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

generates a code of length 84 over F8[X]/(X^2) who´s minimum homogenous weight is 560.

Homogenous weight enumerator: w(x)=1x^0+322x^560+882x^568+672x^576+553x^584+28672x^588+413x^592+343x^600+266x^608+217x^616+126x^624+119x^632+84x^640+70x^648+21x^656+7x^672

The gray image is a linear code over GF(8) with n=672, k=5 and d=560.
This code was found by Heurico 1.16 in 0.978 seconds.