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

generates a code of length 87 over F8[X]/(X^2) who´s minimum homogenous weight is 576.

Homogenous weight enumerator: w(x)=1x^0+119x^576+560x^584+777x^592+679x^600+3584x^602+455x^608+25088x^610+364x^616+308x^624+259x^632+196x^640+112x^648+105x^656+84x^664+49x^672+14x^680+14x^688

The gray image is a linear code over GF(8) with n=696, k=5 and d=576.
This code was found by Heurico 1.16 in 1.03 seconds.