The generator matrix

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

generates a code of length 44 over Z3[X]/(X^3) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+616x^78+1494x^79+5034x^80+8158x^81+11598x^82+20196x^83+26136x^84+34134x^85+51450x^86+60188x^87+65370x^88+70524x^89+63058x^90+46728x^91+35400x^92+18060x^93+7110x^94+4368x^95+1320x^96+204x^97+114x^98+78x^99+48x^100+24x^101+18x^102+12x^103

The gray image is a linear code over GF(3) with n=396, k=12 and d=234.
This code was found by Heurico 1.16 in 331 seconds.