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In this paper, we are interested in linear codes invariant under the (σ , δ) -action of a given matrix M , that we call (M , σ , δ) -skew codes. Various subclasses of these codes, depending on the (σ , δ) -cyclical property of M , are derived. In particular, we distinguish between three fundamental types: (M , σ , δ) -cyclic, (M , σ , δ) -quasi-cyclic and (M , σ , δ) -multi-cyclic. These different subclasses allow us to obtain some of the well known types of linear codes as special cases depending on the choice of M. We observe that the (σ , δ) -similarity in M n (F q ) gives rise to various relations between these codes. As common in the study of cyclic codes and their generalizations, we use the skew polynomial ring F q [ x , σ , δ ] to study (M , σ , δ) -skew codes. As an application of the theory developed here, we construct some optimal linear codes and MDS codes over different alphabets with additional desirable properties such as self orthogonality and linear complementary duality (LCD). [ABSTRACT FROM AUTHOR]

Linear complementary dual codes (LCD codes for short) are an important subclass of linear codes which have nice applications in communication systems, cryptography, consumer electronics and information protection. In the literature, it has been proved that an [ n , k , d ] Euclidean LCD code over F q with q > 3 exists if there is an [ n , k , d ] linear code over F q , where q is a prime power. However, the existence of binary and ternary Euclidean LCD codes has not been totally characterized. Hence it is interesting to construct binary and ternary Euclidean LCD codes with new parameters. In this paper, we construct new families of binary and ternary leading-systematic Euclidean LCD codes from some special functions including semibent functions, quadratic functions, almost bent functions, and planar functions. These LCD codes are not constructed directly from such functions, but come from some self-orthogonal codes constructed with such functions. Compared with known binary and ternary LCD codes, the LCD codes in this paper have new parameters. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] for all m ≥ 2 . Their extended codes have parameters [ 3 m , 3 m − 1 − 2 m , 5 ] for all m ≥ 2 , and are also optimal. [ABSTRACT FROM AUTHOR]

Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp–Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidimensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence. Several algorithms solve this problem. The so-called Berlekamp–Massey–Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process. We propose a new algorithm for computing the Gröbner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp–Massey–Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations. A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Padé approximants of this mirror polynomial. As an addition from the paper published at the ISSAC conference, we give an adaptive variant of this algorithm taking into account the shape of the final Gröbner basis gradually as it is discovered. The main advantage of this algorithm is that its complexity in terms of operations and sequence queries only depends on the output Gröbner basis. All these algorithms have been implemented in Maple and we report on our comparisons. [ABSTRACT FROM AUTHOR]

A subset F of a finite transitive group G ≤ Sym (Ω) is intersecting if any two elements of F agree on an element of Ω. The intersection density of G is the number ρ (G) = max ⁡ { | F | / | G ω | | F ⊂ G is intersecting } , where ω ∈ Ω and G ω is the stabilizer of ω in G. It is known that if G ≤ Sym (Ω) is an imprimitive group of degree a product of two odd primes p > q admitting a block of size p or two complete block systems, whose blocks are of size q , then ρ (G) = 1. In this paper, we analyze the intersection density of imprimitive groups of degree pq with a unique block system with blocks of size q based on the kernel of the induced action on blocks. For those whose kernels are non-trivial, it is proved that the intersection density is larger than 1 whenever there exists a cyclic code C with parameters [ p , k ] q such that any codeword of C has weight at most p − 1 , and under some additional conditions on the cyclic code, it is a proper rational number. For those that are quasiprimitive, we reduce the cases to almost simple groups containing Alt (5) or a projective special linear group. We give some examples where the latter has intersection density equal to 1, under some restrictions on p and q. [ABSTRACT FROM AUTHOR]

In this paper, we determine the Hamming weight distribution of several classes of double cyclic codes over the finite non-chain ring F q + u F q with u 2 = u , which is isomorphic to F q × F q. Some two-weight and three-weight linear codes over the finite field F q are constructed by double cyclic codes over this ring. [ABSTRACT FROM AUTHOR]

