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
 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^4*X

generates a code of length 44 over F8[X]/(X^2) who´s minimum homogenous weight is 304.

Homogenous weight enumerator: w(x)=1x^0+455x^304+1680x^305+336x^306+490x^312+1008x^313+112x^314+14x^320

The gray image is a linear code over GF(8) with n=352, k=4 and d=304.
This code was found by Heurico 1.16 in 0.0101 seconds.