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

generates a code of length 55 over F8[X]/(X^2) who´s minimum homogenous weight is 344.

Homogenous weight enumerator: w(x)=1x^0+798x^344+1855x^352+2737x^360+448x^364+3577x^368+9408x^372+4011x^376+65856x^380+4746x^384+153664x^388+4669x^392+4368x^400+3374x^408+1876x^416+644x^424+105x^432+7x^440

The gray image is a linear code over GF(8) with n=440, k=6 and d=344.
This code was found by Heurico 1.16 in 34.3 seconds.