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

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+276x^78+54x^79+108x^80+702x^81+216x^82+216x^83+432x^84+54x^85+42x^87+60x^90+18x^93+6x^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 0.0327 seconds.