Insertion-deletion (insdel for short) codes have received extensive attention due to their ability to correct synchronization errors. It is usually a very challenging problem to determine the insdel distances of linear codes. In this paper, a good lower bound on the insdel distance of a linear code with a certain algebraic structure is provided and it indeed gives an affirmative answer to an open problem proposed by Hao Chen (IEEE Transactions on Information Theory, 68(8): 5126–5132, 2022). Applying this lower bound to binary first-order Reed-Muller codes and binary simplex codes, we obtain that they possess linear subcodes that can correct arbitrary insdel errors while maintaining Hamming distances robustness. The application of this bound on Reed-Muller codes completely solves an open problem left by Lara Dolecek and Venkat Anantharam (IEEE Transactions on Information Theory, 53(4): 1430–1443, 2007). In order to enhance the code rate, we further perform puncturing on these subcodes and determine the Hamming distances of the punctured subcodes while ensuring that their insdel distances remain unchanged.

The recent work by Cotardo and Ravagnani (IEEE Trans. Inf. Theory, vol. 68, no. 5, pp. 2941-2950, May 2022) introduced a class of binary codes endowed with asymmetric discrepancy, which are referred to as Combinatorial Neural codes (CN codes), and are motivated by theoretical neuroscience. The applications in binary asymmetric memoryless channel and neuroscience have spurred interest in constructing binary codes and analyzing the error-correction capabilities under asymmetric discrepancy. In this paper, we first characterize equidistant CN codes and propose several constructions of (equidistant) CN codes based on the Hadamard codes and punctured Hadamard codes. For a binary linear codeC⊆ GF(2)n, we then analyze the minimum asymmetric discrepancy of the nonzero subsetC\{0}, the coset u+C(u ∈ GF(2)n), and the unionC∪(1+C), where 0 denotes the all-zero vector and 1 the all-one vector. Based on these results, we completely determine the exact parameters for several classes of CN codes by combining simplex codes or projective 2-weight codes.

Both maximum distance separable (MDS) codes that are not equivalent to generalized Reed-Solomon (GRS) codes (non-GRS MDS codes) and near MDS (NMDS) codes have nice applications in communication and storage systems. In this paper, we introduce and study a new family of linear codes, including their parameters, weight distributions, and self-orthogonal properties. We prove that such codes are either non-GRS MDS codes or NMDS codes. We also determine their weight distributions with the help of the solutions to some subset sum problems. A sufficient and necessary condition for such codes to be self-orthogonal is characterized. Based on this condition, we further deduce that there are no self-dual codes in this class of linear codes and explicitly construct two new classes of almost self-dual codes.

Maximum distance separable (MDS) codes that are not monomially equivalent to generalized Reed-Solomon (GRS) codes are called non-GRS MDS codes, which have important applications in communication and cryptography. Covering radii and deep holes of linear codes are closely related to their decoding problems. In the literature, the covering radii and deep holes of GRS codes have been extensively studied, while little is known about non-GRS MDS codes. In this paper, we study two classes of extended twisted generalized Reed-Solomon (ETGRS) codes involving their non-GRS MDS properties, covering radii, and deep holes. In other words, we obtain two classes of non-GRS MDS codes with known covering radii and deep holes. As applications, we further directly derive more non-GRS MDS codes, and get some results on the existence of their error-correcting pairs. As a byproduct, we find some connections between the well-known Roth-Lempel codes and these two classes ETGRS codes.

Constacyclic codes over finite fields are an important class of linear codes as they contain distance-optimal codes and linear codes with best known parameters. They are interesting in theory and practice, as they have the constacyclic structure. In this paper, an infinite class of q-ary negacyclic codes of length$(q^{m}-1)/2$and an infinite class of q-ary constacyclic codes of length$(q^{m}-1)/(q-1)$are constructed and analyzed. As a by-product, two infinite classes of ternary negacyclic self-dual codes with a square-root-like lower bound on their minimum distances are presented.

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.

