The generator matrix
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X X X X X X X 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 X X X X X X X 0 0 0 0 0 0 X 0 X 1 1 1 1 X X X 0 X 0 0 0
0 X 0 0 0 X X X 0 0 0 X 0 X X X 0 0 0 X 0 X X X 0 0 0 X X 0 0 X 0 X X X X X X 0 0 0 0 X X X 0 0 0 X 0 X X X 0 0 X X 0 X X X X X X 0 0 0 0 0 0 0 0 0 X X X X X X X X
0 0 X 0 X X X 0 0 0 X X X X 0 0 0 0 X X X X 0 0 0 0 X X 0 0 X X X X 0 0 0 X X X X 0 X X 0 0 0 0 X X X X 0 0 0 X X 0 X X 0 0 0 X X X X 0 0 0 0 0 X X X X 0 0 0 0 X X
0 0 0 X X 0 X X 0 X X 0 0 X X 0 0 X X 0 0 X X 0 0 X X 0 0 X X 0 0 X X 0 X X 0 0 X X 0 X X 0 0 X X 0 0 X X 0 X X 0 0 0 X X 0 X X 0 0 X 0 X 0 0 X X 0 0 X X 0 0 X X 0
generates a code of length 82 over Z2[X]/(X^2) who´s minimum homogenous weight is 84.
Homogenous weight enumerator: w(x)=1x^0+28x^84+2x^88+1x^96
The gray image is a linear code over GF(2) with n=164, k=5 and d=84.
As d=84 is an upper bound for linear (164,5,2)-codes, this code is optimal over Z2[X]/(X^2) for dimension 5.
This code was found by Heurico 1.16 in 0.117 seconds.