The generator matrix

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

generates a code of length 83 over F8[X]/(X^2) who´s minimum homogenous weight is 568.

Homogenous weight enumerator: w(x)=1x^0+539x^568+1344x^569+9408x^573+1624x^576+2912x^577+2688x^581+735x^584+896x^585+9408x^589+1155x^592+2016x^593+42x^600

The gray image is a linear code over GF(8) with n=664, k=5 and d=568.
This code was found by Heurico 1.16 in 0.387 seconds.