The capacity of $C_{5}$ and free sets in $C_{m}^{2}$

Abstract:

In a recent paper, S. K. Stein examined the problem of determining the cardinality, $\tau (C_m^k)$, of the largest subset $S$ of the direct product $C_m^k$ of $k$ copies of ${C_m}$ such that distinct sums of elements of $S$ yield distinct elements of $C_m^k$. In this paper we show that ${\tau ^* }({C_5}) = {\lim _{k \to \infty }}(\tau (C_5^k)/k) = 2$, answering a question raised by Stein. We also produce an infinite set of $m$’s such that $\tau (C_m^2) > 2[{\log _2}m]$.