On self-dual doubly-even extremal codes

Let C be a binary linear self-dual doubly-even code of length n and minimal weight d. Such codes exist only if 12 = 0 (mod 8). We put II = 24r + 8s, s = 0, 1, 2. It follows from the work of Gleason [2] and of Mallows and Sloane [6] that d s 4r + 4. C is called extremal if d = 4r + 4. In the following, an extremal code means a binary linear self-dual doubly-even extremal code. We use the set-theoretical notation: Let I be the set of positions of a code. Then a word in E: considered as a mapping from Z to [F2 will be identified with its support. Hence IFi will be identified with the system of subsets of 1. By the Theorem of Assmus-Mattson (see e.g. [l]), the words of fixed weight k of an extremal code C form a 5 2.r-block design, i.e. for any set a of positions of C with (a( = 5 2s the cardinality of C,(a) : {c E C ( ICI = k, c c} independent the choice of u. fact ]&(a)] depends only on and k. found additional property of the