The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+136x^86+288x^87+186x^88+288x^89+104x^90+18x^92+1x^108+1x^116+1x^120

The gray image is a linear code over GF(2) with n=704, k=10 and d=344.
This code was found by Heurico 1.16 in 0.844 seconds.