A028362 -id:A028362 - cp's OEIS frontend

A323358 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.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 17, 55, 117, 226, 343, 535

Offset: 1

Nathan J. Russell, Jan 12 2019

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.

			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.

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Pages 338-393.

For self-dual codes see A028362, A003179, A106162, A028363, A106163, A269455, A120373; for automorphism groups see A322299, A322339, A323357.

A368493 T(n,m) is the number of m-dimensional isotropic subspaces of a 2n-dimensional symplectic space over Z/2, n >= 0 and 0 <= m <= n.

Original entry on oeis.org

1, 1, 3, 1, 15, 15, 1, 63, 315, 135, 1, 255, 5355, 11475, 2295, 1, 1023, 86955, 782595, 782595, 75735, 1, 4095, 1396395, 50868675, 213648435, 103378275, 4922775, 1, 16383, 22362795, 3268162755, 55558766835, 112909751955, 26883274275, 635037975

Offset: 0

Simon Burton, Dec 27 2023

The number of m-dimensional isotropic subspaces of an n-dimensional symplectic space over Z/2.

			Triangle begins:
  1;
  1,    3;
  1,   15,      15;
  1,   63,     315,      135;
  1,  255,    5355,    11475,      2295;
  1, 1023,   86955,   782595,    782595,     75735;
  1, 4095, 1396395, 50868675, 213648435, 103378275, 4922775;
  ...

J. Baez, This Week's Finds in Mathematical Physics (Week 187)
M. H. Poroch, Bounds on subspace codes based on totally isotropic subspace in symplectic spaces and extended symplectic spaces, Asian-European Journal of Mathematics, 12 (2019).
Z. Wan, Notes on finite geometries and the construction of PBIB designs I, Some Anzahl theorems in symplectic geometry over finite fields, Acta Sci. 13 (1964) 515-516.

Main diagonal gives A028362.

Cf. A022166.

T[n_,m_]:=Product[(2^(2i)-1),{i,n-m+1,n}]/Product[(2^i-1),{i,1,m}]; Table[T[n,m],{n,0,7},{m,0,n}] (* Stefano Spezia, Dec 28 2023 *)

T(n,m) = prod(i=n-m+1, n, 2^(2*i)-1)/prod(i=1, m, 2^i-1); \\ Michel Marcus, Dec 27 2023

from math import prod
q = 2
N = lambda n, m : (prod([q**(2*i)-1 for i in range(n-m+1, n+1)])//prod([q**i-1 for i in range(1, m+1)]))
print([N(n, m) for n in range(8) for m in range(n+1)])

T(n,m) = Product_{i=n-m+1..n} (2^(2i)-1)/Product_{i=1..m} (2^i-1).

cp's OEIS Frontend

A323358 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.

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A368493 T(n,m) is the number of m-dimensional isotropic subspaces of a 2n-dimensional symplectic space over Z/2, n >= 0 and 0 <= m <= n.

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