The weight enumerators for certain subcodes of the second order binary Reed-Muller codes

Abstract

In this paper we obtain formulas for the number of codewords of each weight in several classes of subcodes of the second order Reed-Muller codes. Our formulas are derived from the following results: (i) the weight enumerator of the second order RM code, as given by Berlekamp-Sloane (1970), (ii) the MacWilliams-Pless identities, (iii) a new result we present here (Theorem 1), (iv) the Carlitz-Uchiyama (1957) bound, and, (iv′) the BCH bound.The class of codes whose weight enumerators are determined includes subclasses whose weight enumerators were previously found by Kasami (1967–1969) and Berlekamp(1968a, b)