A028362
Total number of self-dual binary codes of length 2n. Totally isotropic spaces of index n in symplectic geometry of dimension 2n.
Original entry on oeis.org
1, 3, 15, 135, 2295, 75735, 4922775, 635037975, 163204759575, 83724041661975, 85817142703524375, 175839325399521444375, 720413716161839357604375, 5902349576513949856852644375, 96709997811181068404530578084375
Offset: 1
G.f. = x + 3*x^2 + 15*x^3 + 135*x^4 + 2295*x^5 + 75735*x^6 + 4922775*x^7 + ...
- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 630.
- T. D. Noe, Table of n, a(n) for n = 1..50
- C. Bachoc and P. Gaborit, On extremal additive F_4 codes of length 10 to 18, J. Théorie Nombres Bordeaux, 12 (2000), 255-271.
- Steven T. Dougherty and Esengül Saltürk, The neighbor graph of binary self-orthogonal codes, Adv. Math. Comm. (2024). See p. 6.
- Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
-
[1] cat [&*[ 2^k+1: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Dec 24 2015
-
seq(mul(1 + 2^j, j = 1..n-1), n = 1..20); # G. C. Greubel, Jun 06 2020
-
Table[Product[2^i+1,{i,n-1}],{n,15}] (* or *) FoldList[Times,1, 2^Range[15]+1] (* Harvey P. Dale, Nov 21 2011 *)
Table[QPochhammer[-2, 2, n - 1], {n, 15}] (* Arkadiusz Wesolowski, Oct 29 2012 *)
-
{a(n)=polcoeff(sum(m=0,n,2^(m*(m-1)/2)*x^m/prod(k=0,m-1,1-2^k*x+x*O(x^n))),n)} \\ Paul D. Hanna, Sep 16 2009
-
{a(n) = if( n<1, 0 , prod(k=1, n-1, 2^k + 1))}; /* Michael Somos, Jan 28 2018 */
-
{a(n) = sum(k=0, n-1, 2^(k*(k+1)/2) * prod(j=1, k, (2^(n-j) - 1) / (2^j - 1)))}; /* Michael Somos, Jan 28 2018 */
-
for n in range(2,40,2):
product = 1
for i in range(1,n//2-1 + 1):
product *= (2**i+1)
print(product)
# Nathan J. Russell, Mar 01 2016
-
from math import prod
def A028362(n): return prod((1<Chai Wah Wu, Jun 20 2022
-
from ore_algebra import *
R. = QQ['x']
A. = OreAlgebra(R, 'Qx', q=2)
print((Qx - x - 1).to_list([0,1], 10)) # Ralf Stephan, Apr 24 2014
-
from sage.combinat.q_analogues import q_pochhammer
[q_pochhammer(n-1,-2,2) for n in (1..20)] # G. C. Greubel, Jun 06 2020
-
;; With memoization-macro definec.
(define (A028362 n) (A028362off0 (- n 1)))
(definec (A028362off0 n) (if (zero? n) 1 (+ (A028362off0 (- n 1)) (* (expt 2 n) (A028362off0 (- n 1))))))
;; Antti Karttunen, Apr 15 2017
A003179
Number of self-dual binary codes of length 2n (up to column permutation equivalence).
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 3, 4, 7, 9, 16, 25, 55, 103, 261, 731, 3295, 24147, 519492
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- R. T. Bilous and G. H. J. van Rees, An enumeration of binary self-dual codes of length 32, preprint.
- R. T. Bilous and G. H. J. van Rees, An enumeration of binary self-dual codes of length 32, Designs, Codes Crypt., 26 (2002), 61-86.
- J. H. Conway and V. S. Pless, On the enumeration of self-dual codes, J. Comb. Theory, A28 (1980), 26-53. MR0558873
- J. H. Conway, V. Pless and N. J. A. Sloane, The Binary Self-Dual Codes of Length Up to 32: A Revised Enumeration, J. Comb. Theory, A28 (1980), 26-53 (Abstract, pdf, ps, Table A, Table D).
- Masaaki Harada and Akihiro Munemasa, Classification of Self-Dual Codes of Length 36, arXiv:1012.5464 [math.CO], 2010-2012.
- W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic. 11 (2005), 451-490.
- W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Pages 7,252-282,338-393.
