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  X  1  1  X  X  0  X  1
 0  X 2X  0 2X^2+X 2X 2X^2+X X^2+2X  0 X^2  0 X^2 2X^2+X X^2+X X^2 2X 2X^2 X^2+2X 2X^2+X X^2+2X X^2+X X^2+2X X^2+X 2X^2+2X X^2+X 2X X^2 2X^2+X 2X^2+X  X 2X^2 X^2+X  0  0 2X^2 X^2 2X^2+X 2X  X 2X 2X
 0  0 X^2  0 X^2  0 2X^2 2X^2 X^2 2X^2 X^2 2X^2 2X^2  0 X^2  0 2X^2 X^2 X^2 2X^2  0  0 X^2 2X^2 2X^2 X^2  0 2X^2 X^2  0  0  0 2X^2  0 X^2 X^2 X^2  0 X^2  0 2X^2
 0  0  0 X^2 2X^2 X^2 X^2 2X^2 2X^2 X^2  0 2X^2 2X^2 2X^2 X^2 2X^2  0 2X^2  0 X^2  0 X^2  0 X^2 2X^2 2X^2 2X^2  0 X^2 X^2  0 X^2 X^2 X^2 2X^2  0  0 X^2 X^2  0  0

generates a code of length 41 over Z3[X]/(X^3) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+310x^78+324x^80+414x^81+648x^83+354x^84+64x^87+42x^90+24x^93+4x^96+2x^108

The gray image is a linear code over GF(3) with n=369, k=7 and d=234.
This code was found by Heurico 1.16 in 20.3 seconds.