Optimal binary codes from $\mathcal{C}_{D}$-codes over a non-chain ring

TL;DR

This work builds binary subfield codes $C_D^{(2)}$ from $C_D$-codes over a ring $R$ using trace maps and defining sets $D$ derived from simplicial complexes, achieving detailed weight distributions and identifying infinite families that are distance-optimal. The authors show that these subfield codes often improve parameter trade-offs over the binary Gray images of the original ring-based codes and establish sufficient conditions for minimality and self-orthogonality, enabling applications to secret sharing and quantum coding. They also connect two-weight codes to strongly regular graphs via Calderbank’s construction, yielding explicit SRG families from the resulting codes. Overall, the work advances explicit constructions of distance-optimal binary codes from ring-based frameworks and strengthens ties between coding theory, combinatorial designs, and graph theory.

Abstract

In \cite{shi2022few-weight}, Shi and Li studied $\mathcal{C}_D$-codes over the ring $\mathcal{R}:=\mathbb{F}_2[x,y]/\langle x^2, y^2, xy-yx\rangle$ and their binary Gray images, where $D$ is derived using certain simplicial complexes. We study the subfield codes $\mathcal{C}_{D}^{(2)}$ of $\mathcal{C}_{D}$-codes over $\mathcal{R},$ where $D$ is as in \cite{shi2022few-weight} and more. We find the Hamming weight distribution and the parameters of $\mathcal{C}_D^{(2)}$ for various $D$, and identify several infinite families of codes that are distance-optimal. Besides, we provide sufficient conditions under which these codes are minimal and self-orthogonal. Two families of strongly regular graphs are obtained as an application of the constructed two-weight codes.

Theorems & Definitions (28)

• Lemma 2.1
• Theorem 2.2
• Lemma 2.3
• Definition 2.4
• Lemma 2.5
• Theorem 2.6
• Definition 3.1
• Theorem 3.2
• Theorem 3.3
• proof