- G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
- E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
A106162
Number of indecomposable Type II binary self-dual codes of length 8n.
Original entry on oeis.org
1, 1, 1, 7, 75, 94251
Offset: 0
- J. H. Conway and V. S. Pless, On the enumeration of self-dual codes, J. Comb. Theory, A28 (1980), 26-53.
- V. S. Pless, The children of the (32,16) doubly even codes, IEEE Trans. Inform. Theory, 24 (1978), 738-746.
- Koichi Betsumiya, Masaaki Harada and Akihiro Munemasa, A Complete Classification of Doubly Even Self-Dual Codes of Length 40, arXiv:1104.3727v3 [math.CO], v3, Aug 02, 2012.
- J. H. Conway, V. Pless and N. J. A. Sloane, The Binary Self-Dual Codes of Length Up to 32: A Revised Enumeration, J. Comb. Theory, A28 (1980), 26-53 (Abstract, pdf, ps, Table A, Table D).
- G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
- E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
a(4) corrected by John van Rees, Jul 21 2005. It was given as 76 by Conway and Pless and as 74 by Rains and Sloane.
a(5) = 94251 = 94343 - 75 - 7 - 7 - 1 - 1 - 1 (cf.
A106163) from Koichi Betsumaya, Aug 11 2012
A028363
Total number of doubly-even self-dual binary codes of length 8n.
Original entry on oeis.org
1, 30, 9845550, 171634285407048750, 193419995622362136809061156168750, 14272693289804307141953423466197932293533748208968750
Offset: 0
- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 631.
-
Join[{1},Table[2*Product[2^i+1,{i,4n-2}],{n,6}]] (* Harvey P. Dale, May 08 2013 *)
Table[Product[2^i + 1, {i, 0, n/2 - 2}], {n, 8, 40, 8}] (* Nathan J. Russell, Mar 04 2016 *)
-
for n in range(8, 50, 8):
product = 1
for i in range(n//2 - 1):
product *= 2**i + 1
print(product, end=", ")
# Nathan J. Russell, Mar 01 2016
There is an error in Eq. (75) of F. J. MacWilliams and N. J. A. Sloane, the lower subscript should be 1 not 0.
A106163
Total number of (indecomposable or decomposable) Type II binary self-dual codes of length 8n.
Original entry on oeis.org
1, 1, 2, 9, 85, 94343
Offset: 0
- Koichi Betsumiya, Masaaki Harada and Akihiro Munemasa, A Complete Classification of Doubly Even Self-Dual Codes of Length 40, arXiv:1104.3727v3 [math.CO], v3, Aug 02, 2012. - From _Jonathan Vos Post_, Aug 06 2012
- J. H. Conway and V. S. Pless, On the enumeration of self-dual codes, J. Comb. Theory, A28 (1980), 26-53. [DOI] MR0558873
- J. H. Conway, V. Pless and N. J. A. Sloane, The Binary Self-Dual Codes of Length Up to 32: A Revised Enumeration, J. Comb. Theory, A60 (1992), 183-195 (Abstract, pdf, ps, Table A, Table D).
- W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic. 11 (2005), 451-490. [DOI]
- G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
- V. S. Pless, The children of the (32,16) doubly even codes, IEEE Trans. Inform. Theory, 24 (1978), 738-746. [DOI] MR0514353
- E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
A269455
Number of Type I (singly-even) self-dual binary codes of length 2n.
Original entry on oeis.org
1, 3, 15, 105, 2295, 75735, 4922775, 625192425, 163204759575, 83724041661975, 85817142703524375, 175667691114114395625, 720413716161839357604375, 5902349576513949856852644375, 96709997811181068404530578084375, 3168896498278970068411253452090715625, 207692645973961964120828372930661061284375, 27222898185745116523209337325140537285726884375, 7136346644902153570976711733098966146766874104484375, 3741493773415815389266667264411257664189964123617799515625
Offset: 1
- 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.
- 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).
-
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 *)
-
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
-
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)
A106165
Number of inequivalent (indecomposable or decomposable) Type I but not Type II binary self-dual codes of length 2n.
Original entry on oeis.org
0, 1, 1, 1, 1, 2, 3, 4, 5, 9, 16, 25, 46, 103, 261, 731, 3210, 24147
Offset: 0
- R. T. Bilous and G. H. J. van Rees, An enumeration of binary self-dual codes of length 32, Designs, Codes Crypt., 26 (2002), 61-86.
