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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+206x^82+742x^83+544x^84+620x^85+496x^86+430x^87+256x^88+288x^89+128x^90+152x^91+57x^92+72x^93+55x^94+32x^95+13x^96+1x^98+2x^102+1x^104

The gray image is a linear code over GF(2) with n=688, k=12 and d=328.
This code was found by Heurico 1.16 in 1.86 seconds.