A034414 -id:A034414 - cp's OEIS frontend

A001380 Weight distribution of binary Golay code of length 24.

1, 0, 759, 2576, 759, 0, 1

Offset: 0

			The weight enumerator is x^24+759*x^16*y^8+2576*x^12*y^12+759*x^8*y^16+y^24.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 84.
W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 71.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 67.

J. H. Conway and N. J. A. Sloane, Orbit and coset analysis of the Golay and related codes, IEEE Trans. Inform. Theory, 36 (1990), 1038-1050.
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. A002289, A034414, A034415.

g24 := x^24+759*x^16*y^8+759*x^8*y^16+2576*x^12*y^12+y^24; e8 := x^8+14*x^4*y^4+y^8; d:=n->x^(n mod 2)*(1/2)*( (x^2+y^2)^floor((n)/2)+(x^2-y^2)^floor((n)/2));

A034597 Leading coefficient of extremal theta series of even unimodular lattice in dimension 24n.

Original entry on oeis.org

1, 196560, 52416000, 6218175600, 565866362880, 45792819072000, 3486157968384000, 256206274225902000, 18422726047165440000, 1305984407917646096640, 91692325887531393024000

Offset: 0

N. J. A. Sloane

nonn

			When n=1 we get the theta series of the 24-dimensional Leech lattice: 1+196560*q^4+16773120*q^6+... (see A008408). For n=2 we get A004672 and for n=3, A004675.

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

N. J. A. Sloane, Table of n, a(n) for n = 0..100
C. L. Mallows, A. M. Odlyzko and N. J. A. Sloane, Upper bounds for modular forms, lattices and codes, J. Alg., 36 (1975), 68-76.
C. L. Mallows and N. J. A. Sloane, An Upper Bound for Self-Dual Codes, Information and Control, 22 (1973), 188-200.
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).
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Cf. A034598 (second coefficient, which eventually becomes negative), A034414, A034415.

# Extremal theta series:
with(numtheory): B := 1:
# set mu:
for mu from 1 to 10 do
   # set max deg:
   md := mu+3;
   f := 1+240*add(sigma[3](i)*x^i, i=1..md);
   f := series(f, x, md);
   f := series(f^3, x, md);
   g := series(x*mul((1-x^i)^24, i=1..md), x, md);
   W0 := series(f^mu, x, md):
   h := series(g/f, x, md):
   A := series(W0, x, md):
   Z := A:
   for i from 1 to mu do
      Z := series(Z*h, x, md);
      A := series(A-coeff(A, x, i)*Z, x, md);
   od:
   B := B, coeff(A,x,mu+1);
od:
lprint(B);

terms = 11; Reap[For[mu = 1, mu <= terms, mu++, md = mu + 3; f = 1 + 240*Sum[DivisorSigma[3, i]*x^i, {i, 1, md}]; f = Series[f, {x, 0, md}];  f = Series[f^3, {x, 0, md}]; g = Series[x*Product[ (1 - x^i)^24, {i, 1, md}], {x, 0, md}]; W0 = Series[f^mu, {x, 0, md}]; h = Series[g/f, {x, 0, md}]; A = Series[W0, {x, 0, md}]; Z = A; For[ i = 1 , i <= mu, i++, Z = Series[Z*h, {x, 0, md}]; A = Series[A - SeriesCoefficient[A, {x, 0, i}]*Z, {x, 0, md}]]; an = SeriesCoefficient[A, {x, 0, mu+1}]; Print[an]; Sow[an]]][[2,1]] (* Jean-François Alcover, Jul 08 2017, adapted from Maple *)

A034415 Second term in extremal weight enumerator of doubly-even binary self-dual code of length 24n.

Original entry on oeis.org

1, 2576, 535095, 18106704, 369844880, 6101289120, 90184804281, 1251098739072, 16681003659936, 216644275600560, 2763033644875595, 34784314216176096, 433742858109499536, 5369839142579042560

Offset: 0

N. J. A. Sloane

sign

The terms become negative at n=154 and so certainly by that point the extremal codes do not exist (see references).

Up to n = 250 the terms steadily increase in magnitude, but their sign changes from positive to negative at n = 154.

			At length 24, the weight enumerator (of the Golay code) is 1+759*x^8+2576*x^12+..., with leading coefficient 759 and second term 2576.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, see Theorem 13, p. 624.

N. J. A. Sloane, Table of n, a(n) for n = 0..250
C. L. Mallows and N. J. A. Sloane, An Upper Bound for Self-Dual Codes, Information and Control, 22 (1973), 188-200.
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).
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Cf. A034414 (leading coefficient), A001380, A034597, A034598.

Maple
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For Maple program see A034414.
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A001380 Weight distribution of binary Golay code of length 24.

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A034597 Leading coefficient of extremal theta series of even unimodular lattice in dimension 24n.

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A034415 Second term in extremal weight enumerator of doubly-even binary self-dual code of length 24n.

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