Necessary conditions for the existence of 3-designs over finite fields\n with nontrivial automorphism groups

2015/09/30 by Maarten De Boeck, De Boeck, Maarten, Anamari Nakić +1
Engineering · Computer Science · #graph theory and CDMA systems #Cooperative Communication and Network Coding #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.1509.09158

Abstract

A q-design with parameters t-(v,k,lambda_t)_q is a pair (V, B) of the\nv-dimensional vector space V over GF(q) and a collection B of k-dimensional\nsubspaces of V, such that each t-dimensional subspace of V is contained in\nprecisely lambda_t members of B. In this paper we give new general necessary\nconditions on the existence of designs over finite fields with parameters 3-(v,\nk , lambda₃)_q with a prescribed automorphism group. These necessary\nconditions are based on a tactical decomposition of such a design over a finite\nfield and are given in the form of equations for the coefficients of tactical\ndecomposition matrices. In particular, they represent necessary conditions on\nthe existence of q-analogues of Steiner systems admitting a prescribed\nautomorphism group.\n

Related

• Tactical decompositions of designs over finite fields
• q-Analogs of t-Wise Balanced Designs from Borel Subgroups
• Existence of q-Analogs of Steiner Systems
• Large Sets of t-Designs over Finite Fields
• On the automorphisms of designs constructed from finite simple groups