- J. H. Conway and V. S. Pless, On the enumeration of self-dual codes, J. Comb. Theory, A28 (1980), 26-53. MR0558873
- J. H. Conway, V. Pless and N. J. A. Sloane, The Binary Self-Dual Codes of Length Up to 32: A Revised Enumeration, J. Comb. Theory, A28 (1980), 26-53 (Abstract, pdf, ps, Table A, Table D).
- W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic., 11 (2005), 451-490.
- G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
- V. S. Pless, The children of the (32,16) doubly even codes, IEEE Trans. Inform. Theory, 24 (1978), 738-746.
- E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
A106164
Number of indecomposable Type I but not Type II binary self-dual codes of length 2n.
Original entry on oeis.org
0, 1, 0, 0, 0, 0, 1, 1, 1, 2, 6, 8, 19, 45, 148, 457, 2448, 20786
Offset: 0
- R. T. Bilous, Enumeration of binary self-dual codes of length 34, Preprint, 2005.
- R. T. Bilous and G. H. J. van Rees, An enumeration of binary self-dual codes of length 32, Designs, Codes Crypt., 26 (2002), 61-86.
- J. H. Conway and V. S. Pless, On the enumeration of self-dual codes, J. Comb. Theory, A28 (1980), 26-53.
- V. S. Pless, The children of the (32,16) doubly even codes, IEEE Trans. Inform. Theory, 24 (1978), 738-746.
- G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
- J. H. Conway, V. Pless and N. J. A. Sloane, The Binary Self-Dual Codes of Length Up to 32: A Revised Enumeration, J. Comb. Theory, A28 (1980), 26-53 (Abstract, pdf, ps, Table A, Table D).
- E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
a(34) computed by
N. J. A. Sloane, based on data in Bilous's paper, Sep 06 2005
A215219
Number of (indecomposable or decomposable) Type II binary self-dual codes of length 8n with the highest minimal distance.
Original entry on oeis.org
1, 1, 2, 1, 5, 16470, 1
Offset: 0
- Koichi Betsumiya, Masaaki Harada and Akihiro Munemasa, A Complete Classification of Doubly Even Self-Dual Codes of Length 40, arXiv:1104.3727v3 [math.CO], v3, Aug 02, 2012. - From _Jonathan Vos Post_, Aug 06 2012
- J. H. Conway and V. S. Pless, On the enumeration of self-dual codes, J. Comb. Theory, A28 (1980), 26-53. [DOI] MR0558873
- J. H. Conway, V. Pless and N. J. A. Sloane, The Binary Self-Dual Codes of Length Up to 32: A Revised Enumeration, J. Comb. Theory, A60 (1992), 183-195 (Abstract, pdf, ps, Table A, Table D).
- S. K. Houghten, C. W. H. Lam, L. H. Thiel and J. A. Parker, The extended quadratic residue code is the only (48,24,12) self-dual doubly-even code, IEEE Trans. Inform. Theory, 49 (2003), 53-59.
- W. C. Huffman, On the classification and enumeration of self-dual codes, Finite Fields Applic. 11 (2005), 451-490. [DOI]
- G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
- V. S. Pless, The children of the (32,16) doubly even codes, IEEE Trans. Inform. Theory, 24 (1978), 738-746. [DOI] MR0514353
- E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
A322429
Number of decomposable binary self-dual codes of length 2n (up to permutation equivalence).
Original entry on oeis.org
0, 1, 1, 1, 2, 2, 3, 5, 7, 10, 17, 29, 58, 113, 274, 772, 3361
Offset: 1
There are A003179(17) = 24147 binary self-dual codes of length 2*17 = 34 up to permutation equivalence. There are A003178(17) = 2523 binary self-dual codes of length 2*17 = 34 that are indecomposable. This means that there are A003179(17) - A003178(17) = a(17) = 3361 binary self-dual codes of length 2*17=34 that are decomposable.
- J. H. Conway, V. Pless and N. J. A. Sloane, The Binary Self-Dual Codes of Length Up to 32: A Revised Enumeration, J. Comb. Theory, A28 (1980), 26-53.
- W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, pp. 7, 18, 338-393.
- G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
- E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
Showing 1-10 of 11 results.
Comments