Classification of Extremal and $s$-Extremal Binary Self-Dual Codes of Length 38

In this paper we classify all extremal and s-extremal binary self-dual codes of length 38. There are exactly 2744 extremal self-dual codes, two s-extremal codes, and 1730 s-extremal codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of -extremal codes permits to achieve the classification of all -extremal codes with .