A014115 -id:A014115 - cp's OEIS frontend

A003956 Order of complex Clifford group of degree 2^n arising in quantum coding theory.

8, 192, 92160, 743178240, 97029351014400, 203286581427673497600, 6819500449352277792129024000, 3660967964237442812098963052691456000, 31446995505814020383166371418359014222725120000

Offset: 0

N. J. A. Sloane and Peter Shor

T. D. Noe, Table of n, a(n) for n = 0..20
Simon Burton, Elijah Durso-Sabina, and Natalie C. Brown, Genons, Double Covers and Fault-tolerant Clifford Gates, arXiv:2406.09951 [quant-ph], 2024. See p. 18.
A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Quantum error correction via codes over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
G. Nebe, E. M. Rains, and N. J. A. Sloane, The invariants of the Clifford groups, arXiv:math/0001038 [math.CO], 2000; Des. Codes Crypt. 24 (2001), 99-121.
G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Edwin Pednault, An alternative approach to optimal wire cutting without ancilla qubits, arXiv:2303.08287 [quant-ph], 2023.
Tefjol Pllaha, Olav Tirkkonen, and Robert Calderbank, Binary Subspace Chirps, arXiv:2102.12384 [cs.IT], 2021.
Bernhard Runge, Codes and Siegel modular forms, Discrete Math. 148 (1996), 175-204.
Peter Selinger, Generators and relations for n-qubit Clifford operators, arXiv:1310.6813 [quant-ph], 2013; Log. Methods Comput. Sci. 11 (2:10) (2015), 1-17, doi:10.2168/LMCS-11(2:10)2015.
Index entries for sequences related to groups.

Cf. A001309, A014116, A014115, A027672, A100221.

Equals twice A027638.

List([0..10], n-> 2^((n+1)^2 +2)*Product([1..n], j-> 4^j -1) ); # G. C. Greubel, Sep 24 2019

[n eq 0 select 8 else 2^((n+1)^2+2)*(&*[4^j-1: j in [1..n]]): n in [0..10]]; // G. C. Greubel, Sep 24 2019

a(n):= 2^(n^2+2*n+3)*mul(4^j-1, j=1..n); seq(a(n), n=0..10); # modified by G. C. Greubel, Sep 24 2019

Table[2^(n^2+2n+3) Product[4^j-1,{j,n}],{n,0,10}] (* Harvey P. Dale, Nov 03 2017 *)

vector(11, n, 2^(n^2 +2)*prod(j=1,n-1, 4^j-1) ) \\ G. C. Greubel, Sep 24 2019

from math import prod
def A003956(n): return prod((1<Chai Wah Wu, Jun 20 2022

[2^((n+1)^2 +2)*product(4^j -1 for j in (1..n)) for n in (0..10)] # G. C. Greubel, Sep 24 2019

From Amiram Eldar, Jul 06 2025: (Start)
a(n) = 2^(n^2+2*n+3) * Product_{k=1..n} (4^k-1).
a(n) ~ c * 2^(2*n^2+3*n+3), where c = A100221. (End)

A001309 Order of real Clifford group L_n connected with Barnes-Wall lattices in dimension 2^n.

Original entry on oeis.org

2, 16, 2304, 5160960, 178362777600, 96253116206284800, 819651496316379542323200, 110857799304670627788849414144000, 238987988705420266773820308079698247680000

Offset: 0

N. J. A. Sloane, Peter Shor

A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
G. Nebe, E. M. Rains and N. J. A. Sloane, The invariants of the Clifford groups, arXiv:math/0001038 [math.CO], 2000; Des. Codes Crypt. 24 (2001), 99-121.
Index entries for sequences related to Barnes-Wall lattices.
Index entries for sequences related to groups.

2^(2n+2) times order of Chevalley group D_n (2) (cf. A001308). Twice A014115. See also A014116, A003956 (for the complex group).

Cf. A100221.

2^(n^2+n+2) * (2^n - 1) * product('2^(2*i)-1','i'=1..n-1);

a[0] = 2; a[n_] := 2^(n^2+n+2) * (2^n-1) * Product[2^(2*i)-1, {i, 1, n-1}]; Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Jul 16 2015, after Maple *)

from math import prod
def A001309(n): return 2 if n == 0 else ((1<Chai Wah Wu, Jun 20 2022

From Amiram Eldar, Jul 06 2025: (Start)
a(n) = 2^(n^2+n+2) * (2^n-1) * Product_{k=1..n-1} (2^(2*k)-1).
a(n) ~ c * 2^(2*n^2+n+2), where c = A100221. (End)

A014116 Order of automorphism group of Barnes-Wall lattice in dimension 2^n.

Original entry on oeis.org

2, 8, 1152, 696729600, 89181388800, 48126558103142400, 409825748158189771161600, 55428899652335313894424707072000

Offset: 0

N. J. A. Sloane

nonn

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 129.

A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), arXiv:quant-ph/9608006, 1996-1997; IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
G. E. Wall, On the Clifford collineation, transform and similarity groups. IV. An application to quadratic forms, Nagoya Math. J., 21 (1962), pp. 199-222.
Index entries for sequences related to Barnes-Wall lattices

Agrees with A014115 except at n=3. Equals half of A001309. Cf. A003956.

2^(n^2+n+1) * (2^n - 1) * product('2^(2*i)-1','i'=1..n-1); # except for n=3.

from math import prod
def A014116(n): return 2+696729598*(n//3) if n == 0 or n == 3 else ((1<Chai Wah Wu, Jun 20 2022

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A003956 Order of complex Clifford group of degree 2^n arising in quantum coding theory.

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A001309 Order of real Clifford group L_n connected with Barnes-Wall lattices in dimension 2^n.

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A014116 Order of automorphism group of Barnes-Wall lattice in dimension 2^n.

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