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%I #11 Jan 07 2019 05:55:16
%S 2,4,10,28,36,66,144,192,340,570,1200,1656,3456,5616,9072,10752,22176
%N Number of divisors for the automorphism group size having the largest number of divisors for a binary self-dual code of length 2n.
%C A code is usually represented by a generating matrix. The row space of the generating matrix is the code itself.
%C Self-dual codes are codes such all codewords are pairwise orthogonal to each other.
%C Two codes are called permutation equivalent if one code can be obtained by permuting the coordinates (columns) of the other code.
%C The automorphism group of a code is the set of permutations of the coordinates (columns) that result in the same identical code.
%C 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).
%C 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.
%C 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.
%C The number of divisors a(n) does count 1 and the number itself.
%C In general the automorphism group size with the largest number of divisors is not unique.
%C 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.
%H W. Cary Huffman and Vera Pless, <a href="https://doi.org/10.1017/CBO9780511807077">Fundamentals of Error Correcting Codes</a>, Cambridge University Press, 2003, Pages 338-393.
%e 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.
%Y Cf. Self-Dual Codes A028362, A003179, A106162, A028363, A106163, A269455, A120373.
%Y Cf. Self-Dual Code Automorphism Groups A322299, A322339.
%K nonn,more
%O 1,1
%A _Nathan J. Russell_, Dec 12 2018
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