Motivated by high-density storage needs, symbol-pair codes were introduced by Cassuto and Blaum to address channels with overlapping symbol outputs. In this paper, we present a systematic study of two families of reducible cyclic codes under the symbol-pair metric. By employing analytical techniques rooted in cyclotomic numbers and Gaussian period theory over finite fields, we characterize the admissible symbol-pair weights of these codes. Significantly, we demonstrate that their minimum symbol-pair distances attain twice the minimum Hamming distances under specific algebraic constraints. Furthermore, we identify and rigorously determine the symbol-pair weight distributions for several three-weight code families. Notably, we construct a class of MDS symbol-pair codes that achieve optimal distance parameters by the puncturing technique. As supplementary contributions, the paper resolves several computational problems concerning generalized cyclotomic numbers, thereby enriching the mathematical foundation for code parameter analysis.

Constacyclic codes over finite fields are a family of linear codes and contain cyclic codes as a subclass. Constacyclic codes are related to many areas of mathematics and outperform cyclic codes in several aspects. Hence, constacyclic codes are of theoretical importance. On the other hand, constacyclic codes are important in practice, as they have rich algebraic structures and may have efficient decoding algorithms. In this paper, two classes of constacyclic codes are constructed using a general construction of constacyclic codes with cyclic codes. The first class of constacyclic codes is motivated by the punctured Dilix cyclic codes and the second class is motivated by the punctured generalised Reed-Muller codes. The two classes of constacyclic codes contain optimal linear codes. The parameters of the two classes of constacyclic codes are analysed and some open problems are presented in this paper.

BCH codes and their dual codes are two special subclasses of cyclic codes and are the best linear codes in many cases. A lot of progress on the study of BCH cyclic codes has been made, but little is known about the minimum distances of duals of BCH codes. Recently, a concept called dually-BCH code was introduced to investigate the duals of BCH codes and the lower bounds on their minimum distances in Gong et al., (2022). For a prime power$q$and an integer$m \ge 4$, let$n=\frac {q^{m}-1}{q+1}$($m$even), or$n=\frac {q^{m}-1}{q-1}$($q>2$). In this paper, some sufficient and necessary conditions in terms of the designed distance will be given to ensure that the narrow-sense BCH codes of length$n$are dually-BCH codes, which extended the results in Gong et al., (2022). Lower bounds on the minimum distances of their dual codes are developed for$n=\frac {q^{m}-1}{q+1}$($m$even). As byproducts, we present the largest coset leader$\delta _{1}$modulo$n$being of two types, which proves a conjecture in Wu et al., (2019) and partially solves an open problem in Li et al., (2017). We also investigate the parameters of narrow-sense BCH codes of length$n$with design distance$\delta _{1}$. The BCH codes presented in this paper have good parameters in general.

BCH codes are an interesting class of cyclic codes due to their efficient encoding and decoding algorithms. In the past sixty years, a lot of progress on the study of BCH codes has been made, but little is known about the properties of their duals. Recently, in order to study the duals of BCH codes and the lower bounds on their minimum distances, a new concept called dually-BCH code was proposed by (Gong et al., 2022). In this paper, the lower bounds on the minimum distances of the duals of narrow-sense BCH codes with length$\frac {q^{m}-1}{\lambda }$over$\mathbb {F}_{q}$are developed, where$\lambda $is a positive integer satisfying$\lambda =q^{s}-1$and$s\, |\,m$, or$\lambda \, |\, q-1$. In addition, the sufficient and necessary conditions in terms of the designed distances for these codes being dually-BCH codes are presented. Our lower bounds on the minimum distances of the duals of BCH codes include the bounds stated in (Gong et al., 2022) as a special case. Moreover, our lower bounds improve the bounds stated in (Gong et al., 2022), the classical Sidel’nikov bound, and the Carlitz-Uchiyama bound when the designed distances of the BCH codes are in some ranges. Several examples show that our proposed lower bounds are good in some cases.

Let${\mathcal C}$be an$[n, k]$linear code over the finite field${\mathbb F}_{q}$. Let$d_{I}({\mathcal C})$denote its insertion-deletion (insdel for short) distance, which characterizes the insdel error-correcting capability of${\mathcal C}$. To determine the insdel distances of linear codes is a very challenging problem. In this paper we propose a strict half-Singleton upper bound$d_{I}({\mathcal C}) \leq 2(n-2k+1)$if${\mathcal C}$does not contain the codeword with all 1s, which generalizes the half-Singleton bound on the insdel distances of linear codes due to Cheng-Guruswami-Haeupler-Li, and a stronger direct upper bound$d_{I}({\mathcal C}) \leq 2(d_{H}({\mathcal C})-t)$under a weak condition, where$t\geq 1$is a positive integer determined by the generator matrix and$d_{H}({\mathcal C})$denotes the Hamming distance of${\mathcal C}$. A sufficient condition for a linear code attaining the strict half-Singleton bound is given. We prove that the code length of an optimal binary linear insdel code with respect to the (strict) half-Singleton bound is about twice its dimension and conjecture that optimal binary linear insdel codes have exact parameters$[{2k, k, 4}]$or$[{2k+1, k, 4}]$with respect to the half-Singleton bound or the strict half-Singleton bound, respectively. Moreover, interestingly explicit optimal linear insdel codes attaining the (strict) half-Singleton bound, with the code length being independent of the finite field size, are given.

The generalized Hamming weight of linear codes is a natural generalization of the minimum Hamming distance. They convey the structural information of a linear code and determine its performance in various applications, and have become one of important research topics in coding theory. Recently, Li (2021) and Li and Li (2022) obtained the complete weight hierarchy of linear codes from quadratic forms over finite fields of odd characteristic by analysis of the solutions of the restricted quadratic equation in its subspace. In this paper, we further determine the complete weight hierarchy of linear codes from quadratic forms over finite fields of even characteristic by carefully studying the behavior of the corresponding restricted quadratic forms to the subspaces of the field, and complement the results of Li and Li.

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$q=p^{m}$, where$p$is an odd prime and$m \geq 2$is an integer. The objective of this paper is to present an infinite family of cyclic codes over${\mathrm {GF}}(q)$supporting an infinite family of 3-designs and two infinite families of negacyclic codes over${\mathrm {GF}}(q^{2})$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${\mathrm {GF}}(q)$are studied. Three infinite families of almost MDS codes are also presented. A constacyclic code over${\mathrm {GF}}(4)$supporting a 4-design and seven open problems are also presented in this paper.

It is well-known that cyclic codes have efficient encoding and decoding algorithms. In recent years, antiprimitive BCH codes have attracted a lot of attention. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. Some lower bounds on the minimum distance of antiprimitive BCH codes are given. The BCH codes presented in this paper have good parameters in general, containing many optimal linear codes. In particular, two open problems about the minimum distance of BCH codes of this type are partially solved in this paper.

Two general constructions of linear codes with functions over finite fields have been extensively studied in the literature. The first one is given by C(f)={ Tr(af(x)+bx)x ∈ \mathbb Fqm*: a,b ∈ \mathbb Fqm }, where q is a prime power, \mathbb Fqm* = \mathbb Fqm \{0}, Tr is the trace function from \mathbb Fqm to \mathbb Fq, and f(x) is a function from \mathbb Fqm to \mathbb Fqm with f(0)=0. Almost bent functions, quadratic functions and some monomials on \mathbb F2m were used in the first construction, and many families of binary linear codes with few weights were obtained in the literature. This paper studies some punctured codes of these binary codes. Several families of binary linear codes with few weights and new parameters are obtained in this paper. Several families of distance-optimal binary linear codes with new parameters are also produced in this paper.