Kanat Abdukhalikov

Tushar Bag, Daniel Panario

We study linear codes and quantum error-correcting codes (QECCs) constructed from group rings over finite fields. Using the algebraic structure of group rings, we give a single framework for codes over several group structures, including cyclic, dihedral, direct-product, and semidirect-product groups. We establish necessary and sufficient conditions for these group codes to be self-orthogonal under the Euclidean, Hermitian, and symplectic inner products. We show that non-isomorphic groups of the same order can generate inequivalent codes with distinct parameters, and we support this with explicit computational comparisons. Using these structural results, we give explicit constructions of quantum codes and provide new examples that match or improve upon the best known parameters. In particular, we describe explicit block-matrix forms of the generating matrices for dihedral and direct-product groups, and we use a Kronecker-product construction to obtain an infinite family of self-orthogonal group codes together with the corresponding QECCs.

Kanat Abdukhalikov, Gyanendra K. Verma

We construct two new families of linear codes by modifying the generator matrices of generalized Reed-Solomon (GRS) codes. For these codes, we explicitly derive parity-check matrices and establish necessary and sufficient conditions ensuring the MDS property. Additionally, we explore subfamilies within these constructions that are non-GRS MDS codes. We also characterize their self-orthogonal and self-dual properties and present some explicit constructions and examples.

Kanat Abdukhalikov, Duy Ho

Additive codes have attracted considerable attention for their potential to outperform linear codes. However, distinguishing strictly additive codes from those that are equivalent to linear codes remains a fundamental challenge. To resolve this ambiguity, we introduce a deterministic test that requires only the generator matrix of the code. We apply this test to verify the strict additivity of several quaternary additive codes recently reported in the literature. Conversely, we demonstrate that a previously known additive complementary dual (ACD) code is equivalent to a linear Hermitian LCD code, thereby improving the best-known bounds for such linear codes.

Kanat Abdukhalikov, Gyanendra K. Verma

In this paper, we study quasi-twisted codes and their relationship with additive constacyclic codes through a polynomial-based approach. We first present a polynomial characterization of quasi-twisted codes over finite fields analogous to quasi-cyclic codes and determine Euclidean, Hermitian, and symplectic duals of quasi-twisted codes with index $2$. Additionally, we provide necessary and sufficient conditions for the self-orthogonality of appropriate quasi-twisted codes. Next, we explore a one-to-one correspondence between quasi-twisted codes of length $lm$ with index $l$ over $\mathbb{F}_q$ and additive constacyclic codes of length $m$ over $\mathbb{F}_{q^l}$. We establish relationships between trace inner products in the additive setting and Euclidean, symplectic inner products in the quasi-twisted setting. Using these relations and the correspondence, we determine the dual of additive constacyclic codes with respect to the trace inner products. As a consequence, we conclude that determining the trace Euclidean dual and trace Hermitian dual of an additive constacyclic code is equivalent to determining the Euclidean and symplectic dual of the corresponding quasi-twisted code.