Optimal Combinatorial Neural Codes with Matched Metric ${\delta }_r$: Characterization and Constructions

Based on the theoretical neuroscience, G. Cotardo and A. Ravagnavi in \cite{CR} introduced a kind of asymmetric binary codes called combinatorial neural codes (CN codes for short), with a “matched metric” ${\delta }_r$ called asymmetric discrepancy, instead of the Hamming distance $d_H$ for usual error-correcting codes. They also presented the Hamming, Singleton and Plotkin bounds for CN codes with respect to ${\delta }_r$ and asked how to construct the CN codes $\text{cC}$ with large size $\mathrm{\mid }\text{cC}\mathrm{\mid }$ and ${\delta }_r(\text{cC})\mathrm{.}$ In this paper we firstly show that a binary code $\text{cC}$ reaches one of the above bounds for ${\delta }_r(\text{cC})$ if and only if $\text{cC}$ reaches the corresponding bounds for $d_H$ and $r$ is sufficiently closed to 1. This means that all optimal CN codes come from the usual optimal codes. %(perfect codes, MDS codes or the codes meet the usual Plotkin bound). Secondly we present several constructions of CN codes with nice and flexible parameters $(n,K,{\delta }_r(\text{cC}))$ by using bent functions.