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

generates a code of length 89 over Z2[X]/(X^4) who´s minimum homogenous weight is 86.

Homogenous weight enumerator: w(x)=1x^0+82x^86+220x^87+172x^88+180x^89+107x^90+148x^91+57x^92+12x^93+12x^94+16x^95+9x^96+6x^98+1x^100+1x^130

The gray image is a linear code over GF(2) with n=712, k=10 and d=344.
This code was found by Heurico 1.16 in 0.531 seconds.