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

generates a code of length 86 over Z2[X]/(X^4) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+54x^82+8x^83+136x^84+184x^85+278x^86+184x^87+114x^88+8x^89+34x^90+19x^92+2x^94+1x^96+1x^156

The gray image is a linear code over GF(2) with n=688, k=10 and d=328.
This code was found by Heurico 1.16 in 0.563 seconds.