Spherically punctured Reed-Muller codes

Consider a binary Reed-Muller code RM(r, m) defined on the m-dimensional hypercube F^m₂. In this paper, we study punctured Reed-Muller codes P_r(m, b) whose positions form a spherical b-layer and include all m-tuples of a given Hamming weight b. These punctured codes inherit some recursive properties of the original RM codes and can be built from the shorter codes, by decomposing a spherical b-layer into sub-layers of smaller dimensions. However, codes P_r(m, b) cannot be formed by the recursive Plotkin construction. We analyze recursive properties of these codes and find their code distances for arbitrary values of parameters r, m, and b.