A001380 -id:A001380 - 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.

A034414 Leading term in extremal weight enumerator of doubly-even binary self-dual code of length 24n.

Original entry on oeis.org

1, 759, 17296, 249849, 3217056, 39703755, 481008528, 5776211364, 69065734464, 824142912363, 9826364199840, 117145945726810, 1396918583188128, 16665451879695801, 198937019774252928

Offset: 0

N. J. A. Sloane

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).

			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.

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.

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).

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

# 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:

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 *)

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

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
```
For Maple program see A034414.
```

A002289 Weight distribution of [ 23,12,7 ] binary perfect Golay code.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 253, 506, 0, 0, 1288, 1288, 0, 0, 506, 253, 0, 0, 0, 0, 0, 0, 1

Offset: 0

N. J. A. Sloane

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. 635.

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

Cf. A001380.

A105683 Weight distribution of [12,6,6]_3 ternary extended Golay code.

Original entry on oeis.org

1, 0, 264, 440, 24

Offset: 0

N. J. A. Sloane, May 06 2005

			Weight enumerator is x^12 + 264*x^6*y^6 + 440*x^3*y^9 + 24*y^12.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977.

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

Cf. A105676, A001380, A002289, A105684, A109141.

K:=FiniteField(3); C:=GolayCode(K,true); W:=WeightEnumerator(C); W;

A122192 Sum of the n-th powers of the roots of the (reduced) weight enumerator of the extended Golay code (1 + 759x^2 + 2576x^3 + 759*x^4 + x^6).

Original entry on oeis.org

6, 0, -1518, -7728, 1149126, 9775920, -851127150, -10374206304, 619950551814, 10059106207584, -443172509029998, -9223980220220304, 309985135145332422, 8134978519171135632, -211181377213616588526, -6965969413257227260608, 139095682365347347024902

Offset: 0

N. J. A. Sloane, Nov 12 2006

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (0,-759,-2576,-759,0,-1).

Cf. A001380, A123013.

Newt:=proc(f) local t1,t2,t3,t4; t1:=f; t2:=diff(f,x); t3:=expand(x^degree(t1,x)*subs(x=1/x,t1)); t4:=expand(x^degree(t2,x)*subs(x=1/x,t2)); factor(t4/t3); end;
g:=1+759*x^2+2576*x^3+759*x^4+x^6; Newt(g); series(%,x,60);

LinearRecurrence[{0,-759,-2576,-759,0,-1}, {6,0,-1518,-7728,1149126,9775920}, 30] (* G. C. Greubel, Jul 11 2021 *)

polsym(x^6 + 759*x^4 + 2576*x^3 + 759*x^2 + 1, 30) \\ Charles R Greathouse IV, Jul 20 2016

def A122192_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 6*(1+506*x^2+1288*x^3+253*x^4)/(1+759*x^2+2576*x^3+759*x^4 +x^6) ).list()
A122192_list(30) # G. C. Greubel, Jul 11 2021

G.f.: 6*(1 + 506*x^2 + 1288*x^3 + 253*x^4)/(1 + 759*x^2 + 2576*x^3 + 759*x^4 + x^6).

A018235 Weight distribution of (48,2^24,12) binary code obtained from Golay code of length 24 lifted to Z/4Z and mapped to GF(2)^2.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 12144, 61824, 195063, 1133440, 1445136, 4080384, 2921232, 4080384, 1445136, 1133440, 195063, 61824, 12144, 0, 0, 0, 0, 0, 1

Offset: 0

N. J. A. Sloane

K. Tanabe, A criterion for designs in Z_4-codes on the symmetrized weight enumerator, Des. Codes Cryptogr. 30 (2003), 169-185.

A. Bonnecaze and P. Sole, Quaternary Constructions of Formally Self-Dual Binary Codes and Unimodular Lattices, Lecture Notes in Computer Science, pages 194-206, 1994.
A. Bonnecaze, P. Sole and A. R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory, Vol. 41, pages 366-377, 1995.

Cf. A105547, A001380.

A105547 Hamming weight distribution of code obtained by lifting Golay code of length 24 to Z/4Z.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 759, 0, 12144, 0, 172592, 61824, 765072, 1133440, 1239447, 4080384, 1445136, 4080384, 1870176, 1133440, 692208, 61824, 28385

Offset: 0

N. J. A. Sloane, May 07 2005

S. T. Dougherty, S. Y. Kim and Y. H. Park, Lifted codes and their weight enumerators, Discrete Math., 305 (2005), 123-135.
K. Tanabe, A criterion for designs in Z_4-codes on the symmetrized weight enumerator, Des. Codes Cryptogr. 30 (2003), 169-185.

A. Bonnecaze and P. Sole, Quaternary Constructions of Formally Self-Dual Binary Codes and Unimodular Lattices, Lecture Notes in Computer Science, pages 194-206, 1994.
A. Bonnecaze, P. Sole and A. R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory, Vol. 41, pages 366-377, 1995.

Cf. A018235, A001380.

A105684 Weight distribution of [11,6,5]_3 ternary Golay perfect code.

Original entry on oeis.org

1, 0, 0, 0, 0, 132, 132, 0, 330, 110, 0, 24

Offset: 0

N. J. A. Sloane, May 06 2005

			Weight enumerator is x^11 + 132*x^6*y^5 + 132*x^5*y^6 + 330*x^3*y^8 + 110*x^2*y^9 + 24*y^11.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977.

Cf. A105676, A001380, A002289, A105683.

K:=FiniteField(3); C:=GolayCode(K,false); W:=WeightEnumerator(C); W;

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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A034414 Leading term in extremal weight enumerator of doubly-even binary self-dual code of length 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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Maple

A002289 Weight distribution of [ 23,12,7 ] binary perfect Golay code.

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A105683 Weight distribution of [12,6,6]_3 ternary extended Golay code.

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Magma

A122192 Sum of the n-th powers of the roots of the (reduced) weight enumerator of the extended Golay code (1 + 759*x^2 + 2576*x^3 + 759*x^4 + x^6).

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A018235 Weight distribution of (48,2^24,12) binary code obtained from Golay code of length 24 lifted to Z/4Z and mapped to GF(2)^2.

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A105547 Hamming weight distribution of code obtained by lifting Golay code of length 24 to Z/4Z.

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A105684 Weight distribution of [11,6,5]_3 ternary Golay perfect code.

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A122192 Sum of the n-th powers of the roots of the (reduced) weight enumerator of the extended Golay code (1 + 759x^2 + 2576x^3 + 759*x^4 + x^6).