A Construction of Two-Weight Codes and Its Applications

Abstract

Abstract. It is well-known that there exists a constant-weight [sθ k−1 , k, sq k−1 ]q code for any positive integer s, which is an s-fold simplex code, where θ j = (q j+1 − 1)/(q − 1). This gives an upper bound nq(k, sq) is the minimum length n for which an [n, k, d]q code exists. We construct a two-weight [sθ k−1 + 1, k, sq k−1 ]q code for 1 ≤ s ≤ k − 3, which gives a better upper bound nq(k, sqAs another application, we prove that nq(5, d) = 4 i=0 d/q i for q 4 + 1 ≤ d ≤ q 4 + q for any prime power q.

On the minimum length of linear codes overF5

“…In this paper, we tackle the problem to determine n 5 (5, d) for all d. See [24] for the updated table of n 5 (5, d). The following results are already known for n 5 (k, d), see [1][2][3]5,7,9,11,21,20,25,24,30]. (3) n 5 (4, d) = g 5 (4, d) + 2 for d = 25.…”