are also optimal. © 2018 Elsevier Inc.

The punctured binary Reed–Muller code is cyclic and was generalised into the punctured generalised Reed–Muller code over GF(q) in the literature. The first objective of this paper is to present another generalisation of the punctured binary Reed–Muller code and the binary Reed–Muller code, and analyse these codes. The second objective of this paper is to consider two applications of the new codes in constructing LCD codes and 2-designs. The major motivation of constructing and studying the new codes and their extended codes is the construction of 2-designs, which is an interesting topic in combinatorics. It is remarkable that the family of newly generalised cyclic codes contains a subclass of optimal ternary codes with parameters [3m−1,3m−1−2m,4] for all m≥2. Their extended codes have parameters [3m,3m−1−2m,5] for all m≥2, and are also optimal. © 2018 Elsevier Inc.

The punctured binary Reed–Muller code is cyclic and was generalised into the punctured generalised Reed–Muller code over GF(q) in the literature. The first objective of this paper is to present another generalisation of the punctured binary Reed–Muller code and the binary Reed–Muller code, and analyse these codes. The second objective of this paper is to consider two applications of the new codes in constructing LCD codes and 2-designs. The major motivation of constructing and studying the new codes and their extended codes is the construction of 2-designs, which is an interesting topic in combinatorics. It is remarkable that the family of newly generalised cyclic codes contains a subclass of optimal ternary codes with parameters [3m−1,3m−1−2m,4] for all m≥2. Their extended codes have parameters [3m,3m−1−2m,5] for all m≥2, and are also optimal. © 2018 Elsevier Inc.

Cellular Automata (CA) have widely been studied to design cryptographic primitives such as stream ciphers and pseudorandom number generators, focusing in particular on the properties of the underlying local rules. On the other hand, there have been comparatively fewer works concerning the applications of CA to the design of S-boxes and block ciphers, a task that calls for a study of CA global rules in terms of vectorial boolean functions. The aim of this paper is to analyze some of the most basic cryptographic criteria of the global rules of CA. We start by observing that the algebraic degree of a CA global rule equals the degree of its local rule. Then, we characterize the Walsh spectrum of CA induced by permutive local rules, from which we derive a formula for the nonlinearity of such CA. Additionally, we prove that the 1-resiliency property of bipermutive local rules transfers to the corresponding global rules. This result leads us to consider CA global rules from a coding-theoretic point of view: in particular, we show that linear CA are equivalent to linear cyclic codes, observing that the syndrome computation process corresponds to the application of the CA global rule, while the error-correction capability of the code is related to the resiliency order of the global rule.

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=qm−1, n=(qm−1)∕(q−1) and n=qm+1. For narrow-sense primitive BCH codes with designed distance δ, we investigate their dimensions for δ in the range 1≤δ≤q⌈⌉+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=(qm−1)∕(q−1) and n=qm+1 over finite fields. © 2017 Elsevier B.V.

The interplay between coding theory and t-designs started many years ago. While every t-design yields a linear code over every finite field, the largest t for which an infinite family of t-designs is derived directly from a linear or nonlinear code is t=3. Sporadic 4-designs and 5-designs were derived from some linear codes of certain parameters. The major objective of this paper is to construct many infinite families of 2-designs and 3-designs from linear codes. The parameters of some known t-designs are also derived. In addition, many conjectured infinite families of 2-designs are also presented. © 2017 Elsevier B.V.

This paper studies the equivalence problem for cyclic codes of length $p^r$ and quasi-cyclic codes of length $p^rl$. In particular, we generalize the results of Huffman, Job, and Pless (J. Combin. Theory. A, 62, 183--215, 1993), who considered the special case $p^2$. This is achieved by explicitly giving the permutations by which two cyclic codes of prime power length are equivalent. This allows us to obtain an algorithm which solves the problem of equivalency for cyclic codes of length $p^r$ in polynomial time. Further, we characterize the set by which two quasi-cyclic codes of length $p^rl$ can be equivalent, and prove that the affine group is one of its subsets.

This paper studies the equivalence problem for cyclic codes of length $p^r$ and quasi-cyclic codes of length $p^rl$. In particular, we generalize the results of Huffman, Job, and Pless (J. Combin. Theory. A, 62, 183--215, 1993), who considered the special case $p^2$. This is achieved by explicitly giving the permutations by which two cyclic codes of prime power length are equivalent. This allows us to obtain an algorithm which solves the problem of equivalency for cyclic codes of length $p^r$ in polynomial time. Further, we characterize the set by which two quasi-cyclic codes of length $p^rl$ can be equivalent, and prove that the affine group is one of its subsets.

