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

generates a code of length 44 over F8[X]/(X^2) who´s minimum homogenous weight is 280.

Homogenous weight enumerator: w(x)=1x^0+112x^280+756x^288+686x^296+3584x^301+651x^304+25088x^309+595x^312+448x^320+448x^328+245x^336+119x^344+35x^352

The gray image is a linear code over GF(8) with n=352, k=5 and d=280.
This code was found by Heurico 1.16 in 0.433 seconds.