The generator matrix

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

generates a code of length 77 over Z2[X]/(X^4) who´s minimum homogenous weight is 72.

Homogenous weight enumerator: w(x)=1x^0+135x^72+202x^73+587x^74+534x^75+650x^76+248x^77+536x^78+266x^79+448x^80+236x^81+136x^82+14x^83+40x^84+28x^85+15x^86+2x^87+6x^88+6x^89+2x^90+2x^94+1x^106+1x^110

The gray image is a linear code over GF(2) with n=616, k=12 and d=288.
This code was found by Heurico 1.16 in 0.719 seconds.