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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]

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]

Linear complementary dual (LCD) codes provide an optimum linear coding solution for the two-user binary adder channel. LCD codes also can be used against side-channel attacks and fault non-invasive attacks. In this paper, we obtain a lower bound on the distance of binary LCD codes through expanded codes. We give necessary and sufficient conditions to extend binary [ n , k ] LCD codes to binary [ n + 1 , k ] and [ n + 1 , k + 1 ] LCD codes. A sufficient condition for a binary [ n , k , d − 1 ] LCD code constructed from a binary [ n + 1 , k , d ] LC D o , e code is also given. Finally, we construct some new binary LCD codes by using our LCD bounds and some methods such as puncturing, shortening, expanding and extension. In particular, we improve some previously known range of d LCD (n , k) of lengths 38 ≤ n ≤ 40 and dimensions 9 ≤ k ≤ 15. We also obtain some values or range of d LCD (n , k) with 41 ≤ n ≤ 50 and 6 ≤ k ≤ n − 6. [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 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]

Maximum-distance-separable (MDS) codes and self-dual codes are important families of classical codes in coding theory. It is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field F q has been completely solved for q is even. In this paper, for finite field with odd characteristic, we obtain some new results on the existence of MDS self-dual codes by generalized Reed-Solomon (GRS) codes and extended GRS codes. [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]

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes and expurgated Goppa codes (resp., extended Goppa codes) over F q m are GRS codes (resp., extended GRS codes) when m = 1. In this paper, we investigate the Galois dual codes of a special kind of Goppa codes and related codes and provide a necessary and sufficient condition for the Galois dual codes of such codes to be Goppa codes and related codes. Then we determine the Galois hulls of the above codes. In particular, we completely characterize Galois LCD, Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such family of codes. Moreover, we apply the above results to EAQECCs. [ABSTRACT FROM AUTHOR]

Recently, linear codes with few weights have been studied extensively. These linear codes have wide applications in secret sharing schemes and authentication codes. In this paper, we introduce a new construction of defining sets using cyclotomic mappings and construct three new classes of binary linear codes with two or three weights. We also explicitly determine the weight distributions of these codes. [ABSTRACT FROM AUTHOR]

An additive cyclic code of length n over F 4 can be defined equivalently as an F 2 [ x ] / 〈 x n + 1 〉 -submodule of F 4 [ x ] / 〈 x n + 1 〉. In this paper we study additive cyclic and complementary dual codes of odd length over F 4 with respect to the trace Hermitian inner product and the trace Euclidean inner product. We characterize subfield subcodes and trace codes of these codes by their generators as binary cyclic codes. [ABSTRACT FROM AUTHOR]

MDS self-dual codes over finite fields have attracted a lot of attention in recent years by their theoretical interests in coding theory and applications in cryptography and combinatorics. In this paper we present a series of MDS self-dual codes with new length by using generalized Reed-Solomon codes and extended generalized Reed-Solomon codes as the candidates of MDS codes and taking their evaluation sets as a union of cyclotomic classes. The conditions on such MDS codes being self-dual are expressed in terms of cyclotomic numbers. [ABSTRACT FROM AUTHOR]

In this paper, for some fixed q , we construct some new classes of maximum distance separable (MDS) self-dual codes over F q by using (extended) generalized Reed-Solomon codes. Our constructions utilize the additive and/or multiplicative structures of the finite field F q. It reveals that by choosing suitable subsets of F q , more MDS self-dual codes with new parameters can be found. [ABSTRACT FROM AUTHOR]

Linear codes with few weights have applications in data storage systems, secret sharing schemes and authentication codes. In this paper, inspired by the butterfly structure [6,29] and the works of Li, Yue and Fu [21] and Jian, Lin and Feng [19] , we introduce a new defining set with the form of the closed butterfly structure and consequently we obtain three classes of 3-weight binary linear codes and a class of 4-weight binary linear codes whose dual is optimal. The lengths and weight distributions of these four classes of linear codes are completely determined by some detailed calculations on certain exponential sums. Computer experiments show that many (almost) optimal codes can be obtained from our construction. [ABSTRACT FROM AUTHOR]