A215219 Number of (indecomposable or decomposable) Type II binary self-dual codes of length 8n with the highest minimal distance.

1, 1, 2, 1, 5, 16470, 1

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

Table of n, a(n) for n=0..6.

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.

V. S. Pless, The children of the (32,16) doubly even codes, IEEE Trans. Inform. Theory, 24 (1978), 738-746. [DOI] MR0514353

E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).

Crossrefs

Cf. A003178, A003179, A106162-A106167.

Sequence in context: A323500 A277471 A110863 * A335931 A091615 A213279

Adjacent sequences: A215216 A215217 A215218 * A215220 A215221 A215222

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