A028363 -id:A028363 - 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

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

The length 36 binary self dual codes have been classified. - Nathan J. Russell, Feb 14 2016

This is number of binary self-dual codes of length 2n up to column permutation equivalence. Sequence A028362 gives an actual count of all possible binary self-dual codes of length 2n. - Nathan J. Russell, Nov 25 2018

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).

Cf. A003178, A028362, A028363, A105685. Equals A106163 + A106165.

a(18) from Nathan J. Russell, Feb 14 2016

Name clarified by Nathan J. Russell, Nov 26 2018

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

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.

A323357 Number of binary self-dual codes of length 2n (up to permutation equivalence) that have a unique automorphism group size.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 7, 9, 16, 23, 42, 68, 94, 124, 159, 187, 212

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. This sequence only compares automorphism group sizes for codes with the same length.

			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.

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.

A321946 Number of divisors for the automorphism group size having the largest number of divisors for a binary self-dual code of length 2n.

Original entry on oeis.org

2, 4, 10, 28, 36, 66, 144, 192, 340, 570, 1200, 1656, 3456, 5616, 9072, 10752, 22176

Offset: 1

Nathan J. Russell, Dec 12 2018

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.

			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.

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

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

Cf. Self-Dual Code Automorphism Groups A322299, A322339.

A322309 Largest automorphism group size for a binary self-dual code of length 2n.

Original entry on oeis.org

2, 8, 48, 1344, 3840, 46080, 645120, 10321920, 185794560, 3715891200, 81749606400, 1961990553600, 51011754393600, 1428329123020800, 42849873690624000, 1371195958099968000, 46620662575398912000

Offset: 1

Nathan J. Russell, Dec 03 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 that all codewords of the code 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 upper 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.  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.

			The largest automorphism group size a binary self-dual code of length 2*16=32 is a(16) = 1371195958099968000.

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

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

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.

Original entry on oeis.org

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.

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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A003179 Number of self-dual binary codes of length 2n (up to column permutation equivalence).

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

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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.

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A323357 Number of binary self-dual codes of length 2n (up to permutation equivalence) that have a unique automorphism group size.

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A321946 Number of divisors for the automorphism group size having the largest number of divisors for a binary self-dual code of length 2n.