The generator matrix

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

generates a code of length 15 over Z3[X]/(X^3) who´s minimum homogenous weight is 23.

Homogenous weight enumerator: w(x)=1x^0+132x^23+1602x^24+1872x^25+6102x^26+7758x^27+20202x^28+23616x^29+38908x^30+39564x^31+23844x^32+10600x^33+2466x^34+252x^35+170x^36+48x^37+8x^39+2x^42

The gray image is a linear code over GF(3) with n=135, k=11 and d=69.
This code was found by Heurico 1.16 in 30.7 seconds.