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- W/Prof Cheryl Praeger
- Dr Neil Gillespie
- A/Prof Michael Giudici

Jan 2013

Symmetry of Codes in Hamming Graphs

Error correcting coding theory is the study of techniques allowing for preservation of data during transmission or storage, where noise from outside sources may cause corruption. This is done by adding redundancy to a message in a systematic way so that any errors stand out and the required information is able to be recovered.

It is expected that a code which allows efficient correction of many errors will have a high degree of symmetry. Indeed combinatorial symmetry of codes has been studied via completely regular codes and group theoretic symmetry via complete transitivity.

It is the goal of my thesis to classify certain classes in the hierarchy of s-neighbour transitive codes. An s-neighbour transitive code is a code such that there exists a subgroup of the automorphism group of the Hamming graph which is transitive on the set of r-neighbours of the code, for all r less than s.

The other half of my thesis looks at elusive codes, which are codes where the neighbour set has more symmetry than that of the code itself.

Finding efficient codes is important for reliable transmission of information. Additionally, we hope that this research brings its own insights to the different areas of Mathematics it links together. These include: finite group theory, graph theory, design theory and finite geometry.

- Australian Postgraduate Award and UWA Top-up Scholarship.