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.

In this paper we show some applications of algebraic curves to the construction of kernels of polar codes over a discrete memoryless channel which is symmetric w.r.t the field operations. We will also study the minimum distance of the polar codes proposed, their duals and the exponents of the matrices used for defining them. All the restrictions that we make to our curves will be accomplished by the so called Castle Curves.

There are two gradient descent decoding procedures for binary codes proposed independently by Liebler and by Ashikhmin and Barg. Liebler in his paper mentions that both algorithms have the same philosophy but in fact they are rather different. The purpose of this communication is to show that both algorithms can be seen as two ways of understanding the reduction process algebraic monoid structure related to the code. The main tool used for showing this is the Gröbner representation of the monoid associated to the linear code.

The structure of multivariate serial codes over a finite chain ring R is established using the structure of the residue field $\bar R$. Multivariate codes extend in a natural way the univariate cyclic and negacyclic codes and include some nontrivial codes over R. The structure of the dual codes in the serial abelian case is also derived, and some conditions for the existence of self‐dual codes over R are studied.

Niederreiter and Xing (2000) recently proposed a propagation rule for linear codes.O/spl uml/zbudak and Stichtenoth showed that the Niederreiter-Xing construction is a particular construction of a matrix-product code. Cheng, Cheng, and Sun analyzed the case when Niederreiter-Xing rule commutes with duality. The aim of this correspondence is to generalize the propagation rule to a wider class of codes and analyze the case when it commutes with duality.