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

generates a code of length 87 over Z2[X]/(X^4) who´s minimum homogenous weight is 86.

Homogenous weight enumerator: w(x)=1x^0+6x^86+100x^87+7x^88+10x^91+1x^94+2x^99+1x^102

The gray image is a linear code over GF(2) with n=696, k=7 and d=344.
This code was found by Heurico 1.16 in 0.719 seconds.