cp's OEIS Frontend

The inverse of an orthogonal matrix is its transpose. This implies the dot product of a row with itself must be 1. This further implies the number of ones in each row must be odd. Given that orthogonal matrices form a group, it must be the case the transpose is also an orthogonal matrix. This requires every column of a binary orthogonal matrix also have an odd number of ones.

For 1 <= n <= 4 the counts are the same for the total number of binary orthogonal matrices (A003053).

The inverse of an orthogonal matrix is its transpose. This implies the dot product of a row with itself must be 1. This further implies the number of ones in each row must be odd. Given that orthogonal matrices form a group, it must be the case the transpose is also an orthogonal matrix. This requires every column of a binary orthogonal matrix also have an odd number of ones. As a result, there will always be an orthogonal matrix of size n X n having rows with n-1 number of ones if n is an even number, namely an all-ones matrix except for zeros down the main diagonal. An n X n orthogonal matrix cannot exist with n-1 ones in each row if n is odd, since n-1 is even.

a(n) = n-1 if n is even.

a(n) < n-1 if n is odd.

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.

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.

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.

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.

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.

There are codes with a trivial automorphism group of size 1. This sequence does not count those codes.

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.

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 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 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.

There is a notable open problem in coding theory regarding binary self-dual codes. Does there exist a type II binary self-dual 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 self-dual 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 self-dual code grows the minimum size of the automorphism group grows up to a point, namely length 18. It would appear that a binary self-dual 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.

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.