Constructions of binary codes with two distances

Abstract

In this paper, we consider the problem of determining the exact value of A2(n,{d1,d2}) defined as the maximal cardinality of a binary code of length n with two possible distances d1 and d2. We prove that if d2>2d1, one has A2(n,{d1,d2})≤n+1. A similar bound holds for codes with d1≢d2(mod2):A2(n,{d1,d2})≤{n+1 for d1 even,n+2 for d1 odd. Furthermore, we settle two conjectures left open by earlier authors that imply the following exact values:A2(n,{2,d})={(n2)+1 for d=4 and n≥6,n for 4<d<n−1,n+1 for d=n−1, and provide combinatorial constructions that improve on the lower bounds on A2(n,{d1,d2}) known so far. Finally, we prove the general upper boundA2(n,{d1,d2})≤(n+1)(n+2)2.