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 limω→1 f(ω) = limω→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.