Existence of New Extremal Doubly-Even Codes and Extremal Singly-Even Codes

Recently the author and Kimura have considered a construction of doubly-even codes from a given doubly-even code. In this note, we show that the restricutoion of doubly-even can be removed in the above construction. As an application, at least 137 inequivalent extremal doubly-even 56,28,12] codes and at least 1000 inequivalent extremal doubly-even 40,20,8] codes are constructed from known self-dual codes. The existence of new extremal singly-even codes is also described.