OFFSET
0,3
COMMENTS
It is important to distinguish between "extremal" (meaning having the highest possible minimal distance permitted by Gleason's theorem) and "optimal" (meaning having the highest minimal distance that can actually be achieved). This sequence enumerates optimal codes. Extremal codes do not exist when n is sufficiently large. For lengths up to at least 64, "extremal" and "optimal" coincide.
"There are 94343 inequivalent doubly even self-dual codes of length 40, 16470 of which are extremal." [Betsumiya et al.] - Jonathan Vos Post, Aug 06 2012
LINKS
Koichi Betsumiya, Masaaki Harada and Akihiro Munemasa, A Complete Classification of Doubly Even Self-Dual Codes of Length 40, arXiv:1104.3727v3 [math.CO], v3, Aug 02, 2012. - From Jonathan Vos Post, Aug 06 2012
J. H. Conway and V. S. Pless, On the enumeration of self-dual codes, J. Comb. Theory, A28 (1980), 26-53. [DOI] MR0558873
J. H. Conway, V. Pless and N. J. A. Sloane, The Binary Self-Dual Codes of Length Up to 32: A Revised Enumeration, J. Comb. Theory, A60 (1992), 183-195 (Abstract, pdf, ps, Table A, Table D).
S. K. Houghten, C. W. H. Lam, L. H. Thiel and J. A. Parker, The extended quadratic residue code is the only (48,24,12) self-dual doubly-even code, IEEE Trans. Inform. Theory, 49 (2003), 53-59.
W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic. 11 (2005), 451-490. [DOI]
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Aug 08 2012
EXTENSIONS
a(6) = 1 (due to Houghten et al.) from Akihiro Munemasa, Aug 08 2012
STATUS
approved