A269455 Number of Type I (singly-even) self-dual binary codes of length 2n.

1, 3, 15, 105, 2295, 75735, 4922775, 625192425, 163204759575, 83724041661975, 85817142703524375, 175667691114114395625, 720413716161839357604375, 5902349576513949856852644375, 96709997811181068404530578084375, 3168896498278970068411253452090715625, 207692645973961964120828372930661061284375, 27222898185745116523209337325140537285726884375, 7136346644902153570976711733098966146766874104484375, 3741493773415815389266667264411257664189964123617799515625

Offset

1,2

Comments

A self dual binary linear code is either Type I (singly even) or Type II (doubly even). A self dual binary linear code can only be Type II if the length of the code (2n) is a multiple of 8. The total number self dual binary linear codes (including equivalent codes) is equal to the number of Type I self dual binary linear codes (including equivalent codes) when the length (2n) is not a multiple of 8. If the length is a multiple of 8 ( 2n =0 mod 8 ) then the total number of Type I codes is the number of type II codes subtracted from the total number of self dual codes of length 2n.

References

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, 2003, Page 366.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977.

Links

Nathan J. Russell, Table of n, a(n) for n = 1..49

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

P. Gaborit, Tables of Self-Dual Codes

E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998; (Abstract, pdf, ps).

Formula

From Nathan J. Russell, Mar 01 2016: (Start)

If 2n = 0 MOD 8 then a(n) = prod_(2^i+1, i=1,...,n-1) - prod_(2^i+1, i=0,...,n-2);

If 2n != 0 MOD 8 then a(n) = prod_(2^i+1, i=1,...,n-1).

If 2n = 0 MOD 8 then a(n) = A028362(n) - A028363( n/8);

If 2n != 0 MOD 8 then a(n) = A028362(n).

(End)

Mathematica

Table[
If[Mod[2 n, 8] == 0,
  Product[2^i + 1, {i, 1, n - 1}] - Product[2^i + 1, {i, 0, n - 2}] ,
  Product[2^i + 1, {i, 1, n - 1}]],
{n, 1, 10}] (* Nathan J. Russell, Mar 01 2016 *)

Prog

(Python)
for n in range(1, 10):
    product1 = 1
    for i in range(1, n-1 + 1):
        product1 *= (2**i+1)
    if (2*n)%8 == 0:
        product2 = 1
        for i in range(n-2 + 1):
            product2 *= (2**i+1)
        print(product1 - product2)
    else:
        print(product1)
(PARI) a(n) = if (2*n%8==0, prod(i=1, n-1, 2^i+1)-prod(i=0, n-2, 2^i+1), prod(i=1, n-1, 2^i+1))
vector(20, n, a(n)) \\ Colin Barker, Feb 28 2016

Crossrefs

Cf. A003178, A003179, A028363, A028361.

Sequence in context: A070826 A048599 A129731 * A088109 A231634 A353587

Adjacent sequences: A269452 A269453 A269454 * A269456 A269457 A269458

Keyword

nonn

Author

Nathan J. Russell, Feb 27 2016

Extensions

a(20) corrected by Andrew Howroyd, Feb 22 2018

Status

approved