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

generates a code of length 73 over Z2[X]/(X^4) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+376x^69+439x^70+712x^71+386x^72+612x^73+380x^74+460x^75+196x^76+272x^77+99x^78+96x^79+22x^80+12x^81+6x^82+4x^83+1x^84+8x^85+4x^86+8x^87+1x^92+1x^96

The gray image is a linear code over GF(2) with n=584, k=12 and d=276.
This code was found by Heurico 1.16 in 9.92 seconds.