A new series of large sets of subspace designs over the binary field

Titelangaben

Kiermaier, Michael ; Laue, Reinhard ; Wassermann, Alfred:
A new series of large sets of subspace designs over the binary field.
In: Designs, Codes and Cryptography. Bd. 86 (2018) Heft 2 . - S. 251-268.
ISSN 1573-7586
DOI: https://doi.org/10.1007/s10623-017-0349-1

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Abstract

In this article, we show the existence of large sets LS_2[3](2,k,v) for infinitely many values of k and v. The exact condition is v ≥ 8 and 0 ≤ k ≤ v such that for the remainders v' and k' of v and k modulo 6 we have 2 ≤ v' ≤ k' ≤ 5.

The proof is constructive and consists of two parts. First, we give a computer construction for an LS_2[3](2,4,8), which is a partition of the set of all 4-dimensional subspaces of an 8-dimensional vector space over the binary field into three disjoint 2-(8, 4, 217)_2 subspace designs. Together with the already known LS_2[3](2,3,8), the application of a recursion method based on a decomposition of the Graßmannian into joins yields a construction for the claimed large sets.