The generator matrix

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

generates a code of length 40 over Z3[X]/(X^3) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+42x^78+36x^79+540x^80+86x^81+18x^82+6x^93

The gray image is a linear code over GF(3) with n=360, k=6 and d=234.
This code was found by Heurico 1.16 in 0.0102 seconds.