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  X  1  1  1  1
 0  X  0  0  X  X a*X a^3*X  0 2*X a^2*X  X a^2*X a*X a^2*X a^5*X a*X  X a*X 2*X a^6*X a^2*X  0 a*X  0  X 2*X a^6*X a^2*X a^2*X 2*X a^3*X  X a^5*X a^6*X a^6*X a^5*X a^6*X  X a^6*X a^5*X 2*X a*X 2*X a*X a*X  X a^2*X a^6*X a*X a^5*X
 0  0  X  0 a^7*X a^6*X a*X 2*X  X a^6*X a^6*X  X a^3*X  X a^5*X  X a^2*X a^6*X a^5*X a^6*X a^5*X a^7*X a^5*X 2*X a^2*X  0  0  0 a^2*X 2*X 2*X a^2*X a^6*X a*X a^6*X a^7*X a*X a^6*X a*X a^5*X a^3*X  X a^7*X a*X a^5*X  X  X 2*X  X a^7*X  0
 0  0  0  X a^7*X  X a^5*X a*X a*X a^7*X 2*X a^7*X a^3*X a^2*X 2*X a^3*X a^6*X 2*X  X a^6*X a^6*X a*X 2*X  0  X a^6*X a^5*X a^6*X a^3*X  0 2*X a*X a^5*X a^3*X a*X a*X  0 a^7*X a^5*X a^2*X a^7*X 2*X a^7*X  X a^5*X a^3*X  0 2*X a^2*X a^2*X a^5*X

generates a code of length 51 over F9[X]/(X^2) who´s minimum homogenous weight is 378.

Homogenous weight enumerator: w(x)=1x^0+432x^378+1184x^387+1096x^396+5832x^400+976x^405+46656x^409+928x^414+672x^423+640x^432+408x^441+176x^450+48x^459

The gray image is a linear code over GF(9) with n=459, k=5 and d=378.
This code was found by Heurico 1.16 in 1.01 seconds.