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

generates a code of length 82 over Z2[X]/(X^4) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+97x^76+184x^77+340x^78+284x^79+590x^80+402x^81+534x^82+394x^83+428x^84+224x^85+282x^86+92x^87+79x^88+62x^89+58x^90+14x^91+13x^92+8x^93+9x^94+1x^126

The gray image is a linear code over GF(2) with n=656, k=12 and d=304.
This code was found by Heurico 1.16 in 0.953 seconds.