Spherically Punctured Reed-Muller Codes

Abstract

Consider a binary Reed–Muller code RM $(r,m)$ defined on the $m$ -dimensional hypercube $\mathbb {F}_{2}^{m}$ . In this paper, we study punctured Reed–Muller codes $P_{r}(m,b)$ , whose positions are restricted to the $m$ -tuples of a given Hamming weight $b$ . In combinatorial terms, this paper concerns $m$ -variate Boolean polynomials of any degree $r$ , which are evaluated on a Hamming sphere of some radius $b$ in $\mathbb {F}_{2}^{m}$ . Codes $P_{r}(m,b)$ inherit some recursive properties of RM codes. In particular, they can be built from the shorter codes, by decomposing a spherical $b$ -layer into sub-layers of smaller dimensions. However, these sub-layers have different sizes and do not form the classical Plotkin construction. We analyze recursive properties of the spherically punctured codes $P_{r}(m,b)$ and find their distances for the arbitrary values of parameters $r,m$ , and $b$ . Finally, we describe recursive (successive cancellation) decoding of these codes.