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.

l-MDS codes, as a generalization of MDS and NMDS codes, have wide applications in the areas of secret sharing schemes, index coding problems, informed source coding problems and combinatorial designs. Self-dual codes are closely related to combinatorics and lattice theory and have important application in cryptography. In this paper, by considering a class of subcodes of GRS and EGRS codes, we introduce two new classes of linear codes termed deleted GRS (DGRS) codes and extended DGRS (EDGRS) codes, and demonstrate their equivalence under certain conditions. Firstly, we not only determine the parity check matrix of the DGRS codes but also give sufficient and necessary conditions for the Singleton defectS(Ck(a,v,h)) = 0, 1 andl, respectively. In particular, ifl= 1 andCk(a,v,h) is NMDS, we can determine the weight distribution of the DGRS codes. Furthermore, we give the Schur square of the DGRS codes. From this, we not only obtain the non- GRS properties of DGRS codes, but give sufficient and necessary conditions for DGRS codes be self-orthogonal or (almost) self-dual. Finally, we present a criterion for constructingl-MDS self-dual codes via DGRS codes, for a givenl≤n/2 −2. Then based on the currently known constructions of MDS self-dual codes, we explicitly construct many new classes ofl-MDS self-dual codes with flexiblel. In particular, ifq=r2, wherer≡ 1 (mod 4) (resp.r≡ 3 (mod 4)), about 23q% (resp. 27q%)l-MDS self-dual codes with flexiblelcan be constructed.

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.

In 2019, Carletet al. introduced the concept of σ duals of linear codes involving the σ inner product, which generalizes the Euclidean, Hermitian and ℓ-Galois cases. This paper focuses on constructing new and improved classical codes and quantum codes within the framework of the σ inner product. We derive some general properties of linear codes, including matrix-product (MP) codes, with respect to the σ inner product. We develop general methods and design effective routes involving certain optimization problems for constructing σ self-orthogonal (SO) and σ dual-containing (DC) MP codes. Our schemes efficiently generate numerous such codes with new or optimal parameters. We establish the σ construction of quantum stabilizer codes from classical codes. We propose a unified method for constructing two general classes of entanglement-assisted quantum error-correcting codes (EAQECCs) based on the σ hulls of general linear codes. This further yields six types of EAQECCs with flexible parameters based on propagation rules using MP codes under the Euclidean and Hermitian cases. Compared to the best-known ternary EAQECCs, we obtain 17 new ones and 13 of them have improved parameters. Finally, we present two infinite families of q-ary EAQECCs with lengths (q2− 1)(q+ 2) andq2(q+2), respectively. These families include many q-ary QECCs that are not only new according to Grassl’s online database but also surpass those listed in Edel’s online database.

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.

Due to the application of high density data storage systems, symbol-pair codes are proposed to combat errors of the overlapping symbol pairs output over symbol-pair read channels. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that they have the highest pair error-correcting capability. In this paper, we construct two new classes of MDS symbol-pair codes with minimum pair distance seven based on simple-root cyclic codes. Our technique is through the decomposition of cyclic codes and the dual of each component code.

The class of ℓ-maximum distance separable (ℓ-MDS) codes is a generalization of maximum distance separable (MDS) codes that has attracted a lot of attention due to its applications in several areas such as secret sharing schemes, index coding problems, informed source coding problems and combinatorialt-designs. In this paper, for ℓ = 1, we completely solve a conjecture recently proposed by Henget al: (Discrete Mathematics, 346(10): 113538, 2023) and obtain infinite families of 1-MDS codes with general dimensions holding 2-designs. These later codes are also proved to be optimal locally recoverable codes. For general positive integers ℓ and ℓ′, we construct new ℓ-MDS codes from known ℓ′-MDS codes via some classical propagation rules involving the extended, expurgated, and (u, u+v) constructions. Finally, we study some general results including characterization, weight distributions, and bounds on maximum lengths of ℓ-MDS codes, which generalize, simplify, or improve some known results in the literature.

