A034414 Leading term in extremal weight enumerator of doubly-even binary self-dual code of length 24n.
1, 759, 17296, 249849, 3217056, 39703755, 481008528, 5776211364, 69065734464, 824142912363, 9826364199840, 117145945726810, 1396918583188128, 16665451879695801, 198937019774252928
Offset: 0
Examples
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.
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
- Seiichi Manyama, Table of n, a(n) for n = 0..919 (terms 0..250 from N. J. A. Sloane)
- 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).
Programs
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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:
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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-François Alcover, Oct 06 2011, after formula *)
Formula
a(24n) = C(24n, 5)*C(5n-2, n-1)/C(4n+4, 5).
Comments