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 ds 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 eg [l]), the words of fixed weight k of an extremal code C form a 5-2. r-block design, ie for any set a of positions of C with (a (= 5-2s the cardinality of C,(a):{c EC (ICI= k, cc} independent the choice of u. fact] & (a)] depends only on and k. found additional property of the in the that kd is the weight C: