Optimal Combinatorial Neural Codes with Matched Metric $\delta_{r}$: Characterization and Constructions

TL;DR

This paper characterizes optimal combinatorial neural codes with a matched asymmetric metric, showing they derive from classical optimal codes, and introduces new constructions using bent functions for flexible parameters.

Contribution

It establishes the equivalence between optimal CN codes under the matched metric and classical optimal codes, and provides novel constructions using bent functions.

Findings

Optimal CN codes correspond to classical optimal codes when the parameter r is close to 1.

Provides constructions of CN codes with flexible parameters using bent functions.

Shows bounds for CN codes are achieved by codes that meet classical bounds.

Abstract

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.