The generator matrix

 1  0  1  1  1 X^3+X^2+X  1  1  0  1 X^3+X^2+X  1  1  1  1 X^3  1 X^3+X  1  1  0  1  1 X^3+X  1  1  1 X^2+X  1 X^2  1  1  1 X^2  1  1 X^3+X^2+X  1  1 X^2+X  X  1  1 X^2  1  1  1  1 X^3+X^2  1  X  1 X^3+X  1  1  1 X^2  1  1  0  1  1 X^2 X^2  1  1  1  X  0  1  1  1  1 X^3  1 X^2  1
 0  1 X+1 X^3+X^2+X X^2+1  1 X^3+X^2+1  0  1 X^3+X^2+X  1 X+1 X^3+1 X^2+X+1 X^3  1 X^3+X  1 X^3+X^2+X+1  0  1  1 X^3+X  1 X^3+X^2+X+1 X^3+X^2+1 X^2  1 X^2+X  1 X+1 X^3+X^2+X X^2+1  1 X^3+X^2+X+1 X^2  1 X^3+1  X  1  1  1 X^2  1 X+1 X^3+X^2 X^3+X+1 X^3+X^2+1  1  X X^2  1  1 X^3+X^2+1 X^3+X^2+X+1 X^2+X  X X^3+X^2+1 X^2+1  1  0  0  1  1 X^3 X^3+1 X^3+X^2+X+1 X^3+X  1 X^2+1 X^2+X+1  1 X^3  X X^3+1 X^2 X+1
 0  0 X^2  0  0  0  0 X^2 X^3+X^2 X^3+X^2 X^2 X^3+X^2 X^3 X^2 X^3+X^2 X^2 X^3 X^3 X^2 X^3 X^3 X^3 X^2 X^3+X^2 X^3+X^2 X^2  0 X^2 X^2 X^3  0  0 X^3+X^2 X^3+X^2  0 X^3+X^2 X^3 X^3+X^2 X^3 X^2  0 X^2 X^3  0 X^3 X^2 X^3 X^2 X^2 X^3+X^2 X^3  0 X^3+X^2  0 X^3+X^2  0  0 X^3+X^2 X^3 X^3+X^2 X^2 X^3 X^3  0 X^3+X^2  0 X^3+X^2 X^3 X^2  0 X^3 X^2 X^3 X^2  0  0  0
 0  0  0 X^3+X^2 X^3 X^3+X^2 X^2 X^2 X^3+X^2 X^3  0 X^3+X^2  0 X^3  0 X^3 X^3 X^3 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^2 X^3+X^2 X^2  0 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^3  0 X^3+X^2 X^3 X^2 X^3 X^3 X^2 X^3+X^2 X^2 X^2 X^3 X^3  0  0 X^2 X^2 X^3+X^2 X^2  0 X^3+X^2 X^3 X^3 X^3+X^2 X^3+X^2 X^2 X^2 X^2 X^3 X^2 X^3 X^2 X^3 X^3 X^3 X^3+X^2 X^3 X^2 X^3+X^2  0 X^3+X^2  0  0 X^3+X^2 X^2 X^2  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+122x^72+324x^73+480x^74+472x^75+537x^76+490x^77+368x^78+504x^79+323x^80+208x^81+133x^82+28x^83+71x^84+14x^85+8x^86+4x^87+2x^89+1x^90+2x^92+2x^102+2x^105

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 0.593 seconds.