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

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+385x^76+88x^77+514x^78+296x^79+600x^80+528x^81+592x^82+272x^83+353x^84+88x^85+154x^86+8x^87+153x^88+52x^90+6x^92+5x^96+1x^128

The gray image is a linear code over GF(2) with n=648, k=12 and d=304.
This code was found by Heurico 1.16 in 9.41 seconds.