Neighbour transitivity on codes in hamming graphs

Error correcting codes are used to maintain the integrity of data across noisy communication channels and storage systems. The objective of an error correcting code is to encode a message by adding a certain amount of redundancy so that the likelihood of recovering the original message is increased. Messages are sent across a communication channel as strings from a certain alphabet; each string has additional entries added to form a codeword. An assumption frequently made in decoding procedures for error correcting codes is that the probability of a transmission error in some entry is independent both of the position where the error occurs and also of the letter of the alphabet occurring in that position. In other words, the probability of each error occurring is equally likely. We study a group theoretic analogue of this assumption. The codes we consider are subsets of ordered m-tuples from a xed alphabet Q of size q, and as such we interpret codes as subsets of vertices of the Hamming graph. When transmitting a code, C, an error occurs when a single entry in a codeword is changed, and so, in the Hamming graph, errors are adjacent vertices to codewords. We de ne a neighbour of a codeword to be an adjacent vertex of that codeword that is not itself a codeword. We say a code C satis es the neighbour transitivity property if there exists an automorphism group of the Hamming graph that acts transitively on the set of neighbours of C, which is a group theoretic analogue of the assumption that each possible error is equally likely. This thesis is a treatise on codes in Hamming graphs for which the neighbour transitivity property holds. We give constructions of new codes, which have the neighbour transitive property and good error correcting capabilities. Additionally, using permutation group theory we characterise and classify various families of codes with the neighbour transitive property.