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

generates a code of length 85 over Z2[X]/(X^4) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+413x^80+32x^81+524x^82+240x^83+688x^84+480x^85+594x^86+240x^87+493x^88+32x^89+208x^90+79x^92+50x^94+20x^96+1x^100+1x^144

The gray image is a linear code over GF(2) with n=680, k=12 and d=320.
This code was found by Heurico 1.16 in 120 seconds.