A pair (n , q) with q a prime power and n a positive integer for which gcd ⁡ (n , q) = 1 is non-standard if one of the following properties holds, which we will show to be equivalent. − The group U n , q of n -th roots-of-unity in the splitting field F q m of x n − 1 over F q is fixed set-wise by an F q -linear map on F q m that is not of the form x ⟶ α x q j . − There is a non-cyclic linear recurring sequence s of period n (so with s not of the form s i = α ξ i for i ≥ 0) with associated characteristic polynomial being irreducible over F q , such that U n , q = { s 0 , ... , s n − 1 }. In that case, the group U n , q is called non-standard by Brison and Nogueira, who studied this phenomenon in a sequence of papers. − A q -ary irreducible cyclic code of length n has "extra" permutation automorphisms (some well-known examples are the (duals of the) Golay codes and binary simplex codes for n ≥ 7). We will refer to such codes as non-standard irreducible cyclic codes or NSIC-codes. We first investigate non-standard linear recurring sequence subgroups and establish the equivalence between the first and second of the above properties. Then we investigate permutation automorphisms of general (repeated-root) cyclic codes and irreducible cyclic codes and establish the equivalence between the first and third of these properties. We also introduce a notion of non-standardness for general cyclic codes, and relate it to our earlier definition of NSIC-codes. This paper is the first part of an expanded version of the arXiv paper [28]. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative chain ring with unique maximal ideal 〈 γ 〉 , and let n be a positive integer relatively prime to the characteristic of R / 〈 γ 〉 . In this paper, some new necessary and sufficient conditions for the existence of nontrivial self-dual cyclic codes of length n over R are provided. Explicit enumeration formulas for self-dual cyclic codes over R are studied. In particular, several main results of [21] are special cases of the present paper. [ABSTRACT FROM AUTHOR]

Pseudo-geometric designs are combinatorial designs which share the same parameters as a finite geometry design, but are not isomorphic to that design. As far as we know, many pseudo-geometric designs have been constructed by the methods of finite geometries and combinatorics. However, none of pseudo-geometric designs with the parameters S (2 , q + 1 , (q n − 1) / (q − 1)) is constructed by the approach of coding theory. In this paper, we use cyclic codes to construct pseudo-geometric designs. We firstly present a family of ternary cyclic codes from the m -sequences with Welch-type decimation d = 2 ⋅ 3 (n − 1) / 2 + 1 , and obtain some infinite family of 2-designs and a family of Steiner systems S (2 , 4 , (3 n − 1) / 2) using these cyclic codes and their duals. Moreover, the parameters of these cyclic codes and their shortened codes are also determined. Some of those ternary codes are optimal or almost optimal. Finally, we show that one of these obtained Steiner systems is inequivalent to the point-line design of the projective space PG (n − 1 , 3) and thus is a pseudo-geometric design. [ABSTRACT FROM AUTHOR]

In this study, we determine the structure of cyclic codes over the ring Z q [ u ] / 〈 u 2 〉 which is isomorphic to R = Z q + u Z q where q = p s , p is a prime, s is a positive integer, and u 2 = 0. This is equivalent to determining the algebraic structure of ideals of the polynomial quotient ring R [ x ] / 〈 x n − 1 〉 , which is addressed in this paper completely. By establishing the structure of ideals of R [ x ] / 〈 x n − 1 〉 with gcd ⁡ (p , n) = 1 , we present an exact formula that enumerates the number of ideals of this ring that leads to the enumeration of cyclic codes over this ring. Finally, we consider and explore some special families of cyclic codes for some specific q and determine their size. [ABSTRACT FROM AUTHOR]

Let p and q be odd primes and q be a cubic primitive modulo p u for some positive integer u. In this paper, we prove that the solutions of some Diophantine equations provide the weight enumerators of some cubic primitive irreducible cyclic codes of prime length. Bounds on the minimum distances of these codes are also given. [ABSTRACT FROM AUTHOR]

For a simple-root λ -constacyclic code C over F q , let 〈 ρ 〉 and 〈 ρ , M 〉 be the subgroups of the automorphism group of C generated by the cyclic shift ρ , and by the cyclic shift ρ and the scalar multiplication group M , respectively. Let N G (C ⁎) be the number of orbits of a subgroup G of the automorphism group of C acting on C ⁎ = C ﹨ { 0 }. In this paper, we establish explicit formulas for N 〈 ρ 〉 (C ⁎) and N 〈 ρ , M 〉 (C ⁎). Consequently, we derive a upper bound on the number of non-zero weights of C. We present some irreducible and reducible λ -constacyclic codes, which show that the upper bound is tight. A sufficient condition to guarantee N 〈 ρ 〉 (C ⁎) = N 〈 ρ , M 〉 (C ⁎) is presented. [ABSTRACT FROM AUTHOR]

