Zero Error List-Decoding Capacity of the q/(q-1) Channel

Let m, q, ` be positive integers such that m ≥ ` ≥ q. A family H of functions from [m] to [q] is said to be an (m, q, `)-family if for every subset S of [m] with ` elements, there is an h ∈ H such that h(S) = [q]. Let, N(m, q, `) be the size of the smallest (m, q, `)-family. We show that for all q, ` ≤ 1.58q and all sufficiently large m, we have N(m, q, `) = exp(Ω(q)) logm. Special cases of this follow from results shown earlier in the context of perfect hashing: a theorem of Fredman & Komlós (1984) implies that N(m, q, q) = exp(Ω(q)) logm, and a theorem of Körner (1986) shows that N(m, q, q+ 1) = exp(Ω(q)) logm. We conjecture that N(m, q, `) = exp(Ω(q)) logm if ` = O(q). A standard probabilistic construction shows that for all q, ` ≥ q and all sufficiently large m, N(m, q, `) = exp(O(q)) logm. Our motivation for studying this problem arises from its close connection to a problem in coding theory, namely, the problem of determining the zero error listdecoding capacity for a certain channel studied by Elias [IEEE Transactions on Information Theory, Vol. 34, No. 5, 1070–1074, 1988]. Our result implies that for the so called q/(q− 1) channel, the capacity is exponentially small in q, even if the list size is allowed to be as big as 1.58q. The earlier results of Fredman & Komlós and Körner cited above imply that the capacity is exponentially small if the list size is at most q + 1.