A010082 Weight distribution of extended Hamming code of length 64.

1, 0, 10416, 1166592, 69194232, 2366570752, 51316746768, 747741998592, 7633243745820, 56276359749120, 306558278858160, 1255428754917120, 3916392495228360, 9399341113166592, 17480786291963792, 25316999607653376, 28634752793916486, 25316999607653376

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• fini
• full
• changed

Comments

Weight enumerator of [64,57,4] Reed-Muller code RM(4,6).

Examples

			69194232*x^8*y^56 + 25316999607653376*x^30*y^34 + 747741998592*x^14*y^50 + 9399341113166592*x^26*y^38 + 1255428754917120*x^22*y^42 + 306558278858160*x^20*y^44 + 7633243745820*x^16*y^48 + 56276359749120*x^46*y^18 + 306558278858160*x^44*y^20 + y^64 + 28634752793916486*x^32*y^32 + 17480786291963792*x^36*y^28 + 1255428754917120*x^42*y^22 + 17480786291963792*x^28*y^36 + 56276359749120*x^18*y^46 + 10416*x^60*y^4 + 1166592*x^58*y^6 + 1166592*x^6*y^58 + 10416*x^4*y^60 + 51316746768*x^12*y^52 + 9399341113166592*x^38*y^26 + 25316999607653376*x^34*y^30 + 2366570752*x^10*y^54 + 3916392495228360*x^24*y^40 + 51316746768*x^52*y^12 + 747741998592*x^50*y^14 + 7633243745820*x^48*y^16 + 3916392495228360*x^40*y^24 + x^64 + 2366570752*x^54*y^10 + 69194232*x^56*y^8
The weight distribution is:
  i A_i
  0 1
  4 10416
  6 1166592
  8 69194232
  10 2366570752
  12 51316746768
  14 747741998592
  16 7633243745820
  18 56276359749120
  20 306558278858160
  22 1255428754917120
  24 3916392495228360
  26 9399341113166592
  28 17480786291963792
  30 25316999607653376
  32 28634752793916486
  34 25316999607653376
  36 17480786291963792
  38 9399341113166592
  40 3916392495228360
  42 1255428754917120
  44 306558278858160
  46 56276359749120
  48 7633243745820
  50 747741998592
  52 51316746768
  54 2366570752
  56 69194232
  58 1166592
  60 10416
  64 1
		

References

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

Links

• Georg Fischer, Table of n, a(n) for n = 0..32
• M. Terada, J. Asatani and T. Koumoto, Weight Distribution

Programs

Mathematica

m:=63; rt=RecurrenceTable[{n*a[n]==Binomial[m, n-1]-a[n-1]-(m-n+2)*a[n-2], a[0]==1, a[1]==0}, a, {n, 0, m}]; Join[{1}, Table[rt[[i]]+rt[[i+1]], {i, 2, m, 2}], {1}] (* Georg Fischer, Jul 16 2020 *)