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. 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 = 3 m − 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. [ABSTRACT FROM AUTHOR]

Cyclic codes are a subclass of linear codes and have wide applications in data storage systems, communication systems and consumer electronics due to their efficient encoding and decoding algorithms. Let p ≥ 5 be an odd prime and m be a positive integer. Let C (1 , e , s) denote the p -ary cyclic code with three nonzeros α , α e , and α s , where α is a generator of F p m ⁎ , s = p m − 1 2 , and 2 ≤ e ≤ p m − 2. In this paper, by analyzing the solutions of certain equations over finite fields, we present four classes of optimal p -ary cyclic codes C (1 , e , s) with parameters [ p m − 1 , p m − 2 m − 2 , 4 ]. Some known results on optimal quinary cyclic codes with parameters [ 5 m − 1 , 5 m − 2 m − 2 , 4 ] are special cases of our constructions. In addition, by analyzing the irreducible factors of certain polynomials over F 5 m , we present two classes of optimal quinary cyclic codes C (1 , e , s). [ABSTRACT FROM AUTHOR]

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In 2013, Ding, et al. presented nine open problems about optimal ternary cyclic codes. The first two and the 6th problems were completely solved, and the third and last three problems were partially solved. In this paper, we focus on the 7th problem. By determining the root set of some special polynomials over finite fields, we give an incomplete answer and then present a counterexample. Furthermore, we construct two new classes of optimal ternary cyclic codes with respect to the Sphere Packing Bound by studying some special polynomials over finite fields. [ABSTRACT FROM AUTHOR]

Generalized quasi-cyclic (GQC) codes have been investigated as well as quasi-cyclic (QC) codes, e.g., on the construction of efficient low-density parity-check codes. While QC codes have the same length of cyclic intervals, GQC codes have different lengths of cyclic intervals. Similarly to QC codes, each GQC code can be described by an upper triangular generator polynomial matrix, from which the systematic encoder is constructed. In this paper, a complete theory of generator polynomial matrices of GQC codes, including a relation formula between generator polynomial matrices and parity-check polynomial matrices through their equations, is provided. This relation generalizes those of cyclic codes and QC codes. While the previous researches on GQC codes are mainly concerned with 1-generator case or linear algebraic approach, our argument covers the general case and shows the complete analogy of QC case. We do not use Gröbner basis theory explicitly in order that all arguments of this paper are self-contained. Numerical examples are attached to the dual procedure that extracts one from each other. Finally, we provide an efficient algorithm which calculates all generator polynomial matrices with given cyclic intervals. [ABSTRACT FROM AUTHOR]

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, we settle an open problem about a class of optimal ternary cyclic codes which was proposed by Ding and Helleseth [5]. Let C (1 , e) be a cyclic code of length 3 m − 1 over GF (3) with two nonzeros α and α e , where m and e are given integers, and α is a generator of GF (3 m) ⁎. It is shown that C (1 , e) is optimal with parameters [ 3 m − 1 , 3 m − 1 − 2 m , 4 ] if one of the following conditions is met. 1) m ≡ 0 (mod 4) , m ≥ 4 , and e = 3 m 2 + 5. 2) m ≡ 2 (mod 4) , m ≥ 6 , and e = 3 m + 2 2 + 5. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of linear codes over finite fields, viz. skew multi-twisted codes (or skew generalized quasi-twisted codes), which is a generalization of some well-known classes of linear codes such as cyclic codes and quasi-cyclic codes. We also study algebraic structures of skew multi-twisted codes and their Galois duals (i.e., orthogonal complements with respect to the Galois inner product). We also view skew multi-twisted codes as direct sums of certain concatenated codes, which gives rise to a method to construct these codes. We also determine a lower bound on their minimum Hamming distances using their multilevel concatenated structure. Apart from this, we determine the parity-check polynomial of each skew multi-twisted code, and obtain BCH type bounds on their minimum Hamming distances. We also determine generating sets of Galois duals of some skew multi-twisted codes from generating sets of these codes. We also derive necessary and sufficient conditions under which a skew multi-twisted code is (i) Galois self-dual, (ii) Galois self-orthogonal and (iii) Galois LCD (linear with complementary dual). As an application of this result, we also provide enumeration formulae for all Hermitian self-dual, Hermitian self-orthogonal, Hermitian LCD and Euclidean LCD multi-twisted codes over finite fields. We also obtain many linear codes with best known and optimal parameters from 1-generator skew multi-twisted codes over finite fields F 8 and F 9. [ABSTRACT FROM AUTHOR]

