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

generates a code of length 56 over F8[X]/(X^2) who´s minimum homogenous weight is 344.

Homogenous weight enumerator: w(x)=1x^0+189x^344+980x^352+1967x^360+2842x^368+3472x^376+3584x^378+4270x^384+50176x^386+4473x^392+175616x^394+4592x^400+4235x^408+3227x^416+1778x^424+616x^432+126x^440

The gray image is a linear code over GF(8) with n=448, k=6 and d=344.
This code was found by Heurico 1.16 in 34.8 seconds.