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  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  1  1  1  1  1  1  1  1 X^3  1  1  1  1  1  1  1  1 X^2  1  1  1  0  1  X  1
 0  X  0 X^2+X X^2 X^3+X^2+X X^3+X^2  X  0 X^2+X X^3+X^2 X^3+X X^2+X X^3  X X^3+X^2 X^3 X^2+X X^2  X  0 X^3+X^2+X X^3+X^2 X^3+X X^3 X^3+X^2+X X^3+X^2  X X^3+X^2 X^2+X X^3  X X^2+X  0  0  X X^3+X X^2 X^3+X^2+X X^3+X^2 X^3+X^2+X  0  0 X^3+X  X X^2 X^3+X^2+X X^3+X^2 X^3+X^2 X^3+X^2+X X^3+X X^3+X^2  X  X X^2 X^2 X^3+X^2+X X^3+X  0 X^3 X^3  0 X^3+X^2+X X^2+X X^3 X^3+X^2+X  0 X^3 X^3  0 X^2+X X^2+X X^2+X X^2 X^3 X^2 X^2+X  X  X X^3+X^2+X X^3
 0  0 X^3+X^2  0 X^2 X^2  0 X^2  0  0  0  0 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^2 X^3 X^3+X^2 X^2 X^3 X^2 X^3 X^3 X^3 X^3 X^3 X^3 X^3+X^2 X^3+X^2 X^2 X^3+X^2  0  0 X^3+X^2 X^2 X^3+X^2 X^3+X^2  0 X^3 X^2  0 X^3+X^2  0 X^3  0 X^3+X^2 X^2 X^3+X^2 X^3+X^2 X^3  0  0 X^2 X^3 X^2 X^2 X^3+X^2 X^3 X^2 X^3 X^2  0 X^3  0 X^3 X^3 X^3 X^2 X^2 X^3 X^3+X^2 X^2 X^2 X^3 X^3+X^2  0  0 X^2 X^3+X^2 X^3+X^2
 0  0  0 X^3  0  0  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3  0  0  0  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3  0  0 X^3  0  0  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3  0 X^3  0  0 X^3 X^3 X^3
 0  0  0  0 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3  0  0 X^3  0  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0  0  0  0 X^3 X^3  0  0  0 X^3 X^3  0 X^3  0 X^3 X^3  0  0  0 X^3 X^3 X^3  0 X^3  0

generates a code of length 81 over Z2[X]/(X^4) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+28x^76+124x^77+133x^78+236x^79+435x^80+328x^81+319x^82+160x^83+72x^84+100x^85+27x^86+52x^87+8x^88+24x^89+1x^154

The gray image is a linear code over GF(2) with n=648, k=11 and d=304.
This code was found by Heurico 1.16 in 0.875 seconds.