MDS阵列码的解码算法
Decoding Algorithms for MDS Array Codes
AI总结：
本文针对由超正则矩阵与非奇异矩阵克罗内克积构造的MDS阵列码，利用奇偶校验矩阵结构开发了适用于删除信道、q元对称信道等的解码算法，给出了明确步骤并辅以有限域示例说明。
AI中文摘要：
我们研究一类MDS阵列码的解码过程，这类码此前是通过超正则矩阵与有限域上非奇异矩阵的克罗内克积构造得到的。利用其奇偶校验矩阵的特殊结构，我们为不同信道模型开发了解码算法。对于删除信道，我们提供了一种可恢复最多n-k个符号删除的任意模式的算法；对于q元对称信道，我们研究了1个和2个符号错误的解码，给出了确定其位置和值的明确步骤。我们还考虑超正则矩阵为范德蒙德矩阵的特殊情况，展示了如何在解码过程中利用其额外的代数结构。我们提供了不同有限域上的明确示例来说明所提出的步骤。
英文摘要：
We study decoding procedures for a family of MDS array codes previously constructed from the Kronecker product of a superregular matrix and a non-singular matrix over a finite field. By exploiting the particular structure of their parity-check matrices, we develop decoding algorithms for different channel models. For the erasure channel, we provide an algorithm capable of recovering any pattern of up to $n-k$ symbol erasures. For the $q$-ary symmetric channel, we investigate the decoding of one and two symbol errors and give explicit procedures for determining their locations and values. We also consider the particular case in which the superregular matrix is a Vandermonde matrix, showing how its additional algebraic structure can be exploited in the decoding process. Explicit examples over different finite fields are provided to illustrate the proposed procedures.
