The generator matrix

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

generates a code of length 87 over Z2[X]/(X^4) who´s minimum homogenous weight is 83.

Homogenous weight enumerator: w(x)=1x^0+544x^83+343x^84+696x^85+314x^86+512x^87+298x^88+572x^89+292x^90+352x^91+23x^92+100x^93+2x^94+16x^95+2x^96+4x^97+16x^99+2x^100+4x^101+1x^104+2x^120

The gray image is a linear code over GF(2) with n=696, k=12 and d=332.
This code was found by Heurico 1.16 in 45.4 seconds.