The problem of determining Aq(n, d), the maximum cardinality of a q-ary code of length n with minimum distance at least d, is considered in some cases where corresponding MDS codes do not exist. Slight improvements of the Singleton bound are given, including Aq(q + 2, q) q3 − 3 if q is odd, A5(7, 5) 53 − 4 and A16(18, 15) 184 − 4. © 2004 Elsevier B.V. All rights reserved.

In this paper, we study the bound on three kinds of hash family using the Singleton bound. To ε−U(N ;n,m) hash family, in the caes of n > m > 1 and 1 ≥ ε ≥ ε1(n,m), we get that the new bound is better. To ε − △U(N ;n,m) hash family , in the case of n > m > 1 and 1 ≥ ε ≥ ε3(n,m), the new bound is better. To ε − SU(N ;n,m) hash family, in the case of n > 2m > 2 and 1 ≥ ε ≥ ε4(n,m), we get that th...