Spherically Punctured Reed-Muller Codes

Consider a binary Reed-Muller code RM(r, m) defined on the m-dimensional hypercube 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 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.