A018236 -id:A018236 - cp's OEIS frontend

A105675 Highest minimal distance of any Type II doubly-even binary self-dual code of length 8n.

4, 4, 8, 8, 8, 12, 12, 12

Offset: 1

N. J. A. Sloane, May 06 2005

Is a(9) = 12 or 16? This is an open question of long standing.

			At length 8 the only Type II doubly-even self-dual code is the Hamming code e_8, which has d=4, so a(1) = 4. The [24,12,8] Golay code has d=8, so a(3) = 8.

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

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).
N. J. A. Sloane, Is There a (72,36) d = 16 Self-Dual Code?, IEEE Trans. Information Theory, vol. IT-19 (1973), p. 251.

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

Cf. also A001380, A018236.

A120373 Weight distribution of [72,36,12] doubly-even binary self-dual extended quadratic-residue (or QR) code.

Original entry on oeis.org

1, 0, 0, 2982, 214065, 18303516, 462306915, 4398818490, 16600354155, 25759476488, 16600354155, 4398818490, 462306915, 18303516, 214065, 2982, 0, 0, 1

Offset: 0

N. J. A. Sloane, Mar 19 2009

			The actual weight enumerator is x^72 + 2982*x^60*y^12 + 214065*x^56*y^16 + 18303516*x^52*y^20 + 462306915*x^48*y^24 + y^72 + 4398818490*x^44*y^28 + 16600354155*x^40*y^32 + 25759476488*x^36*y^36 + 16600354155*x^32*y^40 + 4398818490*x^28*y^44 + 462306915*x^24*y^48 + 18303516*x^20*y^52 + 214065*x^16*y^56 + 2982*x^12*y^60.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977, Chap. 16.

C. J. Tjhai and Martin Tomlinson, Weight Distributions of Quadratic Residue and Quadratic Double Circulant Codes over GF(2).

Cf. A018236.

Name corrected by Jon-Lark Kim, Aug 16 2024

A004675 Theta series of extremal even unimodular lattice in dimension 72.

Original entry on oeis.org

1, 0, 0, 0, 6218175600, 15281788354560, 9026867482214400, 1989179450818560000, 213006159759990870000, 13144087517631410995200, 525100718690287495741440, 14756609779472604266496000, 310160311536865273422120000

Offset: 0

N. J. A. Sloane

nonn

The construction of such a lattice was announced by G. Nebe, Aug 12 2010. - N. J. A. Sloane, Aug 13 2010

			Theta series begins 1 + 6218175600*q^8 + 15281788354560*q^10 + 9026867482214400*q^12 + 1989179450818560000*q^14 + 213006159759990870000*q^16 + 13144087517631410995200*q^18 + 525100718690287495741440*q^20 + 14756609779472604266496000*q^22 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 195.

N. J. A. Sloane, Table of n, a(n) for n = 0..1000
J.-C. Belfiore and P. Sole, A Type II lattice of norm 8 in dimension 72, arXiv:1010.4484 [cs.IT], 2010. - _N. J. A. Sloane_, Oct 23 2010
G. Nebe and N. J. A. Sloane, Home page for this lattice
G. Nebe, An extremal even unimodular lattice of dimension 72, Preprint, arXiv:1008.2862 [math.NT], Aug 12 2010. - _N. J. A. Sloane_, Aug 13 2010

Cf. A018236.

# get th2, th3, th4 = Jacobi theta constants out to degree maxd
maxd:=2001:
temp0:=trunc(evalf(sqrt(maxd)))+2: a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od: th2:=series(a,q,maxd):
a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od: th3:=series(a,q,maxd):
th4:=series(subs(q=-q,th3),q,maxd):
# get Leech etc
t1:=th2^8+th3^8+th4^8: e8:=series(t1/2,q,maxd):
t1:=th2^8*th3^8*th4^8: delta24:=series(t1/256,q,maxd):
leech:=series(e8^3-720*delta24,q,maxd):
u1:=series(leech^3,q,maxd):
#u2:=series(leech^2*delta24,q,maxd):
u3:=series(leech*delta24^2,q,maxd):
u4:=series(delta24^3,q,maxd):
u5:=series(u1-589680*u3-78624000*u4,q,maxd);

terms = 13;
maxd = 2*terms;
th1 = EllipticTheta[1, 0, q];
th2 = EllipticTheta[2, 0, q];
th3 = EllipticTheta[3, 0, q];
th4 = th3 /. q -> -q;
t1 = th2^8 + th3^8 + th4^8;
e8 = Series[t1/2, {q, 0, maxd}];
t1 = th2^8*th3^8*th4^8;
delta24 = Series[t1/256, {q, 0, maxd}];
leech = Series[e8^3 - 720*delta24, {q, 0, maxd}];
u1 = Series[leech^3, {q, 0, maxd}];
u3 = Series[leech*delta24^2, {q, 0, maxd}];
u4 = Series[delta24^3, {q, 0, maxd}];
u5 = Series[u1 - 589680*u3 - 78624000*u4, {q, 0, maxd}];
CoefficientList[u5, q^2][[1 ;; terms]](* Jean-François Alcover, Jul 08 2017, adapted from Maple *)

A018237 Weight distribution of hypothetical [ 68,34,12 ] code derived from hypothetical [ 72,36,16 ] doubly-even self-dual code.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 442, 14960, 174471, 1478048, 9546537, 46699952, 175078410, 509477760, 1160564636, 2081169376, 2949602799, 3312254400, 2949602799, 2081169376, 1160564636, 509477760, 175078410, 46699952, 9546537, 1478048, 174471, 14960, 442, 0, 0, 0, 0, 0, 1

Offset: 0

N. J. A. Sloane

S. T. Dougherty and M Harada, Extremal and shadow extremal binary self-dual codes, preprint (Masaaki HARADA, harada(AT)math.okayama-u.ac.jp).

S. T. Dougherty, Publications
S. T. Dougherty, A related length 68 code.
S. T. Dougherty and M. Harada, New extremal self-dual codes of length 68, IEEE Transactions on Information Theory, 45 (1999), 2133-2136.

Cf. A018236.

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 15 2001

a(27)-a(28) corrected by Andrey Zabolotskiy, Nov 22 2021

cp's OEIS Frontend

A105675 Highest minimal distance of any Type II doubly-even binary self-dual code of length 8n.

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A120373 Weight distribution of [72,36,12] doubly-even binary self-dual extended quadratic-residue (or QR) code.

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A004675 Theta series of extremal even unimodular lattice in dimension 72.

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A018237 Weight distribution of hypothetical [ 68,34,12 ] code derived from hypothetical [ 72,36,16 ] doubly-even self-dual code.

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