Locally repairable codes (LRCs) have emerged as an important coding scheme in distributed storage systems (DSSs) with relatively low repair costs by accessing fewer non-failure nodes. Theoretical bounds and optimal constructions of LRCs have been widely investigated. Optimal LRCs via cyclic and constacyclic codes provide significant benefits of elegant algebraic structure and efficient encoding procedure. In this paper, we continue to consider the constructions of optimal LRCs via cyclic and constacyclic codes with long code lengths. Specifically, we first obtain two classes of q -ary cyclic Singleton-optimal (n , k , d = 6 ; r = 2) -LRCs with length n = 3 (q + 1) when 3 | (q − 1) and q is even, and length n = 3 2 (q + 1) when 3 | (q − 1) and q ≡ 1 (mod 4) , respectively. To the best of our knowledge, this is the first construction of q -ary cyclic Singleton-optimal LRCs with length n > q + 1 and minimum distance d ≥ 5. On the other hand, an LRC achieving the Hamming-type bound is called a perfect LRC. By using cyclic and constacyclic codes, we construct two new families of q -ary perfect LRCs with length n = q m − 1 q − 1 , minimum distance d = 5 and locality r = 2. [ABSTRACT FROM AUTHOR]

Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. 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. [ABSTRACT FROM AUTHOR]

Abstract: Cyclic codes are a subclass of linear codes and have a lot of applications in consumer electronics, data transmission technologies, broadcast systems, and computer applications as they have efficient encoding and decoding algorithms. In this paper, three cyclotomic sequences of order four are employed to construct a number of classes of cyclic codes over with prime length. Under certain conditions lower bounds on the minimum weight are developed. Some of the codes obtained are optimal or almost optimal. In general, the codes constructed in this paper are very good. Some of the cyclic codes obtained in this paper are closely related to almost difference sets and difference sets. [Copyright &y& Elsevier]

Abstract: Cyclic codes have been an interesting topic of both mathematics and engineering for decades. They are prominently used in consumer electronics, data transmission technologies, broadcast systems, and computer applications. Three classical approaches to the study and construction of cyclic codes are those based on the generator matrix, the generator polynomial and the idempotent. The objective of this paper is to develop another approach – the q-polynomial approach. Fundamental theory of this approach will be developed, and will be employed to construct a new family of cyclic codes in this paper. [Copyright &y& Elsevier]

In this paper, we give an improved bound for mixed exponential sums associated to good cyclic codes. [ABSTRACT FROM AUTHOR]

Highlights • Theoretical minimal RNA rings recapitulate the order of the genetic code's codon-amino acid assignments. • The paper present theoretical minimak RNA rings produced by simulations in silico which suggest an ancestral frameless translation of partially overlapping trinucleotides advancing by single nucleotide steps, constrained by resource scarcity. • Minimal RNA rings reflect the transition from frameless to ribosomal translation and are realistic candidates for ancestral tRNAs. Abstract Background Theoretical minimal RNA rings form stem-loop hairpins coding for each of the 20 amino acids and a stop, presumably mimicking life's first minimal coding and self-replicating RNAs. They resemble consensual tRNAs. Mean amino acid positions in proteins follow the genetic code's consensual amino acid inclusion order, a 5′-late-to-3′-early amino acid gradient. Hypothesis We translated minimal RNA rings to test whether translated peptides share that gradient with modern proteins, using a) ribosomal translation, non-overlapping consecutive codons; and b) frameless translation advancing nucleotide by nucleotide, producing partially overlapping codons. Results For frameless translation, most RNA rings code for a 5′-late-to-3′early amino acid gradient. Gradients indicate decreasing amino acid metabolic costs, from large to small amino acids. For ribosomal translation, the 5′-late-to-3′early amino acid gradient evolves from early to late RNA rings when ranked according to yields in Miller's experiment of their predicted anticodon's cognate amino acid. Conclusions Simulations that produced in silico minimal RNA rings didn't account for coded amino acid properties. Yet, produced peptides remind actual proteins, and suggest ancestral frameless translation of partially overlapping trinucleotides advancing by single nucleotide steps, constrained by resource scarcity. Minimal RNA rings reflect the transition from frameless to ribosomal translation and are realistic candidates for ancestral tRNAs. [ABSTRACT FROM AUTHOR]

