Sum-free sets in vector spaces over GF(2)

Abstract

AbstractA subset S of an abelian group G is said to be sum-free if whenever a, b ∈ S, then a + b ∉ S. A maximal sum-free (msf) set S in G is a sum-free set which is not properly contained in another sum-free subset of G. We consider only the case where G is the vector space (V(n) of dimension n over GF(2). We are concerned with the problem of determining all msf sets in V(n). It is well known that if S is a msf set then |S| ⩽ 2n − 1. We prove that there are no msf sets S in V(n) with 5 × 2n − 4 < |S| < 2n − 1. (This bound is sharp at both ends.) Further, we construct msf sets S in V(n), n ⩾ 4, with |S| = 2n − s + 2s + t − 3 × 2t for 0 ⩽ t ⩽ n − 4 and 2 ⩽ s ⩽ [(n − t)2]. These methods suffice to construct msf sets of all possible cardinalities for n ⩽ 6. We also present some of the results of our computer searches for msf sets in V(n). Up to equivalence we found all msf-sets for n ⩽ 6. For n > 6 our searches used random sampling and, in this case, we find many more msf sets than our present methods of construction can account for