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

generates a code of length 77 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+510x^72+64x^73+416x^74+256x^75+624x^76+384x^77+512x^78+256x^79+606x^80+64x^81+352x^82+48x^84+2x^88+1x^128

The gray image is a linear code over GF(2) with n=616, k=12 and d=288.
This code was found by Heurico 1.16 in 0.797 seconds.