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

Let $\mathcal {X}= \{x_{1},x_{2},\ldots, x_{q}\}$ and let notation="LaTeX">$n(m,q,\ell)$ be the smallest notation="LaTeX">$n$ for which there is a code notation="LaTeX">$C \subseteq \mathcal {X} ^{n}$ of notation="LaTeX">$m$ elements such that every list notation="LaTeX">$w_{1}, w_{2}, \ldots, w_{\ell +1}$ distinct codewords from notation="LaTeX">$C$ , coordinate notation="LaTeX">$j \in [n]$ notation="LaTeX">$\{w_{1}[j], w_{2}[j], +1}[j]\} = {X}$ . We show constant notation="LaTeX">$A>0$ notation="LaTeX">$\epsilon < 1/5$ all large notation="LaTeX">$q$ enough ( notation="LaTeX">$m>q^{5}$ ), we have notation="LaTeX">$n(m,q, \lceil \epsilon q\ln {q}\rceil) \geq \exp {(Aq^{1-5\epsilon })}\log _{2}{m}$ This bound has consequences zero-error list-decoding capacity notation="LaTeX">$q/(q-1)$ channel studied by Elias (1988). Our result implies notation="LaTeX">$A$ $ as above, with list-size {q}$ at most notation="LaTeX">$\exp (-Aq^{1-5\epsilon })$ is, it falls exponentially increases. confirms conjecture Chakraborty et al. (2006).