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  X  1  1  1  1  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 2X X^2 2X^2+X X^2+X X^2+2X  X X^2  X 2X^2  X 2X X^2+2X X^2+2X 2X^2+2X  0  0 X^2 X^2 2X^2+X 2X^2+X X^2+X X^2+X 2X^2+X 2X X^2+X 2X^2 2X^2+2X 2X^2+X  0
 0  0 X^2  0 X^2  0 2X^2 2X^2 X^2 2X^2 X^2 2X^2 2X^2  0  0 X^2  0 X^2 X^2 X^2 2X^2 2X^2  0 X^2 2X^2  0 2X^2  0 X^2 X^2  0 2X^2 X^2  0 2X^2  0 2X^2 X^2  0 2X^2 2X^2 X^2
 0  0  0 X^2 2X^2 X^2 X^2 2X^2 2X^2 X^2  0 2X^2 2X^2  0 2X^2  0 2X^2 X^2 X^2 X^2  0  0 X^2 2X^2 X^2 2X^2  0 2X^2 X^2  0  0  0 X^2  0 X^2 2X^2 X^2 X^2  0 X^2 2X^2 2X^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+48x^78+192x^80+58x^81+372x^83+1030x^84+324x^86+30x^87+60x^89+16x^90+24x^92+24x^93+6x^96+2x^120

The gray image is a linear code over GF(3) with n=378, k=7 and d=234.
This code was found by Heurico 1.16 in 0.0408 seconds.