Abstract Let m 0 and m 1 be two odd positive integers. In this paper, the weight distribution of Z 2 -double cyclic codes of length m 0 + m 1 for a special case are determined explicitly. [ABSTRACT FROM AUTHOR]

Abstract Subspace codes and particularly cyclic subspace codes have attracted a lot of research due to their applications in error correction for random network coding. In this paper, we explore the parameters of a class of the cyclic subspace codes whose subspace polynomials are more generalized. Therefore, some corollaries that extend the previously known results are obtained. Finally, a construction of k -dimensional cyclic subspace codes of size q N − 1 q − 1 and minimal distance 2 k − 2 is provided. [ABSTRACT FROM AUTHOR]

Abstract Under research for nearly sixty years, Bose–Chaudhuri–Hocquenghem (BCH) codes have played increasingly important roles in many applications such as communication, data storage and information security. However, the dimension and minimum distance of BCH codes have been seldom solved by now because of their intractable characteristics. The objective of this paper is to study the dimensions of some binary BCH codes with length n = 2 m + 1. Many new techniques are employed to investigate the coset leaders modulo n. For m = 2 t + 1 , 4 t + 2 , 8 t + 4 and m ≥ 10 , the first five largest coset leaders modulo n are determined, and the dimensions of some BCH codes of length n with designed distance δ > 2 ⌈ m 2 ⌉ are presented. These new skills and results may be helpful to study other types of cyclic codes over finite fields. [ABSTRACT FROM AUTHOR]

Abstract Self-reciprocal and self-conjugate-reciprocal polynomials over finite fields have been of interest due to their rich algebraic structures and wide applications. Self-reciprocal irreducible monic factors of x n − 1 over finite fields and their applications have been quite well studied. In this paper, self-conjugate-reciprocal irreducible monic (SCRIM) factors of x n − 1 over finite fields of square order are focused on. The characterization of such factors is given together with the enumeration formula. In many cases, recursive formulas for the number of SCRIM factors of x n − 1 are given as well. As applications, Hermitian complementary dual cyclic codes over finite fields and Hermitian self-dual cyclic codes over finite chain rings of prime characteristic are discussed. [ABSTRACT FROM AUTHOR]

Abstract Due to their wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. The objective of this paper is to present a family of optimal ternary cyclic codes from the Niho-type exponent. Specifically, for an odd integer m and a positive integer r with 4 r ≡ 1 (mod m) , a family of cyclic codes C (u , v) of length 3 m − 1 over GF (3) with two nonzeros β u and β v is studied, where β is a generator of GF (3 m) ⁎ , u = (3 m + 1) / 2 and v = 3 r + 2 is the ternary Niho-type exponent. The parameters of this family of cyclic codes are determined. It turns out that C (u , v) is optimal with respect to the Sphere Packing bound if 9 ∤ m and otherwise almost optimal. Thanks to a recent proof of the Dobbertin–Helleseth–Kumar–Martin conjecture by Katz and Langevin, the dual of this family of cyclic codes is shown to have at most five nonzero weights. [ABSTRACT FROM AUTHOR]

