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

generates a code of length 87 over Z2[X]/(X^4) who´s minimum homogenous weight is 81.

Homogenous weight enumerator: w(x)=1x^0+90x^81+197x^82+222x^83+380x^84+470x^85+501x^86+578x^87+498x^88+382x^89+259x^90+190x^91+112x^92+74x^93+55x^94+22x^95+28x^96+8x^97+12x^98+12x^99+4x^100+1x^144

The gray image is a linear code over GF(2) with n=696, k=12 and d=324.
This code was found by Heurico 1.16 in 1.12 seconds.