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...

7 We prove for a large class of parameters t and r that a multiset of at most t d−k + r d−k−2 points in PG(d, q) that blocks every k-dimensional subspace at least t times must contain a sum of t subspaces of codimension k. 9 We use our results to identify a class of parameters for linear codes for which the Griesmer bound is not sharp. Our theorem generalizes the non-existence results from Maru...

The class of poset metrics is very large and contains some interesting families of metrics. A family of metrics, based on posets which are formed from disjoint chains which have the same size, is examined. A necessary and sufficient condition, for the existence of perfect single-errorcorrecting codes for such poset metrics, is proved.

We argue that supersymmetric gluodynamics (theory of gluons and gluinos) has a condensate-free phase. Unlike the standard phase, the discrete axial symmetry of the Lagrangian is unbroken in this phase, and the gluino condensate does not develop. Extra unconventional vacua are supersymmetric and are characterized by the presence of (bosonic and fermionic) massless bound states. A set of argument...