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

Author

N. J. A. Sloane

Keywords

• nonn
• fini
• full

Examples

			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
		

References

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

Links

• M. Terada, J. Asatani and T. Koumoto, Weight Distribution
• List of weight distributions

Crossrefs

Row 5 of A340030.

Programs

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 *)

PARI

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

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