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

generates a code of length 53 over F8[X]/(X^2) who´s minimum homogenous weight is 344.

Homogenous weight enumerator: w(x)=1x^0+217x^344+805x^352+742x^360+637x^368+28672x^371+434x^376+483x^384+280x^392+245x^400+161x^408+77x^416+14x^424

The gray image is a linear code over GF(8) with n=424, k=5 and d=344.
This code was found by Heurico 1.16 in 0.543 seconds.