The generator matrix

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

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+133x^72+616x^73+1168x^74+1010x^75+1198x^76+944x^77+767x^78+632x^79+563x^80+384x^81+368x^82+194x^83+126x^84+40x^85+23x^86+20x^87+3x^88+2x^90

The gray image is a linear code over GF(2) with n=616, k=13 and d=288.
This code was found by Heurico 1.16 in 4.69 seconds.