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  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  1  1  1  1  1  1  1  1  1  X  1  1  1  1  1  X  1  1
 0  X  0  0  X  X a*X a^2*X a^4*X a^5*X a^5*X  X a^5*X a^6*X  0 a*X a^5*X a*X a^5*X  0 a^4*X a*X a*X a^2*X a*X  X a^4*X a^2*X  X a^3*X a^6*X a^4*X  0  0 a^4*X a^2*X a^2*X a^6*X a^2*X a^3*X a^2*X a^5*X a^2*X a^5*X a^6*X a^4*X a*X a^4*X  X a^4*X  X a^5*X a*X  X a^3*X  X  0 a^2*X a^2*X a*X a^5*X a*X a^4*X a^3*X a^4*X a^2*X a^6*X a^6*X a^5*X  X a^3*X a^4*X a^5*X a^4*X  0  0  X  X  0 a^4*X  0  X  0  0 a^2*X a^3*X
 0  0  X  0 a^6*X a^5*X a*X a^6*X a^6*X a*X a^6*X a^2*X a*X a^3*X  X a^5*X  X  X a^4*X a^3*X  X a*X a^4*X  0 a^2*X  0  0 a^2*X a^2*X a^6*X  X a^5*X a^4*X a^6*X  X a^2*X  0 a^6*X a^2*X a^5*X  X a^2*X a^6*X a^5*X a^4*X a^5*X a^4*X a*X a^5*X  0 a^6*X a^2*X  X  0 a^3*X a^3*X a*X a^4*X a^4*X a^5*X a^4*X a^4*X  0 a^3*X a^3*X a^6*X a^4*X a^6*X  0 a^4*X a*X a^2*X  X a^2*X a^3*X a*X a^6*X a^2*X  0 a^2*X a*X a^6*X a*X  X a^6*X  0
 0  0  0  X  X a^4*X a^4*X  X a^6*X a^4*X a^6*X a^6*X a^3*X a*X a^4*X a*X a^3*X a^5*X a^6*X a^2*X a^4*X  X  0 a^5*X  0 a^5*X a*X a^6*X a^3*X  0 a^3*X a^6*X a^3*X a^4*X a^2*X a*X  0 a^4*X a^2*X  X a^2*X a^3*X  0 a^5*X a^3*X a^5*X a^2*X a^5*X a^3*X a^4*X a^6*X  0 a^2*X a*X a^2*X  X a^3*X a*X a^6*X a^3*X a^4*X a^3*X a^5*X a^5*X a^4*X a*X a^4*X  X a^4*X a^3*X  0  X a*X a^5*X  0 a*X a^2*X  X a^4*X a^2*X a^2*X a^5*X  0 a^3*X a^4*X a^6*X

generates a code of length 86 over F8[X]/(X^2) who´s minimum homogenous weight is 576.

Homogenous weight enumerator: w(x)=1x^0+595x^576+707x^584+448x^588+679x^592+6272x^596+595x^600+21952x^604+441x^608+294x^616+196x^624+175x^632+161x^640+84x^648+70x^656+42x^664+42x^672+7x^680+7x^688

The gray image is a linear code over GF(8) with n=688, k=5 and d=576.
This code was found by Heurico 1.16 in 1.36 seconds.