Abstract: We revisit in this paper the concept of decoding binary cyclic codes with Gröbner bases. These ideas were first introduced by Cooper, then Chen, Reed, Helleseth and Truong, and eventually by Orsini and Sala. We discuss here another way of putting the decoding problem into equations: the Newton identities. Although these identities have been extensively used for decoding, the work was done manually, to provide formulas for the coefficients of the locator polynomial. This was achieved by Reed, Chen, Truong and others in a long series of papers, for decoding quadratic residue codes, on a case-by-case basis. It is tempting to automate these computations, using elimination theory and Gröbner bases. Thus, we study in this paper the properties of the system defined by the Newton identities, for decoding binary cyclic codes. This is done in two steps, first we prove some facts about the variety associated with this system, then we prove that the ideal itself contains relevant equations for decoding, which lead to formulas. Then we consider the so-called online Gröbner basis decoding, where the work of computing a Gröbner basis is done for each received word. It is much more efficient for practical purposes than preprocessing and substituting into the formulas. Finally, we conclude with some computational results, for codes of interesting length (about one hundred). [Copyright &y& Elsevier]

Linear codes with a few weights have many nice applications including combinatorial design, distributed storage system, secret sharing schemes and so on. In this paper, we construct two families of linear codes with a few weights based on special polynomials over finite fields. The first family of linear codes are extended primitive cyclic codes which are affine-invariant. The second family of linear codes are reducible cyclic codes. The parameters of these codes and their duals are determined. As the first application, we prove that these two families of linear codes hold t -designs, where t = 2 , 3. As the second application, the minimum localities of the codes are also determined and optimal locally recoverable codes are derived. [ABSTRACT FROM AUTHOR]

Binary odd-like duadic codes have parameters [ n , (n + 1) / 2 ] , where n is odd. It is well known that the minimum odd weight of odd-like duadic codes has the lower bound n. The binary quadratic-residue codes and the punctured binary Reed-Muller codes of certain order are two families of binary cyclic codes with parameters [ n , (n + 1) / 2 , d ≥ n ]. It is very hard to construct an infinite family of binary cyclic codes with length n and dimension near (n + 1) / 2 whose minimum distances have a square-root bound. Recently, Tang and Ding constructed an infinite family of binary cyclic codes with parameters [ 2 m − 1 , 2 m − 1 , d ] whose minimum distances have lower bounds close to the square-root bound. The objective of this paper is to construct several infinite families of binary cyclic codes of length 2 m − 1 and dimension near 2 m − 1 whose minimum distance d much exceeds the square-root bound. For m ≥ 9 being odd, an infinite family of binary cyclic codes with parameters [ 2 m , 2 m − 1 , d ≥ 3 ⋅ 2 m − 1 2 − 1 ] is presented. For m ≥ 8 being even, an infinite family of binary cyclic codes with parameters [ 2 m , 2 m − 1 − 1 + m 2 , d ≥ 3 ⋅ 2 m − 2 2 + 1 ] is presented. [ABSTRACT FROM AUTHOR]

Linear codes and t -designs are interactive with each other. It is well known that some t -designs have been constructed by using certain linear codes in recent years. However, only a small number of infinite families of the extended codes of linear codes holding an infinite family of t -designs with t ≥ 3 are reported in the literature. In this paper, we study the extended codes of the augmented codes of a class of binary cyclic codes with three zeros and their dual codes, and show that those codes hold 3-designs. Furthermore, we obtain some shortened codes from the studied cyclic codes and explicitly determine their parameters. Some of those shortened codes are optimal or almost optimal. [ABSTRACT FROM AUTHOR]

In this paper, we characterize hulls of double cyclic codes over Z 2. Firstly, we give the generators of hulls of double cyclic codes under the separable case and the non-separable case with coprime odd block lengths. Then, using the concept of a generating function in combinatorics, the solution of dimensions of hulls of double cyclic codes over Z 2 is given. Furthermore, the enumeration of double cyclic codes over Z 2 with hulls of a fixed dimension is determined. As an application, some good entanglement-assisted quantum error-correcting codes (EAQECCs) are obtained by the hulls of non-separable double cyclic codes. [ABSTRACT FROM AUTHOR]

In this paper, by Gauss sums, we determine Hamming weight distributions of generalized quasi-cyclic (GQC) codes over the finite chain ring F q + u F q , where u 2 = 0. As applications, some new infinite families of minimal linear codes with W min W max ≤ q − 1 q and projective two-weight codes are constructed. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. The main purpose of this paper is to establish and exploit the algebraic structure of some repeated-root two dimensional constacyclic codes of length 2 p s . 2 k over the finite field F p m . Moreover, we study all self-dual repeated-root two-dimensional ( λ , 1 ) -constacyclic codes of length 2 p s . 2 or 2 p s . 2 2 over F p m for λ ∈ { − 1 , 1 } . [ABSTRACT FROM AUTHOR]

