Algebraic Structures, Combinatorial Structures, Quaternary Linear Codes, Group Rings

Abstract

This paper characterizes the algebraic and combinatorial structures of quaternary linear codes constructed over group rings of the form ℤ₄[G], where G is a finite group. It focuses on the interplay between the hull—defined as the intersection of a code with its dual—and the automorphism group of the code, determining how these structures reflect underlying symmetries and influence code equivalence and decoding complexity. Additionally, the paper investigates the combinatorial designs supported by the codewords of these group ring codes, making use of the Assmus–Mattson Theorem and related results to determine conditions under which the supports of codewords form t-designs. Several classes of group rings, including those formed by abelian and non-abelian groups, are analyzed.

Explicit examples of codes are constructed, their hulls and automorphism groups are computed, and new families of quaternary codes giving rise to combinatorial designs are identified. These structures are subjected to characterization, leading to a classification.

The results show that certain group structures lead to codes with trivial hulls and rich automorphism groups, while others yield families of codes supporting non-trivial t-designs—revealing deep connections between algebraic and combinatorial properties.