The generator matrix

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

generates a code of length 32 over Z3[X]/(X^3) who´s minimum homogenous weight is 55.

Homogenous weight enumerator: w(x)=1x^0+864x^55+1266x^56+4934x^57+9084x^58+14454x^59+27052x^60+33840x^61+48504x^62+73074x^63+74646x^64+80076x^65+76690x^66+45972x^67+23742x^68+12272x^69+4104x^70+546x^71+114x^72+108x^73+54x^74+18x^75+24x^76+2x^81

The gray image is a linear code over GF(3) with n=288, k=12 and d=165.
This code was found by Heurico 1.16 in 225 seconds.