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  X  1  1  1  1  1  1  1
 0 X^2  0  0  0  0 X^2 2X^2 2X^2  0  0 X^2 X^2 X^2 X^2 X^2 X^2 X^2  0 2X^2  0 2X^2  0 2X^2  0 X^2 X^2 2X^2 X^2  0 X^2  0 2X^2 2X^2 X^2  0 2X^2 X^2  0 X^2  0 X^2
 0  0 X^2  0  0 X^2 2X^2  0 2X^2  0 X^2 X^2 2X^2 2X^2  0 X^2  0 X^2 X^2 X^2 X^2  0  0 2X^2 2X^2 X^2 2X^2  0 X^2 2X^2 2X^2  0  0 2X^2  0 2X^2  0 X^2 2X^2 2X^2 2X^2  0
 0  0  0 X^2  0 2X^2 2X^2 X^2  0 X^2 X^2  0  0 X^2 2X^2 X^2 X^2 2X^2 2X^2  0  0 2X^2 2X^2 2X^2 2X^2 2X^2  0  0 X^2 X^2 X^2 X^2  0 2X^2  0 X^2 2X^2 2X^2  0 2X^2  0  0
 0  0  0  0 X^2 2X^2 2X^2 2X^2 2X^2 2X^2  0 2X^2  0  0 2X^2 2X^2  0 X^2  0  0 2X^2 2X^2 X^2 2X^2 X^2 2X^2 2X^2  0  0  0 X^2 X^2 X^2 X^2 X^2 X^2 X^2  0  0 X^2 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+82x^78+58x^81+162x^82+1458x^84+324x^85+58x^87+8x^90+20x^96+14x^99+2x^123

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 0.0438 seconds.