Optimal combinatorial neural codes via symmetric designs

Combinatorial neural (CN) codes are binary codes introduced firstly by Curto et al. for asymmetric channel, and then are further studied by Cotardo and Ravagnani under the metric \(\delta _r\) (called asymmetric discrepancy) which measures the differentiation of codewords in CN codes. When \(r>1\), CN codes are different from the usual error-correcting codes in symmetric channel (\(r=1\)). In this paper, we focus on the optimality of some CN codes with \(r>1\). An upper bound for the size of CN codes with \(\delta _r=r+1\) is deduced, by discussing the relationship between such CN codes and error-detecting codes for asymmetric channels, which is shown to be tight in this case. We also propose an improved Plotkin bound for CN codes. Notably, by applying symmetric designs related with Hadamard matrices, we not only generalize one former construction of optimal CN codes by bent functions obtained by Zhang et al. (IEEE Trans Inf Theory 69:5440–5448, 2023), but also obtain seven classes of new optimal CN codes meeting the improved Plotkin bound.