On Self-Dual Codes
by
Vera Pless
University of Illinois at Chicago

Self-dual codes are an interesting class of codes which include many very good codes. These codes are also related to interesting designs, interesting groups and lattices. The extended Golay codes over GF(2) and GF(3) are self-dual as is the [8, 4, 4] binary Hamming code and the [6, 3, 4] Hexacode over GF(4). The weight enumerators of these codes are given by the Gleason polynomials which also provide an upper bound on their minimum weights. Codes which meet this bound are called extremal and vectors of a fixed weight in them contain t-designs where t can be 5, 3 or 1 ( depending on the code's length (mod 8)). In order to find the extremal codes, all self-dual codes of a particular length have been classified for many modest lengths. This has been possible because there are formulas for the number of self-dual codes of a fixed length. We give a short history of these classifications. We also describe the instances where it was possible to classify only the extremal self-dual codes . If a binary code has the same weight distribution as its dual code, it is called formally self-dual (f.s.d.). These codes include the self-dual codes and when they are not self-dual, their weight enumerators are combinations of Gleason polynomials . Extremal f.s.d. codes exist which have higher minimum weights than self-dual codes with the same parameters. We describe recent work (as yet unpublished) which has classified the extremal even f.s.d codes through length 30.

Date received: January 27, 1999