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

generates a code of length 84 over Z2[X]/(X^4) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+70x^78+236x^79+305x^80+312x^81+426x^82+484x^83+513x^84+572x^85+406x^86+256x^87+193x^88+84x^89+94x^90+56x^91+22x^92+40x^93+8x^94+8x^95+5x^96+4x^98+1x^140

The gray image is a linear code over GF(2) with n=672, k=12 and d=312.
This code was found by Heurico 1.16 in 1.02 seconds.