In this work, we study and determine the dimensions of Euclidean and Hermitian hulls of two classical propagation rules, namely, the$(u,u+v)$-construction and the direct sum construction. Some new criteria for the resulting codes derived from these two propagation rules being self-dual, self-orthogonal, or linear complementary dual (LCD) codes are given. As applications, we employ the$(u,u+v)$-construction to obtain (almost) self-orthogonal codes; employ the direct sum construction to provide lower bounds on the minimum distance of FSD (LCD) codes; and employ both these two constructions to derive linear codes with prescribed hull dimensions. Many (almost) optimal codes are presented. In particular, a family of binary almost Euclidean self-orthogonal Griesmer codes is constructed. We also obtain many binary, ternary Euclidean and quaternary Hermitian FSD LCD codes of larger lengths and improve some lower bounds on the minimum distance of known ternary Euclidean LCD codes.

MDS self-dual codes have nice algebraic structures and are uniquely determined by lengths. Recently, the construction of MDS self-dual codes of new lengths has become an important and hot issue in coding theory. In this paper, we construct six new classes of MDS self-dual codes by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes. Together with our constructions, the proportion of all known MDS self-dual codes relative to possible MDS self-dual codes generally exceed 57%. As far as we know, this is the largest known ratio. Moreover, some new families of MDS self-orthogonal codes are also constructed.

In this paper, we construct several classes of maximum distance separable (MDS) codes via generalized Reed-Solomon (GRS) codes and extended GRS codes, where we can determine the dimensions of their Euclidean hulls or Hermitian hulls. It turns out that the dimensions of Euclidean hulls or Hermitian hulls of the codes in our constructions can take all or almost all possible values. As a consequence, we can apply our results to entanglement-assisted quantum error-correcting codes (EAQECCs) and obtain several new families of MDS EAQECCs with flexible parameters. The required number of maximally entangled states of these MDS EAQECCs can take all or almost all possible values. Moreover, several new classes of q-ary MDS EAQECCs of length n > q+1 are also obtained.

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.

Recently, symbol-pair codes are proposed to protect against pair errors in symbol-pair read channels. One main task in symbol-pair coding theory is to design codes with large minimum pair distance. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense they attain maximal minimum pair distance. In this paper, based on constacyclic codes, we construct some new MDS symbol-pair codes with minimum pair-distance five and six. Compared with classical q-ary MDS codes, the constructed MDS symbol-pair codes have length up to q2+q+1.

One central theme in quantum error-correction is to construct quantum codes that have a large minimum distance. Quantum maximal distance separable (MDS) codes are optimal in the sense they attain maximal minimum distance. Recently, constructing quantum MDS codes has received much attention and seems to become more and more difficult. In this paper, based on classical constacyclic codes, we construct some new quantum MDS codes by employing the Hermitian construction. Compared with the known quantum MDS codes, these quantum MDS codes have much larger minimum distance.

Letqbe an odd prime power. Based on classical negacyclic codes, we construct two classes of quantum maximum-distance-separable (MDS) codes with parameters [[q2+1,q2-2d+3,d]]qwhereq≡ 1 (mod 4) and 2 ≤d≤q+1 is even, and [[(q2+1)/2,(q2+1)/2-2d+2,d]]qwhere 3 ≤d≤qis odd. Some of these quantum MDS codes are new in the sense that their parameters are different from all the previously known ones.

In this paper, we investigate the structure and properties of cyclic codes over the ringF2+vF2. We first study the relationship between cyclic codes overF2+vF2and binary cyclic codes. Then we prove that cyclic codes over the ring are principally generated, and give the generator polynomial of cyclic codes over the ring. Finally, we obtain the unique idempotent generators for cyclic codes of odd length and determine the number of cyclic codes for a given lengthnoverF2+vF2.

Using Reed-Solomon codes and cyclic MDS codes of length 2m+ 1 over F2m, we obtain several classes of negacyclic MDS codes over GR(2a, m) of lengths 2m- 1 and 2m+ 1 as analogues of Hensel lifts of these codes.

In this paper, we investigate the structure and properties of cyclic codes over the ring F2+ vF2. We first study the relationship between cyclic codes over F2+ vF2and binary cyclic codes. Then we prove that cyclic codes over the ring are principally generated, and give the generator polynomial of cyclic codes over the ring. Finally, we obtain the unique idempotent generators for cyclic codes of odd length over F2+ vF2.