A linear code with parameters [n,k,n−k+1] is called a maximum distance separable (MDS for short) code. A linear code with parameters [n,k,n−k] is said to be almost maximum distance separable (AMDS for short). A linear code is said to be near maximum distance separable (NMDS for short) if both the code and its dual are AMDS. MDS codes are very important in both theory and practice. There is a classical construction of a [q+1,2u−1,q−2u+3] MDS code for eachuwith 1 ≤u≤ ⌊q+1/2⌋, which is a reversible and cyclic code. The objective of this paper is to study the extended codes of this family of MDS codes. Two families of MDS codes and several families of NMDS codes are obtained. The NMDS codes have applications in finite geometry, cryptography and distributed and cloud data storage systems. The weight distributions of some of the extended codes are determined.

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.

Binary cyclic codes with parameters$[n,(n+1)/2, d\geq \sqrt {n}]$are very interesting, as their minimum distances have a square-root bound. The binary quadratic residue codes and the punctured binary Reed-Muller codes of order$(m-1)/2$for odd$m$are two infinite families of binary cyclic codes with such parameters. The objective of this paper is to present and analyse an infinite family of binary BCH codes${\mathcal {C}}(m)$with parameters$[2^{m}-1,2^{m-1},d]$whose minimum distance$d$much exceeds the square-root bound when$m \geq 11$is a prime. The binary BCH code${\mathcal {C}}(3)$is the binary Hamming code and distance-optimal. The binary BCH code${\mathcal {C}}(5)$has parameters$[{31,16,7}]$and is distance-almost-optimal. The binary BCH code${\mathcal {C}}(7)$has parameters$[{127,64,21}]$and has the best known parameters. In addition, there is no known$[2^{m}-1,2^{m-1}]$binary cyclic code whose minimum distance is better than the minimum distance of this binary BCH code${\mathcal {C}}(m)$with parameters$[2^{m}-1,2^{m-1}]$for any odd prime$m$.

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] codeCin this family, there is no known [n,k,d′] linear code over GF(q) such thatd′ >d. An infinite family of cyclic codes over GF(q) is said to have cyclicly-best-known parameters if for any [n,k,d] codeCin this family, there is no known [n,k,d′] cyclic code over GF(q) such thatd′ >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 byC(2,m,2), and their dual codesC⊥(2,m,2). The weight distribution ofC⊥(2,m,2)is settled and the parameters ofC(2,m,2)are investigated in this paper. A larger family of binary cyclic codesC(2,m,r)and their duals are also constructed and studied in this paper, where 0 ≤r≤m– 1.

The construction of self-dual codes with large minimum distances has been an active topic in coding theory. The construction and classification of extremal self-dual codes over small fields are related to other fields of mathematics, such as lattices and invariant theory as well as combinatorialt-designs. It is well-known thatq-ary self-dual cyclic codes exist only whenqis an even prime power andq-ary self-dual negacyclic codes exist for any odd prime powerq. In 2009 a family of binary self-dual cyclic codes with lengthsniand minimum distancesdi≥ 1/2√ni, wherenigoes to the infinity ifigoes to the infinity, was constructed. In this paper, we construct several families ofq-ary self-dual negacyclic codes of lengthsnwith their minimum distances larger than or equal ton1/2for various lengthsnand any given odd prime powerq. Whenq∈ {3, 5} and the length is small, the minimum distances of the constructed self-dual negacyclic codes are comparable with these self-dual codes with largest known minimum distances in the literature.

BCH codes are an important class of cyclic codes which have applications in satellite communications, DVDs, disk drives, and two-dimensional bar codes. Although BCH codes have been widely studied, their parameters are known for only a few special classes. Recently, Ding et al. made some new progress in BCH codes. However, we still have very limited knowledge on the dimension of BCH codes, not to mention the weight distribution of BCH codes. In this paper, we generalize the results on BCH codes from several previous papers. 1) The dimension of narrow-sense BCH codes of length ((qm-1)/λ) with designed distance 2 ≤ δ ≤ ((qΓ(m+1)/2⌉- 1)/(λ) + 1) is settled, where λ is any factor of (q - 1). 2) The weight distributions of two classes of narrow-sense BCH codes of length ((qm- 1)/2) with designed distance δ = (((q - 1)qm-1- q⌊(m-1)12⌋- 1)/2) and δ = (((q - 1)qm-1- q⌊(m+1)/2⌋- 1)/2) are determined. 3) The weight distribution of a class of BCH codes of length ((qm- 1)/(q - 1)) is determined. In particular, a subclass of this class of BCH codes is optimal with respect to the Griesmer bound. Some optimal linear codes obtained from this class of BCH codes are characterized.