Improved partial permutation decoding for Reed–Muller codes

Abstract

It is shown that for $n\ge 5$ and $r\le \frac{n-1}{2}$, if an $(n,M,2r+1)$ binary code exists, then the $r$th-order Reed–Muller code $\mathcal{R}(r,n)$ has $s$-PD-sets of the minimum size $s+1$ for $1\le s\le M-1$, and these PD-sets correspond to sets of translations of the vector space ${\mathbb{F}}_2^n$. In addition, for the first order Reed–Muller code $\mathcal{R}(1,n)$, $s$-PD-sets of size $s+1$ are constructed for $s$ up to the bound $\lfloor \frac{2^n}{n+1}\rfloor -1$. The results apply also to generalized Reed–Muller codes.