cp's OEIS Frontend

All binary self-dual codes must have even length. This is due to the fact that all binary self-dual codes contain the "all 1's codeword". If a binary self-dual code could have odd length, the dot product of the "all 1's codeword" with itself would be equal to 1. A dot product of 1 would imply it is not orthogonal to itself and so the code could not be self-dual.

It is important to distinguish between "extremal" (meaning having the highest possible minimal distance permitted by Gleason's theorem) and "optimal" (meaning having the highest minimal distance that can actually be achieved). This sequence enumerates optimal codes. Extremal codes do not exist when n is sufficiently large. For lengths up to at least 64, "extremal" and "optimal" coincide.

"There are 94343 inequivalent doubly even self-dual codes of length 40, 16470 of which are extremal." [Betsumiya et al.] - Jonathan Vos Post, Aug 06 2012

Every binary self-dual code is either indecomposable or decomposable. A decomposable binary self-dual code is the direct sum of a set of indecomposable binary self-dual codes of smaller length.

Two codes are said to be permutation equivalent if permuting the columns of one code results in the other code.

If permuting the columns of a code results in the same identical code the permutation is called an automorphism.

The automorphisms of a code form a group called the automorphism group.

Some codes have automorphism groups that contain the same number of elements. There are situations, both trivial and otherwise, that codes of different lengths can have the same size automorphism groups.

Some codes have automorphism group sizes that are unique to the code. This sequence only compares automorphism group sizes for codes with the same length.

In fact all such codes of length <= 42 are indecomposable.

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.

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.

Two codes are said to be permutation equivalent if permuting the columns of one code results in the other code.

If permuting the columns of a code results in the same identical code the permutation is called an automorphism.

The automorphisms of a code form a group called the automorphism group.

Some codes have automorphism groups that contain the same number of elements. There are situations, both trivial and otherwise, that codes of different lengths can have the same size automorphism groups.

Some codes have automorphism group sizes that are unique to the code for a given length.

There are instances where more than one code can share the same automorphism group size yet have different weight distributions (weight enumerator). This sequence provides the number of automorphism group sizes where this is true for a given length.