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

generates a code of length 89 over Z2[X]/(X^4) who´s minimum homogenous weight is 83.

Homogenous weight enumerator: w(x)=1x^0+88x^83+189x^84+284x^85+418x^86+436x^87+427x^88+540x^89+540x^90+376x^91+265x^92+160x^93+130x^94+108x^95+43x^96+36x^97+16x^98+16x^99+2x^100+4x^101+16x^102+1x^144

The gray image is a linear code over GF(2) with n=712, k=12 and d=332.
This code was found by Heurico 1.16 in 1.2 seconds.