### Neil Gillespie

#### Thesis

Neighbour Transitive Codes in Hamming Graphs

#### Summary

Error correcting codes are used to protect data against errors caused by interference that occurs during the transmission through a noisy communication channel, or for reliable storage of data in media such as CDs and DVDs.

An underlying principle that is often assumed in error- correcting coding theory is that each error has an equal probability of occurring during transmission. This principle is fundamental to minimum distance decoding, which is one of the main methods used for decoding. This thesis looks at this principle from a group theoretic perspective. We consider a \emph{code} to be a subset of the vertex set of a \emph{Hamming graph}. Moreover, in this setting a \emph{neighbour} of the code is a vertex which differs in exactly one entry from some codeword, that is, exactly the result of one error occuring. This thesis examines codes with the property that some group of automorphisms acts transitively on the \emph{set of neighbours} of the code. This is a strong way of saying that each single error is equivalent, or equally likely.

#### Why my research is important

Many results on coding theory assume the principle that each error has an equal probability of occurring. Therefore an investigation into neighbour transitive codes would lead to a better understanding of constructions of very symmetrical codes for which this principle is evident.