On Lowest Density MDS Codes
On Lowest Density MDS Codes
复制标题
DOI：
10.1109/18.746771
复制
发表时间：
1999
期刊：
影响因子：
--
通讯作者：
M. Blaum;R. Roth
中科院分区：
文献类型：
--
作者：
M. Blaum;R. Roth
设F/sub q/表示有限域GF（q），h为正整数.考虑符号字母表F/sub q/sup B/上的MDS（最大距离可分离）码，其在F/sub q/上是线性的，并且在F/sub q/上具有稀疏（“低密度”）奇偶校验和生成矩阵，其在F/sub q/sup B/上是系统的。给出了F/sub q//sup B/上的F/sub q/-线性MDS码的任意系统奇偶校验矩阵或生成矩阵中非零元素个数的下界，沿着给出了达到这些下界的任意MDS码的长度的上界.提出了一种构造，实现这些界限的某些冗余值。构造的基础是F/sub q/上的一组稀疏非奇异矩阵，它们的两两差也是非奇异的。也给出了系统的奇偶校验和生成矩阵的条件放宽到F/sub q/，而不是F/sub q//sup B/的情况下的界限和结构。
Let F/sub q/ denote the finite field GF(q) and let h be a positive integer. MDS (maximum distance separable) codes over the symbol alphabet F/sub q//sup b/ are considered that are linear over F/sub q/ and have sparse ("low-density") parity-check and generator matrices over F/sub q/ that are systematic over F/sub q//sup b/. Lower bounds are presented on the number of nonzero elements in any systematic parity-check or generator matrix of an F/sub q/-linear MDS code over F/sub q//sup b/, along with upper bounds on the length of any MDS code that attains those lower bounds. A construction is presented that achieves those bounds for certain redundancy values. The building block of the construction is a set of sparse nonsingular matrices over F/sub q/ whose pairwise differences are also nonsingular. Bounds and constructions are presented also for the case where the systematic condition on the parity-check and generator matrices is relaxed to be over F/sub q/, rather than over F/sub q//sup b/.