Generalized Steiner Systems

Optimal constant weight codes over Zk and generalized designs

We consider optimal constant weight codes over arbitrary alphabets. Some of these codes are derived from good codes over the same alphabet, and some of these codes are derived from block design. Generalizations of Steiner systems play an important role in this context. We give several construction methods for these generalizations. An interesting class of codes are those which form generalized ...

Generalized Steiner systems GS5(2, 5, v, 5)

Constructions for generalized Steiner systems GS (3, 4, v , 2)

Generalized Steiner systems GS (3, 4, v, 2) were first discussed by Etzion and used to construct optimal constant weight codes over an alphabet of size three with minimum Hamming distance three, in which each codeword has length v and weight four. Not much is known for GS (3, 4, v, 2)s except for a recursive construction and two small designs for v = 8, 10 given by Etzion. In this paper, more s...

Vasiliev codes of length n = 2 m and Steiner systems S ( n , 4 , 3 )

We study the extended binary perfect nonlinear Vasiliev codes of length n = 2m and Steiner systems S(n, 4, 3) of rank n − m over F2. The generalized concatenation (GC) construction of Vasiliev codes induces a variant of the doubling construction of Steiner systems S(n, 4, 3) of rank n − m over F2. We prove that any Steiner system S(n = 2m, 4, 3) of rank n − m is obtained by such doubling constr...

On the Number of Blocks in a Generalized Steiner System

We consider t-designs with *=1 (generalized Steiner systems) for which the block size is not necessarily constant. An inequality for the number of blocks is derived. For t=2, this inequality is the well known De Bruijn Erdo s inequality. For t>2 it has the same order of magnitude as the Wilson Petrenjuk inequality for Steiner systems with constant block size. The point of this note is that the ...