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A034414 Leading term in extremal weight enumerator of doubly-even binary self-dual code of length 24n. 4
1, 759, 17296, 249849, 3217056, 39703755, 481008528, 5776211364, 69065734464, 824142912363, 9826364199840, 117145945726810, 1396918583188128, 16665451879695801, 198937019774252928 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The term after the leading nonzero term eventually becomes negative and so for large n the extremal codes do not exist (see references, also A034415).

REFERENCES

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

C. L. Mallows and N. J. A. Sloane, An Upper Bound for Self-Dual Codes, Information and Control, 22 (1973), 188-200.

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..250

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).

FORMULA

a(24n) = C(24n, 5)*C(5n-2, n-1)/C(4n+4, 5).

EXAMPLE

At length 24, the extremal weight enumerator is 1+759*x^8+2576*x^12+..., with leading coefficient 759; this is the weight enumerator of the binary Golay code.

MAPLE

# Extremal weight enumerators:

kernelopts(printbytes=false): interface(screenwidth=200);

W0:=1; f:=1+14*x+x^2; f:=f^3; g:=x*(1-x)^4;

for mu from 1 to 100 do

# set max deg

md:=mu+3; 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: lprint(A);

od:

MATHEMATICA

a[n_] := 18(6n-1)(8n-1)(12n-1)(24n-1)Binomial[5n-2, n-1]/((n+1)(2n+1)(4n+1)(4n+3)); a[0] = 1; Table[a[n], {n, 0, 14}](* Jean-Fran├žois Alcover, Oct 06 2011, after formula *)

CROSSREFS

Cf. A034415 (second coefficient, which becmes negative), A001380, A034597.

Sequence in context: A001380 A001920 A225022 * A157983 A014747 A258135

Adjacent sequences:  A034411 A034412 A034413 * A034415 A034416 A034417

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified November 19 04:01 EST 2017. Contains 294912 sequences.