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

generates a code of length 85 over Z2[X]/(X^4) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+170x^79+65x^80+360x^81+186x^82+564x^83+274x^84+954x^85+260x^86+560x^87+179x^88+264x^89+26x^90+128x^91+21x^92+52x^93+8x^94+14x^95+2x^96+4x^99+1x^100+2x^101+1x^144

The gray image is a linear code over GF(2) with n=680, k=12 and d=316.
This code was found by Heurico 1.16 in 114 seconds.