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

generates a code of length 31 over Z3[X]/(X^3) who�s minimum homogenous weight is 58.

Homogenous weight enumerator: w(x)=1x^0+162x^58+278x^60+174x^61+972x^62+414x^63+36x^64+30x^66+72x^67+4x^69+24x^70+18x^73+2x^90

The gray image is a linear code over GF(3) with n=279, k=7 and d=174.
This code was found by Heurico 1.16 in 9.64 seconds.