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

generates a code of length 42 over Z3[X]/(X^3) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+398x^78+108x^80+630x^81+810x^82+432x^83+1406x^84+1620x^85+432x^86+292x^87+220x^90+142x^93+66x^96+2x^99+2x^117

The gray image is a linear code over GF(3) with n=378, k=8 and d=234.
This code was found by Heurico 1.16 in 0.35 seconds.