The generator matrix

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

generates a code of length 83 over Z2[X]/(X^4) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+198x^79+654x^80+684x^81+636x^82+468x^83+378x^84+306x^85+268x^86+130x^87+114x^88+118x^89+33x^90+36x^91+40x^92+8x^93+13x^94+4x^95+4x^96+2x^98+1x^100

The gray image is a linear code over GF(2) with n=664, k=12 and d=316.
This code was found by Heurico 1.16 in 1.78 seconds.