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

generates a code of length 88 over F8[X]/(X^2) who´s minimum homogenous weight is 584.

Homogenous weight enumerator: w(x)=1x^0+161x^584+609x^592+763x^600+448x^602+721x^608+6272x^610+399x^616+21952x^618+413x^624+196x^632+224x^640+224x^648+119x^656+119x^664+84x^672+42x^680+21x^688

The gray image is a linear code over GF(8) with n=704, k=5 and d=584.
This code was found by Heurico 1.16 in 1.05 seconds.