Completely regular codes with different parameters giving the same distance-regular coset graphs

We construct several classes of completely regular codes with different parameters, but identical intersection array. Given a prime power q and any two natural numbers a,b, we construct completely transitive codes over different fields with covering radius ρ=min{a,b}ρ=min{a,b} and identical intersection array, specifically, one code over F_q^r for each divisor r of a or b. As a corollary, for any prime power qq, we show that distance regular bilinear forms graphs can be obtained as coset graphs from several completely regular codes with different parameters

About non equivalent completely regular codes with identical intersection array

We obtain several classes of completely regular codes with different parameters, but identical intersection array. Given a prime power q and any two natural numbers a, b, we construct completely transitive codes over different fields with covering radius ρ = min{a, b} and identical intersection array, specifically, we construct one code over F_{q^r} for each divisor r of a or b. As a corollary, for any prime power q, we show that distance regular bilinear forms graphs can be obtained as coset graphs from several completely regular codes with different parameters

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