On the non-existence of perfect and nearly perfect codes

AbstractThe main result of the paper is the proof of the non-existence of a class of completely regular codes in certain distance-regular graphs. Corollaries of this result establish the non-existence of perfect and nearly perfect codes in the infinite families of distance-regular graphs J(2b + 1, b) and J(2b+2,b)

Characterisation of a family of neighbour transitive codes

We consider codes of length $m$ over an alphabet of size $q$ as subsets of the vertex set of the Hamming graph $\Gamma=H(m,q)$. A code for which there exists an automorphism group $X\leq Aut(\Gamma)$ that acts transitively on the code and on its set of neighbours is said to be neighbour transitive, and were introduced by the authors as a group theoretic analogue to the assumption that single errors are equally likely over a noisy channel. Examples of neighbour transitive codes include the Hamming codes, various Golay codes, certain Hadamard codes, the Nordstrom Robinson codes, certain permutation codes and frequency permutation arrays, which have connections with powerline communication, and also completely transitive codes, a subfamily of completely regular codes, which themselves have attracted a lot of interest. It is known that for any neighbour transitive code with minimum distance at least 3 there exists a subgroup of $X$ that has a $2$-transitive action on the alphabet over which the code is defined. Therefore, by Burnside's theorem, this action is of almost simple or affine type. If the action is of almost simple type, we say the code is alphabet almost simple neighbour transitive. In this paper we characterise a family of neighbour transitive codes, in particular, the alphabet almost simple neighbour transitive codes with minimum distance at least $3$, and for which the group $X$ has a non-trivial intersection with the base group of $Aut(\Gamma)$. If $C$ is such a code, we show that, up to equivalence, there exists a subcode $\Delta$ that can be completely described, and that either $C=\Delta$, or $\Delta$ is a neighbour transitive frequency permutation array and $C$ is the disjoint union of $X$-translates of $\Delta$. We also prove that any finite group can be identified in a natural way with a neighbour transitive code.Comment: 30 Page

Arithmetic completely regular codes

In this paper, we explore completely regular codes in the Hamming graphs and related graphs. Experimental evidence suggests that many completely regular codes have the property that the eigenvalues of the code are in arithmetic progression. In order to better understand these "arithmetic completely regular codes", we focus on cartesian products of completely regular codes and products of their corresponding coset graphs in the additive case. Employing earlier results, we are then able to prove a theorem which nearly classifies these codes in the case where the graph admits a completely regular partition into such codes (e.g, the cosets of some additive completely regular code). Connections to the theory of distance-regular graphs are explored and several open questions are posed.Comment: 26 pages, 1 figur

Diameter Perfect Lee Codes

Lee codes have been intensively studied for more than 40 years. Interest in these codes has been triggered by the Golomb-Welch conjecture on the existence of the perfect error-correcting Lee codes. In this paper we deal with the existence and enumeration of diameter perfect Lee codes. As main results we determine all $q$ for which there exists a linear diameter-4 perfect Lee code of word length $n$ over $Z_{q},$ and prove that for each $n\geq 3$ there are uncountable many diameter-4 perfect Lee codes of word length $n$ over $Z.$ This is in a strict contrast with perfect error-correcting Lee codes of word length $n$ over $Z\,$\ as there is a unique such code for $n=3,$ and its is conjectured that this is always the case when $2n+1$ is a prime. We produce diameter perfect Lee codes by an algebraic construction that is based on a group homomorphism. This will allow us to design an efficient algorithm for their decoding. We hope that this construction will turn out to be useful far beyond the scope of this paper