Spherically punctured biorthogonal codes

Consider a binary Reed-Muller code RM(r, m) defined on the full set of binary m-tuples and let this code be punctured to the spherical layer S(b) that includes only m-tuples of a given Hamming weight b. More generally, we can consider punctured RM codes RM(r, m, B) restricted to some set B of several spherical layers S(b), b ϵ B. In this paper we specify this construction for the biorthogonal codes RM(1, m) and the Hadamard codes H(m). It is shown that the overall weight of any code vector in a punctured code H(m, B) is determined by the weight w of its information block. More specifically, this weight depends only on the values of the Krawtchouk polynomials K_b^m(w) for all b ϵ B. We further refine our codes by limiting the possible weights w of the input information blocks. As a result, we obtain sequences of codes that meet or closely approach the Griesmer bound.