The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming distance is gained, and it is possible to design efficient decoding algorithms. In this paper, we give a version of the Hartmann–Tzeng bound that works for a wide class of skew cyclic codes. We also provide a practical method for constructing them with designed distance. For skew BCH codes, which are covered by our constructions, we discuss decoding algorithms. Detailed examples illustrate both the theory as the constructive methods it supports. [ABSTRACT FROM AUTHOR]

Relative generalized Hamming weights (RGHWs) of a linear code with respect to a linear subcode determine the security of the linear ramp secret sharing scheme based on the linear codes. They can be used to express the information leakage of the secret when some keepers of shares are corrupted. Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems. In this paper, we investigate the RGHWs of cyclic codes of two nonzeros with respect to its irreducible cyclic subcodes. We give two formulae for RGHWs of the cyclic codes. As applications of the formulae, explicit examples are computed. Moreover, RGHWs of cyclic codes in the examples are very large, comparing with the generalized Plotkin bound of RGHWs. So it guarantees very high security for the secret sharing scheme based on the dual codes. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the linear complementary dual codes (LCD codes for short) to k -Galois LCD codes, and study them by a uniform method. A necessary and sufficient condition for linear codes to be k -Galois LCD codes is obtained, two classes of k -Galois LCD MDS codes are exhibited. Then, necessary and sufficient conditions for λ -constacyclic codes being k -Galois LCD codes are characterized. Some classes of k -Galois LCD λ -constacyclic MDS codes are constructed. Finally, we study Hermitian LCD λ -constacyclic codes, and present a class of Hermitian LCD λ -constacyclic MDS codes. [ABSTRACT FROM AUTHOR]

Recently some special type of mixed alphabet codes that generalize the standard codes has attracted much attention. Besides Z 2 Z 4 -additive codes, Z 2 Z 2 [ u ] -linear codes are introduced as a new member of such families. In this paper, we are interested in a new family of such mixed alphabet codes, i.e., codes over Z 2 Z 2 [ u 3 ] where Z 2 [ u 3 ] = { 0 , 1 , u , 1 + u , u 2 , 1 + u 2 , u + u 2 , 1 + u + u 2 } is an 8-element ring with u 3 = 0 . We study and determine the algebraic structures of linear and cyclic codes defined over this family. First, we introduce Z 2 Z 2 [ u 3 ] -linear codes and give standard forms of generator and parity-check matrices and later we present generators of both cyclic codes and their duals over Z 2 Z 2 [ u 3 ] . Further, we present some examples of optimal binary codes which are obtained through Gray images of Z 2 Z 2 [ u 3 ] -cyclic codes. [ABSTRACT FROM AUTHOR]

Cyclic subspace codes gained a lot of attention especially because they may be used in random network coding for correction of errors and erasures. Roth, Raviv and Tamo in 2018 established a connection between cyclic subspace codes (with certain parameters) and Sidon spaces. These latter objects were introduced by Bachoc, Serra and Zémor in 2017 in relation with the linear analogue of Vosper's Theorem. This connection allowed Roth, Raviv and Tamo to construct large classes of cyclic subspace codes with one or more orbits. In this paper we will investigate cyclic subspace codes associated to a set of Sidon spaces, that is cyclic subspace codes with more than one orbit. Moreover, we will also use the geometry of linear sets to provide some bounds on the parameters of a cyclic subspace code. Conversely, cyclic subspace codes are used to construct families of linear sets which extend a class of linear sets recently introduced by Napolitano, Santonastaso, Polverino and the author. This yields large classes of linear sets with a special pattern of intersection with the hyperplanes, defining rank metric and Hamming metric codes with only three distinct weights. [ABSTRACT FROM AUTHOR]

