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

generates a code of length 77 over Z2[X]/(X^4) who´s minimum homogenous weight is 73.

Homogenous weight enumerator: w(x)=1x^0+436x^73+444x^74+642x^75+353x^76+556x^77+372x^78+520x^79+293x^80+288x^81+50x^82+96x^83+8x^84+4x^85+4x^86+4x^87+8x^89+8x^90+2x^91+4x^93+1x^94+1x^100+1x^102

The gray image is a linear code over GF(2) with n=616, k=12 and d=292.
This code was found by Heurico 1.16 in 1.47 seconds.