The interplay between coding theory and t-designs started many years ago. While every t-design yields a linear code over every finite field, the largest t for which an infinite family of t-designs is derived directly from a linear or nonlinear code is t=3. Sporadic 4-designs and 5-designs were derived from some linear codes of certain parameters. The major objective of this paper is to construct many infinite families of 2-designs and 3-designs from linear codes. The parameters of some known t-designs are also derived. In addition, many conjectured infinite families of 2-designs are also presented. © 2017 Elsevier B.V.

This paper studies the equivalence problem for cyclic codes of length $p^r$ and quasi-cyclic codes of length $p^rl$. In particular, we generalize the results of Huffman, Job, and Pless (J. Combin. Theory. A, 62, 183--215, 1993), who considered the special case $p^2$. This is achieved by explicitly giving the permutations by which two cyclic codes of prime power length are equivalent. This allows us to obtain an algorithm which solves the problem of equivalency for cyclic codes of length $p^r$ in polynomial time. Further, we characterize the set by which two quasi-cyclic codes of length $p^rl$ can be equivalent, and prove that the affine group is one of its subsets.

This paper studies the equivalence problem for cyclic codes of length $p^r$ and quasi-cyclic codes of length $p^rl$. In particular, we generalize the results of Huffman, Job, and Pless (J. Combin. Theory. A, 62, 183--215, 1993), who considered the special case $p^2$. This is achieved by explicitly giving the permutations by which two cyclic codes of prime power length are equivalent. This allows us to obtain an algorithm which solves the problem of equivalency for cyclic codes of length $p^r$ in polynomial time. Further, we characterize the set by which two quasi-cyclic codes of length $p^rl$ can be equivalent, and prove that the affine group is one of its subsets.

The generalized Hamming weights (GHWs) of linear codes are fundamental parameters, the knowledge of which is of great interest in many applications. However, to determine the GHWs of linear codes is difficult in general. In this paper, we study the GHWs for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases. This is achieved by extending the idea of Yang et al. into higher dimension and by employing some interesting combinatorial arguments. It shall be noted that these cyclic codes may have arbitrary number of nonzeros. © 2016 IEEE.

The generalized Hamming weights (GHWs) of linear codes are fundamental parameters, the knowledge of which is of great interest in many applications. However, to determine the GHWs of linear codes is difficult in general. In this paper, we study the GHWs for a family of reducible cyclic codes and obtain the complete weight hierarchy in several cases. This is achieved by extending the idea of Yang et al. into higher dimension and by employing some interesting combinatorial arguments. It shall be noted that these cyclic codes may have arbitrary number of nonzeros.

Let m >= 3 be an odd integer and p be an odd prime. In this paper, a number of classes of three-weight cyclic codes C-(1,C-e) over F-p, which have parity-check polynomial m(1)(x)m(e)(x), are presented by examining general conditions on the parameters p, m, and e, where m(i)(x) is the minimal polynomial of pi(-i) over F-p for a primitive element pi of F(p)m. Furthermore, for p equivalent to 3 (mod 4) and a positive integer e satisfying (p(k) + 1) . e equivalent to 2 (mod p(m) - 1) for some positive integer k with gcd(m, k) = 1, the value distributions of the exponential sums T(a, b) = Sigma(x is an element of)F(p)m omega(Tr(ax+bxe)) and S(a, b, c) = Sigma(x is an element of)F(p)m omega(Tr(ax+bxe+cxs)), where s = (p(m) - 1)/2, are determined. As an application, the value distribution of S(a, b, c) is utilized to derive the weight distribution of the cyclic codes C((1, e, s)) with parity-check polynomial m(1)(x)m(e)(x)m(s)(x). In the case of p = 3 and even e satisfying the above condition, the dual of the cyclic code C-(1,C- e,C- s) has optimal minimum distance.

Let m >= 3 be an odd integer and p be an odd prime. In this paper, a number of classes of three-weight cyclic codes C-(1,C-e) over F-p, which have parity-check polynomial m(1)(x)m(e)(x), are presented by examining general conditions on the parameters p, m, and e, where m(i)(x) is the minimal polynomial of pi(-i) over F-p for a primitive element pi of F(p)m. Furthermore, for p equivalent to 3 (mod 4) and a positive integer e satisfying (p(k) + 1) . e equivalent to 2 (mod p(m) - 1) for some positive integer k with gcd(m, k) = 1, the value distributions of the exponential sums T(a, b) = Sigma(x is an element of)F(p)m omega(Tr(ax+bxe)) and S(a, b, c) = Sigma(x is an element of)F(p)m omega(Tr(ax+bxe+cxs)), where s = (p(m) - 1)/2, are determined. As an application, the value distribution of S(a, b, c) is utilized to derive the weight distribution of the cyclic codes C((1, e, s)) with parity-check polynomial m(1)(x)m(e)(x)m(s)(x). In the case of p = 3 and even e satisfying the above condition, the dual of the cyclic code C-(1,C- e,C- s) has optimal minimum distance.

