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

generates a code of length 55 over F8[X]/(X^2) who´s minimum homogenous weight is 360.

Homogenous weight enumerator: w(x)=1x^0+406x^360+854x^368+735x^376+511x^384+28672x^385+406x^392+441x^400+329x^408+210x^416+126x^424+63x^432+14x^440

The gray image is a linear code over GF(8) with n=440, k=5 and d=360.
This code was found by Heurico 1.16 in 0.557 seconds.