A321946 Number of divisors for the automorphism group size having the largest number of divisors for a binary self-dual code of length 2n.

2, 4, 10, 28, 36, 66, 144, 192, 340, 570, 1200, 1656, 3456, 5616, 9072, 10752, 22176

Offset: 1

Author

Nathan J. Russell, Dec 12 2018

Keywords

• nonn
• more

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 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.

The values in the sequence are not calculated lower 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. Sequence A028362gives this count. 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. Sequence A003179 gives number of binary self-dual codes up to permutation equivalence.

The values in the sequence are not calculated by a formula or algorithm. They are the result of calculating the number of divisors for every automorphism group of every binary self-dual code.

The number of divisors a(n) does count 1 and the number itself.

In general the automorphism group size with the largest number of divisors is not unique.

In general the automorphism group size with the largest number of divisors is not the largest group automorphism group size for a given binary self-dual code length.

Examples

			There is one binary self-dual code of length 2*14=28 having an automorphism group size of 1428329123020800.  This number has a(14) = 5616 divisors (including 1 and 1428329123020800).  The automorphism size of 1428329123020800 represents the automorphism size with the largest number of divisors for a binary self-dual code of length 2*14=28.
		

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, A269455, A120373.

Cf. Self-Dual Code Automorphism Groups A322299, A322339.