两类扭曲里德-所罗门码的变体

中文摘要 AI 辅助

广义里德-所罗门码和扭曲广义里德-所罗门码是最大距离可分码的重要来源。本文研究了通过在里德-所罗门型求值码中引入列扭曲和行列同时扭曲而得到的两个变体。对于列扭曲族，我们以显式子集乘积条件的形式给出了该码为MDS的充分必要条件。在所给的参数假设下，Schur平方的维数为2k+1，这导致得到的MDS码不等价于里德-所罗门码。对于行列扭曲族，我们以初等对称函数的形式建立了MDS性质的充分必要条件。更大的Schur平方维数使其进一步区别于里德-所罗门码和已知的扭曲族，从而产生新的非RS MDS码。最后，我们为这两个码族获得了显式的奇偶校验矩阵和对偶描述。这些结果为后续研究自正交性、壳维数以及量子码构造的应用奠定了基础。

英文摘要

Generalized Reed-Solomon codes and twisted generalized Reed-Solomon codes provide important sources of maximum distance separable codes. In this paper, we study two variants obtained by introducing column twists and simultaneous row-column twists into Reed-Solomon-type evaluation codes. For the column-twisted family, we provide necessary and sufficient conditions for the code to be MDS in terms of explicit subset product conditions. Under the stated parameter assumptions, the Schur square has dimension 2k+1, which leads to MDS codes that are not equivalent to Reed-Solomon codes. For the row-column twisted family, we establish necessary and sufficient conditions for the MDS property in terms of elementary symmetric functions. The larger Schur-square dimension provides a further distinction from both Reed-Solomon codes and known twisted families, thereby yielding new non-RS MDS codes. Finally, explicit parity-check matrices and dual descriptions are obtained for both code families. These results provide a foundation for subsequent studies of self-orthogonality, hull dimensions, and applications to quantum-code constructions.

发表机构

• School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）

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