MDS codes with determined Hermitian hull dimensions have attracted significant attention for their application in quantum error correction. From an MDS code over Fq2with fixed Hermitian hull dimension ℓ, whereqis a prime power larger than 2, one can obtain an MDS code with any smaller ℓ′-dimensional Hermitian hull for 0 ≤ ℓ′ ≤ ℓ. Then it is natural to consider the problem of determining the maximum Hermitian hull dimension, denoted byLq(n, k), among all MDS codes with the same lengthnand dimensionkover Fq2. Some constructions of Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes had been proposed, which addressed this problem for certain parameter regimes. However, it is still unknown for many cases, in particular fork≥q+ 1. In this paper, we study the Hermitian hulls of a class of codes which generalizes GRS codes, called twisted generalized Reed-Solomon (TGRS) codes. TGRS codes contain MDS subclasses that are not linearly equivalent to GRS codes (called non-GRS codes). We give a bound on the Hermitian hull dimensions of certain TGRS codes of general twists. In addition, we derive a lower bound onLq(n, k) forn|q2− 1 and 1 ≤k≤n, which generalizes and improves some previous results. For some parameter regimes wheren≥q+ 1 andk≥q+ 1, we prove thatLq(n, k) ≥k/2 and explicitly construct [n, k]q2MDS codes whose Hermitian hulls have dimension at leastk/2. This result solves partially an open problem pointed out in the literature. The constructed MDS codes arise from either GRS or non-GRS TGRS codes. Furthermore, some sufficient conditions for TGRS codes with general twists to be Hermitian self-orthogonal are given, and Hermitian self-orthogonal non-GRS MDS codes are constructed. Based on our constructions, we provide several families of MDS entanglement-assisted quantum error-correcting codes.

In a recent work, quantum locally recoverable codes (qLRCs) have been introduced for their potential application in large-scale quantum data storage and implication for quantum LDPC codes. This work focuses on the bounds and constructions of qLRCs derived from the Hermitian construction, which solves an open problem proposed by Luo $et~al.$ (IEEE Trans. Inf. Theory, 71 (3): 1794-1802, 2025). We present four bounds for qLRCs and give comparisons in terms of their asymptotic formulas. We construct several new infinite families of NMDS codes, with general and flexible dimensions, that support t-designs for $t\in \{2,3\}$, and apply them to obtain Hermitian dual-containing classical LRCs (cLRCs). As a result, we derive three explicit families of optimal qLRCs. Compared to the known qLRCs obtained by the CSS construction, our optimal qLRCs offer new and more flexible parameters. It is also worth noting that the constructed cLRCs themselves are interesting as they are optimal with respect to four distinct bounds for cLRCs.

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.

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.

In eight published papers in IEEE Transactions on Information Theory, infinite families of optimal few-weight binary andq-ary linear codes were constructed and their weight distributions were determined. These codes are linear codes meeting the Griesmer bound. We indicate that many Griesmer codes constructed in these papers are not new. They are actually Solomon-Stiffler codes invented in 1965. Therefore weight distributions of some special binary orq-ary Solomon-Stiffler codes were determined in the papers mentioned above. From a similar geometric approach as Solomon-Stiffler codes, we construct ten infinite families of binary, ternary and quaternary few-weight, optimal, almost optimal and near-optimal linear codes close to the Griesmer bound and their weight distributions are determined. These linear codes have positive Griesmer defects up to five, and thus not Solomon-Stiffler codes and Griesmer codes from minihypers. Moreover, many optimal, best known and almost optimal quantum codes of small lengths, comparing with Grassl's table on quantum codes, are constructed from the same geometric approach as binary Solomon-Stiffler codes.

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.

Perfect deletion-correcting codes of the same length over the same alphabet can have different sizes. The interesting problem of determining the possible sizes of perfect deletion-correcting codes has previously been studied. In this paper, we study the corresponding problem for burst deletion-correcting codes. We completely determine the spectrum of sizes for perfect burst deletion-correcting codes for certain classes of parameters and also construct new classes of perfect deletion-correcting codes.