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 $\cC$ with large size $|\cC|$ and $\delta_{r}(\cC).$ In this paper we firstly show that a binary code $\cC$ reaches one of the above bounds for $\delta_{r}(\cC)$ if and only if $\cC$ 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 $(n,K, \delta_r(\cC))$ by using bent functions.