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Extremal self-dual codes

In the present thesis we consider extremal self-dual codes. We mainly concentrate on Type II codes (binary doubly-even codes), which may theoretically exist for lengths n = 8k ≤ 3928. It is noteworthy that extremal Type II codes have been actually constructed only for 13 lengths, 136 being the largest. Over the last decades the study of extremal codes became inseparable from the study of their automorphisms. For example, one of the few methods to construct a new “good” code C is to assume that C is invariant under a certain automorphism and to use the restrictions imposed by this fact. Making use of the non-trivial automorphism groups we classify extremal extended quadratic residue codes and quadratic double circulant codes. The two families provide examples of extremal codes for 9 of the 13 lengths, for which extremal codes are constructed. Another of our main results is the classification of extremal Type II codes C with 2-transitive automorphism groups. We show that C is either a quadratic residue code, a Reed-Muller code, or a putative code of length 1024. Similar classification results are also obtained in case of ternary and quaternary extremal codes. In the thesis we also provide a new easy-to-handle criterion to determine possible cycle structures of the automorphisms of binary extremal codes. Using this result we show that a binary extremal code of length n with an automorphism of prime order p > 2 is equivalent to an extended cyclic code. We classify extremal extended cyclic codes of length n ≤ 1000. Moreover, we prove that for all but 11 values of n > 1000 no extremal extended cyclic code can exist. In the final part of the thesis we consider singly-even and doubly-even binary extremal codes in terms of their decoding performance. We are able to determine the best singly-even codes and prove that these always perform better than doubly-even codes with the same parameters.