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

Let m, q, ℓ be positive integers such that m ≥ℓ≥q. A family \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal H}$\end{document} of functions from [m] to [q] is said to be an (m,q,ℓ)-family if for every subset S of [m] with ℓ elements, there is an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h \in {\cal H}$\end{document} such that h(S) = [q]. Let, N(m,q,ℓ) be the size of the smallest (m,q,ℓ)-family. We show that for all q, ℓ≤1.58q and all sufficiently large m, we have N(m,q,ℓ) = exp(Ω(q)) log m. 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N(m,q,q+1) = \exp(\Omega(q)) \log m$\end{document}. We conjecture that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N(m,q,\ell) = \exp(\Omega(q)) \log m$\end{document} if ℓ= O(q). A standard probabilistic construction shows that for all q, ℓ≥q and all sufficiently large m, N(m,q,ℓ) = exp(O(q)) log m. 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 list-decoding 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.