The generator matrix

 1  0  1  1  1 X^3+X^2+X  1  X  1 X^3  1  1 X^2  1  1  1 X^2+X  1  1 X^3+X^2  1 X^3+X  1  1  1  1  1 X^3+X  1  0  1  1 X^3+X^2  1 X^3+X^2+X  1 X^2+X  1  X  1 X^3  1  1  0  1  1  1  X  1  1  1  1  1  1  X  1 X^2  1  1 X^3+X^2+X X^2  1  1 X^3+X  0  1  1  1  X X^3+X  1  X X^3+X^2 X^3+X^2  X  0  1  1
 0  1 X+1 X^2+X X^3+X^2+1  1 X^3+X^2  1 X^2+X+1  1 X^3+X  1  1 X^3 X+1 X^3+X^2+X  1 X^3+X^2+X+1 X^2  1  X  1 X+1 X^3+X^2+X+1 X^2+1 X^3+1  0  1 X^2+1  1 X^3+X^2+X X^3+X+1  1 X^3+1  1 X^3  1 X^3+X^2+1  1 X^3+X^2+X+1  1  X X^2+1  1 X^3+X X^2 X^2 X^3+X^2+X X^3+X X^3+X  0 X^3+X^2+X  X X^2 X^3+X X^3+X^2+X  1 X^3+1 X+1  1  1  1 X^3+X^2+X+1  1  X X^2+1 X^2+X+1 X^2+X  0  1 X^3+X^2+1  0  1  1  0  1 X+1 X^3+X^2+X
 0  0 X^2  0 X^3+X^2 X^2  0 X^2 X^3+X^2 X^3+X^2  0 X^2 X^3+X^2 X^2 X^3 X^3+X^2  0 X^3 X^2  0 X^3+X^2  0 X^3 X^3  0 X^3  0  0 X^3  0 X^3+X^2 X^2  0 X^3 X^3+X^2 X^3+X^2 X^3 X^2 X^2 X^3+X^2 X^2  0 X^2 X^2 X^2 X^3+X^2 X^3 X^2  0 X^3 X^3+X^2  0 X^3+X^2 X^2 X^3 X^2 X^3+X^2  0 X^2 X^3  0 X^2 X^3 X^2 X^3 X^3  0  0 X^3+X^2  0 X^3 X^3+X^2 X^3 X^2 X^3 X^2 X^3+X^2 X^3+X^2
 0  0  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0  0  0 X^3  0  0  0  0 X^3  0  0 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0  0 X^3  0 X^3  0 X^3  0 X^3 X^3 X^3
 0  0  0  0 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3  0 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0  0  0  0 X^3  0  0  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3  0 X^3  0  0  0 X^3 X^3 X^3  0 X^3  0  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 73.

Homogenous weight enumerator: w(x)=1x^0+88x^73+441x^74+386x^75+550x^76+396x^77+625x^78+272x^79+566x^80+246x^81+271x^82+90x^83+78x^84+32x^85+23x^86+18x^87+2x^88+2x^93+2x^95+3x^96+2x^97+2x^101

The gray image is a linear code over GF(2) with n=624, k=12 and d=292.
This code was found by Heurico 1.16 in 0.75 seconds.