The generator matrix

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

generates a code of length 53 over F8[X]/(X^2) who´s minimum homogenous weight is 368.

Homogenous weight enumerator: w(x)=1x^0+567x^368+2016x^369+392x^370+378x^376+672x^377+56x^378+14x^384

The gray image is a linear code over GF(8) with n=424, k=4 and d=368.
This code was found by Heurico 1.16 in 0.0178 seconds.