A010086 Weight distribution of d=3 Hamming code of length 31.

1, 0, 0, 155, 1085, 5208, 22568, 82615, 247845, 628680, 1383096, 2648919, 4414865, 6440560, 8280720, 9398115, 9398115, 8280720, 6440560, 4414865, 2648919, 1383096, 628680, 247845, 82615, 22568, 5208, 1085, 155, 0, 0, 1

Offset

0,4

References

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 129.

Links

Table of n, a(n) for n=0..31.

M. Terada, J. Asatani and T. Koumoto, Weight Distribution

List of weight distributions

Formula

Recurrence: a(n) = (binomial(m,n-1) - a(n-1) - (m-n+2)*a(n-2))/n for n > 1, a(0)=1, a(1)=0 with m = 31. - Georg Fischer, Apr 14 2020

Example

Weight distribution:
i A_i
0 1
3 155
4 1085
5 5208
6 22568
7 82615
8 247845
9 628680
10 1383096
11 2648919
12 4414865
13 6440560
14 8280720
15 9398115
16 9398115
17 8280720
18 6440560
19 4414865
20 2648919
21 1383096
22 628680
23 247845
24 82615
25 22568
26 5208
27 1085
28 155
31 1

Mathematica

m:=31; RecurrenceTable[{a[n]==(Binomial[m, n-1]-a[n-1]-(m-n+2)*a[n-2])/n,
a[0]==1, a[1]==0}, a, {n, 0, 127}] (* Georg Fischer, Apr 14 2020 *)

Prog

(PARI) Vecrev((1+x)^31 + 31*(1-x)*(1-x^2)^15)/32 \\ Andrew Howroyd, Jan 11 2021

Crossrefs

Row 5 of A340030.

Sequence in context: A224336 A218328 A101535 * A110834 A110842 A289300

Adjacent sequences: A010083 A010084 A010085 * A010087 A010088 A010089

Keyword

nonn,fini,full

Author

N. J. A. Sloane

Status

approved