Perfect codes in direct products of cycles - a complete characterization
Janez Žerovnik
Advances in applied mathematics Volume 41, Number 2, pp. 197-205, 2008. ISSN 0196-8858

Abstract

Let $G = \times^n_{i=1}C_{\ell_i}$ be a direct product of cycles. It is known that for any $r \le 1$, and any $n \le 2$, each connected component of $G$ contains a so-called canonical $r$-perfect code provided that each $\ell_i$ is a multiple of $r^n + (r+1)^n$. Here we prove that up to a reasonably defined equivalence, these are the only perfect codes that exist.

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