The generator matrix

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

generates a code of length 76 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+152x^72+690x^73+668x^74+720x^75+362x^76+390x^77+292x^78+280x^79+187x^80+132x^81+65x^82+108x^83+24x^84+16x^85+5x^86+1x^88+2x^90+1x^92

The gray image is a linear code over GF(2) with n=608, k=12 and d=288.
This code was found by Heurico 1.16 in 0.328 seconds.