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In this paper we study a special type of linear codes, called skew cyclic codes, in the most general case. This set of codes is a generalization of cyclic codes but constructed using a non-commutative ring called the skew polynomial ring. In previous works these codes have been studied with certain restrictions on their length. This work examines their structure for an arbitrary length without any restriction. Our results show that these codes are equivalent to either cyclic 1 codes or quasi-cyclic codes, hence establish strong connections with well-known classes of codes.

A classification of all four-circulant extremal codes of length 32 over F 2 + uF 2 is done by using four-circulant binary self-dual codes of length 32 of minimum weights 6 and 8. As Gray images of these codes, a substantial number of extremal binary self-dual codes of length 64 are obtained. In particular a new code with β = 80 in W 64,2 is found. Then applying an extension method from the literature to extremal self-dual codes of length 64, we have found many extremal binary self-dual codes of length 66. Among those, five of them are new codes in the sense that codes with these weight enumerators are constructed for the first time. These codes have the values β = 1, 30, 34, 84, 94 in W 66,1 .

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