In this paper, a new two-dimensional spectral/spatial codes family, named two dimensional dynamic cyclic shift codes (2D-DCS) is introduced. The 2D-DCS codes are derived from the dynamic cyclic shift code for the spectral and spatial coding. The proposed system can fully eliminate the multiple access interference (MAI) by using the MAI cancellation property. The effect of shot noise, phase-induced intensity noise and thermal noise are used to analyze the code performance. In comparison with existing two dimensional (2D) codes, such as 2D perfect difference (2D-PD), 2D Extended Enhanced Double Weight (2D-Extended-EDW) and 2D hybrid (2D-FCC/MDW) codes, the numerical results show that our proposed codes have the best performance. By keeping the same code length and increasing the spatial code, the performance of our 2D-DCS system is enhanced: it provides higher data rates while using lower transmitted power and a smaller spectral width. [ABSTRACT FROM AUTHOR]

Cyclic Reed–Solomon codes, a type of BCH codes, are widely used in consumer electronics, communication systems, and data storage devices. This fact demonstrates the importance of BCH codes – a family of cyclic codes – in practice. In theory, BCH codes are among the best cyclic codes in terms of their error-correcting capability. A subclass of BCH codes are the narrow-sense primitive BCH codes. However, the dimension and minimum distance of these codes are not known in general. The objective of this paper is to determine the dimension and minimum distances of two classes of narrow-sense primitive BCH codes with designed distances δ = ( q − 1 ) q m − 1 − 1 − q ⌊ ( m − 1 ) / 2 ⌋ and δ = ( q − 1 ) q m − 1 − 1 − q ⌊ ( m + 1 ) / 2 ⌋ . The weight distributions of some of these BCH codes are also reported. As will be seen, the two classes of BCH codes are sometimes optimal and sometimes among the best linear codes known. [ABSTRACT FROM AUTHOR]

The generalized Hamming weights (GHWs) are fundamental parameters of linear codes. GHWs are of great interest in many applications since they convey detailed information of linear codes. In this paper, we continue the work of [10] to study the GHWs of a family of cyclic codes with arbitrary number of nonzeroes. The weight hierarchy is determined by employing a number-theoretic approach. [ABSTRACT FROM AUTHOR]

Let α be a primitive element of a finite field F r , where r = q m 1 m 2 and gcd ⁡ ( m 1 , m 2 ) = d , so α 1 = α r − 1 q m 1 − 1 and α 2 = α r − 1 q m 2 − 1 are primitive elements of F q m 1 and F q m 2 , respectively. Let e be a positive integer such that gcd ⁡ ( e , q m 2 − 1 q d − 1 ) = 1 , F q m 2 = F q ( α 2 e ) , and α 1 and α 2 e are not conjugates over F q . We define a cyclic code C = { c ( a , b ) : a ∈ F q m 1 , b ∈ F q m 2 } , c ( a , b ) = ( T 1 ( a α 1 i ) + T 2 ( b α 2 e i ) ) i = 0 n − 1 , where T i denotes the trace function from F q m i to F q for i = 1 , 2 . In this paper, we use Gauss sums to investigate the weight distribution of C , which generalizes the results of C. Li and Q. Yue in [13,14] . Furthermore, we explicitly determine the weight distribution of C if d = 1 , 2 . Moreover, we prove it is optimal three-weight achieving the Griesmer bound if d = 1 and gcd ⁡ ( m 2 − e m 1 , q − 1 ) = 1 . [ABSTRACT FROM AUTHOR]

Cyclic codes have attracted a lot of research interest for decades as they have efficient encoding and decoding algorithms. In this paper, for an odd prime p , we investigate two classes of p -ary cyclic codes for special cases and determine their weight distributions explicitly. The results show that both codes have at most five nonzero weights. [ABSTRACT FROM AUTHOR]

The main results of this paper are in two directions. First, the family of finite local Frobenius non-chain rings of length 4 (hence of nilpotency index 3) is determined. As a by-product all finite local Frobenius non-chain rings with p 4 elements, ( p a prime) are given. Second, the number and structure of γ -constacyclic codes over finite local Frobenius non-chain rings with nilpotency index 3, of length relatively prime to the characteristic of the residue field of the ring, are determined. [ABSTRACT FROM AUTHOR]

In this paper, a class of binary cyclic codes with three generalized Niho-type nonzeros is introduced. Based on some techniques in solving certain equations over finite fields, the proposed cyclic codes are shown to have six nonzero weights and the weight distribution is also completely determined. [ABSTRACT FROM AUTHOR]