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

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+4x^86+88x^87+7x^88+24x^89+2x^90+1x^98+1x^106

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 0.453 seconds.