The generator matrix

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

generates a code of length 78 over Z2[X]/(X^4) who´s minimum homogenous weight is 74.

Homogenous weight enumerator: w(x)=1x^0+444x^74+408x^75+690x^76+344x^77+536x^78+440x^79+424x^80+296x^81+328x^82+48x^83+78x^84+24x^86+20x^88+12x^90+1x^96+2x^104

The gray image is a linear code over GF(2) with n=624, k=12 and d=296.
This code was found by Heurico 1.16 in 1.09 seconds.