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

generates a code of length 74 over Z2[X]/(X^4) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+296x^69+188x^70+482x^71+366x^72+618x^73+499x^74+490x^75+278x^76+364x^77+150x^78+174x^79+37x^80+86x^81+11x^82+34x^83+4x^84+12x^85+4x^87+1x^88+1x^112

The gray image is a linear code over GF(2) with n=592, k=12 and d=276.
This code was found by Heurico 1.16 in 99 seconds.