Designs and self-dual codes with long shadows

AbstractIn this paper we introduce the notion of s-extremal codes for self-dual binary codes and we relate this notion to the existence of 1-designs or 2-designs in these codes. We extend the classification of codes with long shadows of Elkies (Math. Res. Lett. 2(5) (1995) 643) to codes with minimum distance 6, for which we give partial classification

Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page