Let m >= 3 be an odd integer and p be an odd prime. In this paper, a number of classes of three-weight cyclic codes C-(1,C-e) over F-p, which have parity-check polynomial m(1)(x)m(e)(x), are presented by examining general conditions on the parameters p, m, and e, where m(i)(x) is the minimal polynomial of pi(-i) over F-p for a primitive element pi of F(p)m. Furthermore, for p equivalent to 3 (mod 4) and a positive integer e satisfying (p(k) + 1) . e equivalent to 2 (mod p(m) - 1) for some positive integer k with gcd(m, k) = 1, the value distributions of the exponential sums T(a, b) = Sigma(x is an element of)F(p)m omega(Tr(ax+bxe)) and S(a, b, c) = Sigma(x is an element of)F(p)m omega(Tr(ax+bxe+cxs)), where s = (p(m) - 1)/2, are determined. As an application, the value distribution of S(a, b, c) is utilized to derive the weight distribution of the cyclic codes C((1, e, s)) with parity-check polynomial m(1)(x)m(e)(x)m(s)(x). In the case of p = 3 and even e satisfying the above condition, the dual of the cyclic code C-(1,C- e,C- s) has optimal minimum distance.

It is considered two variants of syndrome decoding of unsystematic cyclic codes with decoding errors correction for output symbols of decoder, which is recursive or non-recursive filter of decoded coding symbols.

Bakalářská práce seznamuje s principy protichybového zabezpečení lineárními blokovými a cyklickými systematickými kódy a jejich vizualizace pomocí výukového programu. První část práce obsahuje ucelený teoretický základ principů protichybového zabezpečení a vlastnosti lineárních blokových a cyklických systematických kódů. Druhá část popisuje samotný výukový program, realizovaný v grafickém prostředí GUI programu MATLAB. Tato aplikace krok za krokem vizualizuje dílčí části výpočtů při zabezpečení, detekci i opravě chyb. Jejím hlavním přínosem je využití jako názorné výukové pomůcky., This bachelor thesis provides an introduction to the principles of forward error correction securing by linear block and cyclical systematic codes and its visualization by educational application. First part of the thesis contains comprehensive theoretical base of principles of forward error correction securing and capability of linear block and cyclical systematic codes. Second part describes the educational application, realized in graphical user interface GUI of MATLAB. This application step by step visualizes parts of procedures of securing, error detection and correction. Its main benefit is its usability as illustrative teaching aid.

Tato Bakalářská práce se zaměřuje na vybrané korekční kódy (ECC) a jejich problematiku. Cílem práce je pomoci čtenářům pochopit ECC snadnou a srozumitelnou formou. Najdeme zde mnoho názorných tabulek a obrázků pro lepší pochopení jak kódy pracují. každá sekce obsahuje jeden kód a většinou je naplněna způsobem zabezpečení a principem dekódování. Pro každou sekci byl vytvořen interaktivní aplet a program pro experimentální účely., This Bachelor thesis is mainly focused on issues of the chosen error correcting codes (ECC) and their demonstrative. Goal of this work is help to understand ECC to readers, easy and reasonable form. We can find here a lot of illustrative tables and figures for better understanding how codes works. Every section constains one code and mostly is filled by code securing and code decoding. For every section interactive Aplet and program for experimental uses were created.

Práca obsahuje základné pojmy potrebné pre pochopenie teórie kódovania. Následne vysvetľuje parametre, ktoré budú u opísaných kódov sledované. Na základe kritérií vyberá BCH kód a Rozšírený Hammingov Kód, za vhodné a ďalej porovnáva ich vlastnosti pre výber efektívnejšieho kódu z tejto dvojice. Výsledkom porovnania vyberá ako vhodnejší Rozšírený Hammingov Kód pre následný návrh kódu. Následne je zobrazený postup návrhu kódu, testy, ktoré overujú funkčnosť kódu a využité zdroje pre implementáciu kódu pre obvod Spartan-3., The thesis contains basic concepts necessary for understanding the theory of coding. It then explains the parameters that will be monitored for the described codes. Based on the criteria, it selects the BCH code and the Extended Hamming Code, as appropriate, and further compares their properties to select a more efficient code from this pair. As a result of the comparison, it selects the Extended Hamming Code as more suitable for the subsequent design of the program. The code design procedures, tests that verify the functionality of the code, and the resources used for the Spartan-3 circuit to implement the code for the Spartan-3 circuit are then shown.