The generator matrix

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

generates a code of length 31 over Z3[X]/(X^3) who´s minimum homogenous weight is 55.

Homogenous weight enumerator: w(x)=1x^0+372x^55+792x^56+2106x^57+3420x^58+2970x^59+6700x^60+8046x^61+6198x^62+11732x^63+7878x^64+3726x^65+3000x^66+1602x^67+396x^68+24x^69+54x^70+12x^71+6x^72+12x^73+2x^78

The gray image is a linear code over GF(3) with n=279, k=10 and d=165.
This code was found by Heurico 1.16 in 3.22 seconds.