A322309 Largest automorphism group size for a binary self-dual code of length 2n.

2, 8, 48, 1344, 3840, 46080, 645120, 10321920, 185794560, 3715891200, 81749606400, 1961990553600, 51011754393600, 1428329123020800, 42849873690624000, 1371195958099968000, 46620662575398912000

Offset: 1

Author

Nathan J. Russell, Dec 03 2018

Keywords

• nonn

Comments

A code is usually represented by a generating matrix. The row space of the generating matrix is the code itself.

Self-dual codes are codes such that all codewords of the code are pairwise orthogonal to each other.

Two codes are called permutation equivalent if one code can be obtained by permuting the coordinates (columns) of the other code.

The automorphism group of a code is the set of permutations of the coordinates (columns) that result in the same identical code.

The values in the sequence are not calculated upper bounds. For each n there exists a binary self-dual code of length 2n with an automorphism group of size a(n).

Binary self-dual codes have been classified (accounted for) up to a certain length. The classification process requires the automorphism group size be known for each code. There is a mass formula to calculate the number of distinct binary self-dual codes of a given length. The automorphism group size allows researchers to calculate the number of codes that are permutationally equivalent to a code. Each new binary self-dual code C of length m that is discovered will account for m!/aut(C) codes in the total number calculated by the mass formula. Aut(C) represents the automorphism size of the code C.

Examples

			The largest automorphism group size a binary self-dual code of length 2*16=32 is a(16) = 1371195958099968000.
		

Links

• W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Pages 338-393.

Crossrefs

Cf. Self-Dual Codes A028362, A003179, A106162, A028363, A106163.