The generator matrix

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

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+144x^73+362x^74+462x^75+450x^76+492x^77+460x^78+418x^79+455x^80+330x^81+256x^82+152x^83+38x^84+40x^85+22x^86+6x^87+2x^98+2x^99+2x^102+2x^105

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 0.657 seconds.