基于公共零点集的二维常循环码结构研究 (On the Structure of Two-Dimensional Constacyclic Codes using Common Zero Sets)

from arxiv, The manuscript has been withdrawn in order to correct errors and significantly revise the results. A substantially improved version will be submitted separately

We consider two-dimensional $(λ_1, λ_2)$-constacyclic codes over $\mathbb{F}_{q}$ of area $M N$, where $q$ is some power of prime $p$ with $\gcd(M,p)=1$ and $\gcd(N,p)=1$. With the help of common zero (CZ) set, we characterize 2-D constacyclic codes. Further, we provide an algorithm to construct an ideal basis of these codes by using their essential common zero (ECZ) sets. We also describe the dual of 2-D constacyclic codes. Finally, we provide an encoding scheme for generating 2-D constacyclic codes from the generator tensor, implementable in a parallel fashion. Through examples, we illustrate that 2-D constacyclic codes can have better minimum distance compared to their cyclic counterparts with the same code area and code rate, generalizing prior work over 2-D binary cyclic coded arrays.

翻译：本文研究定义在有限域 $\mathbb{F}_{q}$ 上、面积为 $M N$ 的二维 $(λ_1, λ_2)$-常循环码，其中 $q$ 是素数 $p$ 的幂，且满足 $\gcd(M,p)=1$ 与 $\gcd(N,p)=1$。借助公共零点集，我们对二维常循环码进行了刻画。进一步，我们提出了一种算法，利用其本质公共零点集来构造这些码的理想基。我们还描述了二维常循环码的对偶码。最后，我们提出了一种可从生成张量并行生成二维常循环码的编码方案。通过实例，我们证明了在相同码面积和码率下，二维常循环码相较于其循环码对应物可能具有更优的最小距离，从而推广了先前关于二维二进制循环码阵列的研究。