| Displaying 1-5 of 5 results found.
page
1
Smallest automorphism group size for a binary self-dual code of length 2n.
+10
4
2, 8, 48, 384, 2688, 10752, 46080, 73728, 82944, 82944, 36864, 12288, 3072, 384, 30, 2, 1
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 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.
REFERENCES
N.J.A. Sloane, Is there a (72,36) d=16 self-dual code, IEEE Trans. Inform. Theory, 19 (1973), 251.
EXAMPLE
The smallest automorphism group size a binary self-dual code of length 2*16 = 32 is a(16) = 2.
CROSSREFS
Cf. Self-Dual Code Automorphism Groups A322299.
Number of binary self-dual codes of length 2n (up to permutation equivalence) that have a unique automorphism group size.
+10
1
1, 1, 1, 2, 2, 3, 4, 7, 9, 16, 23, 42, 68, 94, 124, 159, 187, 212
COMMENTS
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.
EXAMPLE
There are a(18) = 212 binary self-dual codes (up to permutation equivalence) of length 2*18 = 36 that have a unique automorphism group size.
Number of binary self-dual codes of length 2n having an automorphism group size that is a prime power.
+10
0
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 9, 66, 738, 10760
COMMENTS
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.
EXAMPLE
There are a(17)=10760 binary self-dual codes of length 2*17=34 having an automorphism group size that is a prime power.
Number of divisors for the automorphism group size having the largest number of divisors for a binary self-dual code of length 2n.
+10
0
2, 4, 10, 28, 36, 66, 144, 192, 340, 570, 1200, 1656, 3456, 5616, 9072, 10752, 22176
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.
EXAMPLE
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.
Number of distinct automorphism group sizes for binary self-dual codes of length 2n such that multiple same length binary self-dual codes with different weight distributions share the same automorphism group size.
+10
0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 17, 55, 117, 226, 343, 535
COMMENTS
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.
EXAMPLE
There are a(18) = 535 automorphism group sizes for the binary self-dual codes of length 2*18 = 36 where codes having different weight distributions share the same automorphism group size.
Search completed in 0.007 seconds
|