The generator matrix

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

generates a code of length 16 over Z3[X]/(X^3) who´s minimum homogenous weight is 26.

Homogenous weight enumerator: w(x)=1x^0+480x^26+616x^27+1896x^28+2718x^29+5858x^30+7272x^31+16992x^32+11426x^33+7308x^34+3570x^35+760x^36+48x^37+54x^38+46x^39+4x^42

The gray image is a linear code over GF(3) with n=144, k=10 and d=78.
This code was found by Heurico 1.16 in 1.36 seconds.