More on the Minimum and Stopping Distances of RS-LDPC Codes

Abstract

Based on the extended Reed-Solomon (RS) code that has two information symbols over the field ${\mathbb F}_{q}$ , we can construct a binary regular matrix ${ {\boldsymbol{\textstyle H}}}(\gamma,\rho)$ , where $q:=2^{r}$ for some positive integer $r$ , $\gamma $ and $\rho $ are two positive integers such that $\gamma \le q$ and $\rho \le q$ . The matrix ${ {\boldsymbol{\textstyle H}}}(\gamma,\rho)$ specifies a low-density parity-check (LDPC) code ${\mathcal C}(\gamma,\rho)$ , called an RS-LDPC code. In this letter, we provide more results on the minimum distance and stopping distance of this class of codes (denoted by $d({\mathcal C}(\gamma,\rho))$ and $s({ {\boldsymbol{\textstyle H}}}(\gamma,\rho))$ ) for the case $\gamma =6$ . For $\rho =q$ , we derive an upper bound on $s({ {\boldsymbol{\textstyle H}}}(6,q))$ and $d(\mathcal {C}(6,q))$ , which is conjectured to be tight for $q\ge 32$ . For $\rho , we investigate the choices of $\rho $ such that $d(\mathcal {C}(6,\rho))$ (resp., $s({ {\boldsymbol{\textstyle H}}}(6,\rho))$ ) can be improved compared with the original $d(\mathcal {C}(6,q))$ (resp., $s({ {\boldsymbol{\textstyle H}}}(6,q))$ ).