arXiv:2506.03874 · v3

On a class of generalized Roth-Lempel codes

首次提交 2025-06-04 20:08

版本更新 2026-11-12 10:18

雷达观测 2026-11-13 14:16

摘要

Maximum distance separable (MDS), almost MDS (AMDS), and self-dual codes have applications in communications, data storage, and quantum coding. The construction and structural study of MDS codes that are not equivalent to generalized Reed-Solomon (GRS) codes are important problems in coding theory and finite geometry. In this paper, we investigate the generalized Roth-Lempel (GRL) codes $GRL_k(α, v, A_{3\times 3})$ introduced by Li et al. [28] in the case of a $3\times 3$ appended matrix. Using the Cauchy matrix method of Roth and Lempel, we derive a sufficient condition under which these codes are not equivalent to GRS codes. We also establish necessary and sufficient conditions for their duals to be AMDS and for the codes to be self-dual. The self-duality criterion extends the corresponding result in [28], and we provide examples illustrating our results. Under suitable parameter assumptions and an explicit rank condition, we show that the Schur square of an $[n, k]_q$ GRL code $GRL_k(α, v, A_{3\times 3})$ has dimension $2k + 2$. Using this invariant, we show that, under certain conditions, these GRL codes are not equivalent to codes from several known non-GRS families.

作者备注：21 pages