Reminiscences: My study on weight distribution problems of binary cyclic codes from 1965 to 1966

摘要

For two classes of binary codes, I derived the weight distribution formulas in 1966. The first class is the set of binary cyclic codes of 2m - 1 with m ＜ k (the dimension) ≤ 2m such that the extended codes of the dual codes are invariant under the affine group of permutations, and the second class is a certain subset of subcodes of the second order punctured Reed-Muller codes of length 2m-1 . The cross correlation function among a set S of binary sequences can be readily derived from the weight distribution of an associated binary cyclic code C. Let us say that such S is associated with C. In this sense, the original Gold sequences, Gold-like sequences, Dual-BCH sequences and the small set of Kasami sequences are associated with cyclic codes in the first class and the large set of Kasami sequences are associated with cyclic codes in the second class.