A CONSTRUCTION OF TWO-WEIGHT CODES AND ITS APPLICATIONS

Abstract

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