Heavy weight codes

Motivated by certain recent problems in asynchronous communication, we introduce and study B(n,d,w), defined as the maximum number of length n binary sequences with minimum distance d, and such that each sequence has weight at least w. Specifically, we investigate the asymptotic exponential growth rate of B(n, d,w) with respect to n and with fixed ratios δ = d/n and ω = w/n. For ω ∈ [0,1/2], this growth rate function b(δ,ω) is shown to be equal to a(δ), the asymptotic exponential growth rate of A(n, d) - the maximum number of length n binary sequences with minimum distance d. For ω ∈ (1/2,1], we show that b(δ,ω) ≤ a(δ,ω) + f(ω), where a(δ, ω) denotes the asymptotic exponential growth rate of A(n, d, w), the maximum number of length n binary sequences with minimum distance d and constant weight w, and where f(w) is a certain function that satisfies 0 < f(ω) < 0.088 and lim _ω→1 f(ω) = lim_ω→1/2 f(ω) = 0. Based on numerical evidence, we conjecture that b(δ, ω) is actually equal to a(δ,ω) for ω ∈ (1/2,1]. Finally, lower bounds on B(n,d,w) are obtained via explicit code constructions.