The binary weight distribution of the extended (2 m,2 m − 4) code of the Reed-Solomon code over GF(2 m) with generator polynomial ( x − α)( x − α2)( x − α3)

Abstract

Consider an ( n, k) linear code with symbols from GF(2 m ). If each code symbol is represented by a binary m-tuple using a certain basis for GF(2 m ), we obtain a binary ( nm, km) linear code, called a binary image of the original code. In this paper, we present a lower bound on the minimum weight of a binary image of a cyclic code over GF(2 m ) and the weight enumerator for a binary image of the extended (2 m ,2 m − 4) code of the Reed-Solomon code over GF(2 m ) with generator polynomial ( x − α)( x − α 2) ( x − α 3) and its dual code, where α is a primitive element in GF(2 m ).