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

generates a code of length 89 over Z2[X]/(X^4) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+399x^84+88x^85+580x^86+224x^87+630x^88+384x^89+624x^90+256x^91+461x^92+72x^93+236x^94+60x^96+56x^98+15x^100+8x^102+1x^104+1x^148

The gray image is a linear code over GF(2) with n=712, k=12 and d=336.
This code was found by Heurico 1.16 in 3.81 seconds.