

A322339


Smallest automorphism group size for a binary selfdual code of length 2n.


4



2, 8, 48, 384, 2688, 10752, 46080, 73728, 82944, 82944, 36864, 12288, 3072, 384, 30, 2, 1
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OFFSET

1,1


COMMENTS

A code is usually represented by a generating matrix. The row space of the generating matrix is the code itself.
Selfdual 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 selfdual code of length 2n with an automorphism group of size a(n).
Binary selfdual 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 selfdual codes of a given length. Sequence A028362 gives this count. The automorphism group size allows researchers to calculate the number of codes that are permutationally equivalent to a code. Each new binary selfdual 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 selfdual codes up to permutation equivalence.
There is a notable open problem in coding theory regarding binary selfdual codes. Does there exist a type II binary selfdual code of length 72 with minimum weight 16? The founder of OEIS N. J. A. Sloane posed the question in 1973. The question has been posed in several coding theory textbooks since 1973. There are even some rewards regarding the existence and nonexistence of the code. Some of the major work involved with researching the existence of the code has involved calculating possibilities for the automorphism group of the (72, 36, 16) type II binary selfdual code. The weight distribution for the code is listed as the finite sequence A120373. The current research demonstrates that the size of the automorphism group for this code is relatively small, perhaps even trivial with size 1. This sequence shows that as the length of a binary selfdual code grows the minimum size of the automorphism group grows up to a point, namely length 18. It would appear that a binary selfdual code of length 72 would no chance at having a small automorphism group size. However, after length 18 the minimum possible automorphism size stops increasing and starts declining all the way down to trivial a(17) = 1 for length 2*17=34. This demonstrates that a trivial or small sized automorphism group does not rule out the existence of the unknown type II (72, 36, 16) code.


REFERENCES

N.J.A. Sloane, Is there a (72,36) d=16 selfdual code, IEEE Trans. Inform. Theory, 19 (1973), 251.


LINKS

Table of n, a(n) for n=1..17.
W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Pages 338393.
JonLark Kim, A Prize Problem In Coding Theory, University of Louisville


EXAMPLE

The smallest automorphism group size a binary selfdual code of length 2*16 = 32 is a(16) = 2.


CROSSREFS

Cf. SelfDual Codes A028362, A003179, A106162, A028363, A106163, A269455, A120373.
Cf. SelfDual Code Automorphism Groups A322299.
Sequence in context: A219613 A124453 A211827 * A000165 A241122 A109664
Adjacent sequences: A322336 A322337 A322338 * A322340 A322341 A322342


KEYWORD

nonn


AUTHOR

Nathan J. Russell, Dec 04 2018


STATUS

approved



