The weight distribution of irreducible cyclic codes with block lengths n1((q1−1)N)

AbstractWe study the weight distribution of irreducible cyclic (n, k) codeswith block lengths n = n1((q1 − 1)/N), where N|q − 1, gcd(n1,N) = 1, and gcd(l,N) = 1. We present the weight enumerator polynomial, A(z), when k = n1l, k = (n1 − 1)l, and k = 2l. We also show how to find A(z) in general by studying the generator matrix of an (n1, m) linear code, V∗d over GF(qd) where d = gcd (ordn1(q), l). Specifically we study A(z) when V∗d is a maximum distance separable code, a maximal shiftregister code, and a semiprimitive code. We tabulate some numbers Aμ which completely determine the weight distributionof any irreducible cyclic (n1(21 − 1), k) code over GF(2) for all n1 ⩽ 17