The generator matrix

 1  0  0  1  1  1  0 X^3+X^2 X^3+X^2 X^3+X^2  1  1  1  1  X  1 X^2+X  1  1 X^3+X^2+X  1 X^2+X  X  1  1  1 X^3+X^2+X  1  1 X^3+X^2+X  1  1 X^2+X  X  1  0 X^3+X^2  1 X^3 X^3+X  1  1  1  1 X^2+X  1  X X^2 X^2  1  1  1  1 X^2+X  1  1  1  1  1  1 X^3 X^3+X X^2  X X^2+X  1  1  1  1  X  1  1  1  1  1 X^2+X  1 X^2+X X^3+X^2+X X^3+X  0 X^2  1  1
 0  1  0  0 X^2+1 X^3+X^2+1  1  X  1  1 X^3+1 X^3+1 X^3+X^2 X^2 X^3+X^2+X X^2+X  1 X^3+X+1 X^2+X  1 X^2+X+1 X^3+X^2  1 X^3+X^2+X+1 X^2+X X^3+X^2+X  1 X^3+X^2+X+1  0 X^3 X+1 X^2+X+1  1  1 X+1  1 X^2+X X^2  1 X^3 X^3+X^2 X^3+1  X X^3+X^2+1  1 X^2+X  1 X^3+X^2  1  0 X^3+X^2+1 X^2+X X^3 X^3+X X^3+X+1 X^2+1 X^3+X X^3+X+1 X^2+X+1 X^3+X^2+X  1  1 X^2+X  1  1 X^3+X^2+X X^3  X  0  X X^2+1 X^2  1 X^2+1  1 X^3+X  X X^2+X  1  1  1  1 X+1 X^2
 0  0  1 X+1 X^2+X+1 X^2 X^2+X+1  1  X X^3+1 X^3+X^2+1 X^3+X  X X^2+1  1  0 X^3 X^3+X^2  1 X+1 X^3+X^2+X+1  1 X^3+X^2+X X^2+X  X X+1  1 X^2+1  1  1 X^2+1 X^3 X^3+X+1 X^3+X  X  1  1 X^3+X X^3+X^2  1  0 X+1 X^3+X+1 X^2  0 X^3+X^2+1 X^3  1  X X^3+X+1 X^3+X X^2+X  1  1 X^3+X^2  X  0 X^2+X X^3+X^2+X+1 X^3+X^2+X+1 X^2+X+1 X^3+X^2+1  1 X^2+X X^2+1 X^2+1 X^3+X^2+X X^3+X X^3+X^2+X  1  0 X^2 X+1  1 X^3+1  1  X  1 X+1  0  1 X^2+1 X+1  0
 0  0  0 X^2 X^2  0 X^2 X^3+X^2 X^2 X^3  0 X^2 X^2  0 X^3+X^2 X^3 X^3 X^3  0 X^3+X^2  0 X^3 X^2 X^2 X^3+X^2 X^2 X^3 X^3+X^2 X^3  0 X^3 X^2  0  0  0 X^3+X^2 X^2 X^3 X^2 X^3+X^2 X^3+X^2 X^3  0 X^3+X^2 X^3+X^2 X^2 X^2 X^2 X^3+X^2 X^3 X^3 X^3 X^2 X^3  0 X^3+X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^3 X^3+X^2 X^2  0 X^3  0 X^3+X^2  0 X^2 X^3+X^2  0 X^2 X^3+X^2  0 X^3 X^3+X^2 X^2  0 X^3 X^3  0  0 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+121x^78+852x^79+1259x^80+1858x^81+1665x^82+1862x^83+2013x^84+1854x^85+1339x^86+1300x^87+806x^88+576x^89+338x^90+314x^91+81x^92+58x^93+47x^94+24x^95+8x^96+6x^97+1x^98+1x^102

The gray image is a linear code over GF(2) with n=672, k=14 and d=312.
This code was found by Heurico 1.16 in 8.55 seconds.