Abstract: We study the class of codes defined by the row space of the minimum distance relation matrix of $t$th and $l$th layers of Hamming graph $H(m,q)$. By concatenating such matrices we obtain many distance-optimal codes of length up to $128$. For arbitrary $q$, $t$, $n$, $k$ we prove an analogue of a well-known Wilson rank formula \cite{W} and find the dimensions of the codes in this class. For $t=l-1$, the codes are locally recoverable and include the codes from work of \cite{WZ} for $q=2$. We show that the codes with $q=2$ are optimal locally-recoverable codes in our class.

Cite: Mogilnykh I. , Danilko V.
Codes from layers of Hamming graphs
XIX International Symposium on Problems of Redundancy in Information and Control Systems 05-07 Nov 2025