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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 88.

Homogenous weight enumerator: w(x)=1x^0+16x^88+10x^89+2x^90+2x^91+1x^92

The gray image is a linear code over GF(2) with n=688, k=5 and d=352.
This code was found by Heurico 1.16 in 0.437 seconds.