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

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+170x^81+91x^82+342x^83+279x^84+370x^85+675x^86+386x^87+656x^88+334x^89+265x^90+236x^91+48x^92+130x^93+11x^94+52x^95+6x^96+16x^97+12x^98+6x^99+4x^101+2x^102+2x^103+1x^104+1x^140

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 62.2 seconds.