A322299 Number of distinct automorphism group sizes for binary self-dual codes of length 2n.

1, 1, 1, 2, 2, 3, 4, 7, 9, 16, 24, 48, 85, 149, 245, 388

Offset: 1

Author

Nathan J. Russell, Dec 02 2018

Keywords

• nonn
• more

Comments

Codes are vector spaces with a metric defined on them. Specifically, the metric is the hamming distance between two vectors. Vectors of a code are called codewords.

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 all codewords 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.

Examples

			There are a(16) = 388 distinct sizes for the automorphism groups of the binary self-dual codes of length 16.  In general, two automorphism  groups with the same size are not necessarily isomorphic.
		

Links

• W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, 2003, pp. 7, 252-330, 338-393.

Crossrefs

Cf. self-dual codes A028362, A003179, A106162, A028363, A106163.