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

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+136x^78+152x^79+308x^80+248x^81+576x^82+240x^83+874x^84+240x^85+507x^86+248x^87+254x^88+152x^89+98x^90+23x^92+23x^94+10x^96+4x^98+1x^100+1x^144

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 1.34 seconds.