Kasami type codes of higher relative dimension

Let m,n, d, e be fixed positive integers such that m = 2n, e = (n, d) = (m, d), 1 ≤ k ≤ n e . Let s be a fixed maximum-length binary sequence of length 2−1. For 0 ≤ j < k, let sj be the circular decimation of s with decimation factor 2 n e +1. Then s, s1, · · · , sk−1 are maximum-length binary sequences of length 2 − 1, while s0 is a maximum-length binary sequence of length 2−1. Let C be the F2-vector space generated by all circular shifts of s, s0, s1, · · · , sk−1. Then C has an F2n-vector space structure, and is of dimension 2k+1 over F2n . We regard C as a Kasami type code of relative dimension 2k+1. The DC component distribution of C is explicitly calculated out in the present paper. Key phrases: Kasami code, cyclic code, alternating form MSC: 94B15, 11T71.