A003179 -id:A003179 - cp's OEIS frontend

A028362 Total number of self-dual binary codes of length 2n. Totally isotropic spaces of index n in symplectic geometry of dimension 2n.

1, 3, 15, 135, 2295, 75735, 4922775, 635037975, 163204759575, 83724041661975, 85817142703524375, 175839325399521444375, 720413716161839357604375, 5902349576513949856852644375, 96709997811181068404530578084375

Offset: 1

N. J. A. Sloane

These numbers appear in the second column of A155103. - Mats Granvik, Jan 20 2009

a(n) = n terms in the sequence (1, 2, 4, 8, 16, ...) dot n terms in the sequence (1, 1, 3, 15, 135). Example: a(5) = 2295 = (1, 2, 4, 8, 16) dot (1, 1, 3, 15, 135) = (1 + 2 + 12 + 120 + 2160). - Gary W. Adamson, Aug 02 2010

			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.

Cf. A000124, A003178, A003179, A028361, A028363, A048675, A053632, A068052 (XOR-analog), A285101.
Cf. A155103. - Mats Granvik, Jan 20 2009
Cf. A005329, A006088. - Paul D. Hanna, Sep 16 2009

[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

a(n) = Product_{i=1..n-1} (2^i+1).
Letting a(0)=1, we have a(n) = Sum_{k=0..n-1} 2^k*a(k) for n>0. a(n) is asymptotic to c*sqrt(2)^(n^2-n) where c=2.384231029031371724149899288.... = A079555 = Product_{k>=1} (1 + 1/2^k). - Benoit Cloitre, Jan 25 2003
G.f.: Sum_{n>=1} 2^(n*(n-1)/2) * x^n/(Product_{k=0..n-1} (1-2^k*x)). - Paul D. Hanna, Sep 16 2009
a(n) = 2^(binomial(n,2) - 1)*(-1; 1/2){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 23 2015
From Antti Karttunen, Apr 15 2017: (Start)
a(n) = A048675(A285101(n-1)).
a(n) = b(n-1), where b(0) = 1, and for n > 0, b(n) = b(n-1) + (2^n)*b(n-1).
a(n) = Sum_{i=1..A000124(n-1)} A053632(n-1,i-1)*(2^(i-1)) [where the indexing of both rows and columns of irregular table A053632(row,col) is considered to start from zero].
(End)
G.f. A(x) satisfies: A(x) = x * (1 + A(2*x)) / (1 - x). - Ilya Gutkovskiy, Jun 06 2020
Conjectural o.g.f. as a continued fraction of Stieltjes type (S-fraction):
  1/(1 - 3*x/(1 - 2*x/(1 - 10*x/(1 - 12*x/(1 - 36*x/(1 - 56*x/(1 - 136*x/(1 - 240*x/(1 - ... - 2^(n-1)*(2^n + 1)*x/(1 - 2^n*(2^n - 1)*x/(1 - ... ))))))))))). - Peter Bala, Sep 27 2023

A003178 Number of indecomposable self-dual binary codes of length 2n.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 1, 2, 2, 6, 8, 26, 45, 148, 457, 2523, 20786

Offset: 0

N. J. A. Sloane

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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).

Cf. A003179, A028362, A028363. Equals A106162 + A106164.

a(16) corrected and a(17) added by N. J. A. Sloane, based on data in Bilous's paper, Sep 06 2005

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

N. J. A. Sloane, May 09 2005

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).

Cf. A003178, A003179, A106163-A106167.

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

N. J. A. Sloane

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 631.

Cf. A003178, A003179, A028362.

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

a(n) = 2*Product_{i=1..4n-2} (2^i + 1).

There is an error in Eq. (75) of F. J. MacWilliams and N. J. A. Sloane, the lower subscript should be 1 not 0.

Formula corrected by N. J. A. Sloane, May 07 2013 following a suggestion from Harvey P. Dale

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

N. J. A. Sloane, May 09 2005

"There are 94343 inequivalent doubly even self-dual codes of length 40, 16470 of which are extremal" [Betsumiya et al.] - Jonathan Vos Post, Aug 06 2012

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).

Cf. A003178, A003179, A106162, A106163, A106164, A106165, A106166, A106167, A215219.

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

Nathan J. Russell, Feb 27 2016

nonn

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.

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).

Cf. A003178, A003179, A028363, A028361.

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)

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)

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

A322299 Number of distinct automorphism group sizes for binary self-dual codes of length 2n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 7, 9, 16, 24, 48, 85, 149, 245, 388

Offset: 1

Nathan J. Russell, Dec 02 2018

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 a(16) = 388 distinct sizes for the automorphism groups of the binary self-dual codes of length 16.  In general, two automorphism  groups with the same size are not necessarily isomorphic.

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, 2003, pp. 7, 252-330, 338-393.

Cf. self-dual codes A028362, A003179, A106162, A028363, A106163.

A322339 Smallest automorphism group size for a binary self-dual code of length 2n.

Original entry on oeis.org

2, 8, 48, 384, 2688, 10752, 46080, 73728, 82944, 82944, 36864, 12288, 3072, 384, 30, 2, 1

Offset: 1

Nathan J. Russell, Dec 04 2018

nonn

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.

			The smallest automorphism group size a binary self-dual code of length 2*16 = 32 is a(16) = 2.

N.J.A. Sloane, Is there a (72,36) d=16 self-dual code, IEEE Trans. Inform. Theory, 19 (1973), 251.

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Pages 338-393.
Jon-Lark Kim, A Prize Problem In Coding Theory, University of Louisville

Cf. Self-Dual Codes A028362, A003179, A106162, A028363, A106163, A269455, A120373.

Cf. Self-Dual Code Automorphism Groups A322299.

A105685 Number of inequivalent codes attaining highest minimal distance of any Type I (strictly) singly-even binary self-dual code of length 2n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 7, 1, 1, 1, 3, 13, 3

Offset: 1

N. J. A. Sloane, May 06 2005, Aug 23 2008

nonn

			At length 8 the only strictly Type I self-dual code is {00,11}^4, so a(4) = 1.

J. H. Conway and V. S. Pless, On the enumeration of self-dual codes, J. Comb. Theory, A28 (1980), 26-53.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977.
V. S. Pless, The children of the (32,16) doubly even codes, IEEE Trans. Inform. Theory, 24 (1978), 738-746.

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).
P. Gaborit, Tables of Self-Dual Codes
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).

Cf. A105674, A105675, A105676, A105677, A105678, A016729, A066016, A105681, A105682.

A105674 gives the minimal distance of these codes, A106165 the number of codes of any minimal distance and A003179 the number of inequivalent codes allowing Type I or Type II and any minimal distance.

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

N. J. A. Sloane, May 09 2005, Aug 23 2008

The minimal distance of these codes is not constrained. A105685 gives the number with the highest minimal distance.

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).

Cf. A003178, A003179, A106162-A106167.

cp's OEIS Frontend

A028362 Total number of self-dual binary codes of length 2n. Totally isotropic spaces of index n in symplectic geometry of dimension 2n.

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A003178 Number of indecomposable self-dual binary codes of length 2n.

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A106162 Number of indecomposable Type II binary self-dual codes of length 8n.

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A028363 Total number of doubly-even self-dual binary codes of length 8n.

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A106163 Total number of (indecomposable or decomposable) Type II binary self-dual codes of length 8n.

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

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A322299 Number of distinct automorphism group sizes for binary self-dual codes of length 2n.

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A322339 Smallest automorphism group size for a binary self-dual code of length 2n.