The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 X^3  1  X  1  1  1  1  1  1  1  0  1  1  X  1  1 X^3+X^2  1  1 X^2  1  X  X  1 X^3+X^2  1  1  1 X^3 X^3+X^2  X  1  1  0  1  1
 0  X  0 X^3+X^2+X X^2 X^2+X X^3+X^2  X  0 X^2+X X^3 X^2+X X^3+X X^2 X^2  X  0 X^2+X X^2 X^3+X X^3 X^3+X X^3 X^3+X X^2+X  0 X^3  X X^2+X X^3 X^2+X  0 X^2 X^3+X X^3+X^2+X  0  0  X X^3+X^2 X^3+X  X  X X^3 X^2+X X^3+X^2+X X^2+X X^3+X^2  0 X^2 X^3+X^2+X X^3+X^2+X  X X^2 X^2 X^3+X^2+X  X X^3+X^2+X  X X^2 X^2 X^3 X^2 X^2+X  X  X  X X^3 X^3+X X^3+X^2  X  X X^3+X^2+X  X  X  X  X X^3
 0  0 X^3+X^2  0 X^2  0 X^3  0 X^2 X^2 X^3 X^3+X^2 X^3+X^2 X^3+X^2  0 X^2  0 X^3+X^2 X^3 X^3+X^2 X^2 X^3 X^2  0  0  0 X^2  0 X^2 X^3 X^2 X^2 X^3 X^2 X^3  0 X^3  0 X^2 X^3 X^3 X^2  0 X^2 X^2 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^3+X^2  0 X^2 X^3  0 X^2 X^2 X^3 X^3+X^2 X^3+X^2 X^3+X^2 X^2 X^2  0 X^2 X^3+X^2  0 X^2 X^3+X^2 X^3  0 X^2 X^3 X^3  0  0 X^2 X^3+X^2
 0  0  0 X^3+X^2  0 X^3 X^3 X^2 X^2 X^2 X^2  0  0 X^2 X^3+X^2 X^2 X^3 X^3+X^2 X^3+X^2 X^3  0 X^3+X^2 X^3+X^2 X^3  0 X^3+X^2  0 X^3+X^2 X^3  0 X^2 X^2 X^3 X^3+X^2 X^2 X^2 X^2  0  0 X^2  0 X^3  0 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^3 X^3+X^2 X^3  0 X^3+X^2 X^3+X^2 X^2 X^3  0  0 X^3+X^2 X^3  0 X^3 X^2 X^3  0  0  0 X^3+X^2 X^2  0 X^2 X^3 X^2 X^2 X^2 X^3+X^2 X^2 X^3
 0  0  0  0 X^3 X^3 X^3 X^3  0  0  0 X^3  0 X^3 X^3 X^3  0  0 X^3  0  0 X^3  0  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3  0  0  0  0 X^3 X^3  0 X^3  0 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 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3  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 71.

Homogenous weight enumerator: w(x)=1x^0+130x^71+105x^72+336x^73+240x^74+520x^75+433x^76+656x^77+450x^78+496x^79+211x^80+252x^81+44x^82+128x^83+39x^84+24x^85+2x^86+6x^87+10x^88+12x^89+1x^128

The gray image is a linear code over GF(2) with n=616, k=12 and d=284.
This code was found by Heurico 1.16 in 1.27 seconds.