Abstract
Let X be a subgroup of the full automorphism group of the Hamming graph H ( m , q ) , and C a subset of the vertices of the Hamming graph. We say that C is an ( X , 2) -neighbour-transitive code if X is transitive on C , as well as C 1 and C 2 , the sets of vertices which are distance 1 and 2 from the code. It has been shown that, given an ( X , 2) -neighbour-transitive code C , there exists a subgroup of X with a 2 -transitive action on the alphabet; this action is thus almost-simple or affine. This paper completes the classification of ( X , 2) -neighbour-transitive codes, with minimum distance at least 5 , where the subgroup of X stabilising some entry has an almost-simple action on the alphabet in the stabilised entry. The main result of this paper states that the class of ( X , 2) neighbour-transitive codes with an almost-simple action on the alphabet and minimum distance at least 3 consists of one infinite family of well known codes.
Highlights
Ever since Shannon’s 1948 paper [18, 19] there has been a great deal of interest around families of error-correcting codes with a high degree of symmetry. The rationale behind this interest is that codes with symmetry should have good error correcting properties
In an effort to find further classes of efficient codes, Delsarte [4] introduced completely regular codes, a more general class of codes that posses a high degree of combinatorial symmetry
Transitive codes are a class of codes with a high degree of algebraic symmetry and are a subset of completely regular codes. As such a classification of completely transitive codes would be interesting from the point of view of classifying completely regular codes
Summary
Ever since Shannon’s 1948 paper [18, 19] there has been a great deal of interest around families of error-correcting codes with a high degree of symmetry. Transitive codes (first defined in [20], with a generalisation studied in [10]) are a class of codes with a high degree of algebraic symmetry and are a subset of completely regular codes. As such a classification of completely transitive codes would be interesting from the point of view of classifying completely regular codes. The results here are of interest from the point of view of perfect codes over an alphabet of non-prime-power size, since in this case a code cannot be alphabet-affine (and not entry-faithful, by [7]), but may be alphabet-almost-simple.
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