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

generates a code of length 89 over Z2[X]/(X^4) who´s minimum homogenous weight is 85.

Homogenous weight enumerator: w(x)=1x^0+68x^85+88x^86+112x^87+180x^88+180x^89+158x^90+96x^91+64x^92+28x^93+8x^94+16x^95+10x^96+12x^97+2x^106+1x^128

The gray image is a linear code over GF(2) with n=712, k=10 and d=340.
This code was found by Heurico 1.16 in 0.625 seconds.