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

Abstract

A 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| ⩽ 2 n − 1 . We prove that there are no msf sets S in V( n) with 5 × 2 n − 4 < | S| < 2 n − 1 . (This bound is sharp at both ends.) Further, we construct msf sets S in V( n), n ⩾ 4, with | S| = 2 n − s + 2 s + t − 3 × 2 t 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.