New uniqueness proofs for the (5, 8, 24), (5, 6, 12) and related Steiner systems

Mutually Disjoint Steiner Systems S(5, 8, 24) and 5-(24,12,48) Designs

We demonstrate that there are at least 50 mutually disjoint Steiner systems S(5, 8, 24) and there are at least 35 mutually disjoint 5-(24, 12, 48) designs. The latter result provides the existence of a simple 5-(24, 12, 6m) design for m = 24, 32, 40, 48, 56, 64, 72, 80, 112, 120, 128, 136, 144, 152, 160, 168, 200, 208, 216, 224, 232, 240, 248 and 256.

On the Construction of the Steiner System S(5, 8, 24)

Let VI = GF(211), regarded as an 11-dimensional vector space over GF(2), and form V = V, @ (71,). Define vi = 01~ + v, (i = O,..., 22). Since f(X) is irreducible over GF(2), {v. ,..., v 1o, v,} is a basis for V, and so for 11 < i < 22, we can write vi = CzEr, v,, for certain sets Xi _C X = (0 ,..., 10, w}. Moreover, the sets X, can very quickly be computed explicitly by calculating the powers o...

5-Chromatic Steiner Triple Systems

We show that, up to an automorphism, there is a unique independent set in PG(5,2) that meet every hyperplane in 4 points or more. Using this result, we show that PG(5,2) is a 5-chromatic STS. Moreover, we construct a 5-chromatic STS(v) for every admissible v ≥ 127.

Combinatorial 6/5-Approximation of Steiner Tree

The Steiner tree problem is one of the most fundamental and intensively studied NP-hard problems. There have been improved approximation algorithms for this problem starting from the naive 2-approximation algorithm and resulting in the best combinatorial approximation with ratio 1+ ln(3) 2 + ǫ < 1.55 [33]. An LP-based approximation algorithm, and in particular an integrality gap smaller than 2 ...

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