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

generates a code of length 44 over Z3[X]/(X^3) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+222x^79+330x^80+80x^81+576x^82+684x^83+636x^84+684x^85+2502x^86+2130x^87+2814x^88+4092x^89+2092x^90+864x^91+594x^92+84x^93+444x^94+354x^95+66x^96+168x^97+132x^98+12x^99+54x^100+54x^101+6x^103+6x^104+2x^117

The gray image is a linear code over GF(3) with n=396, k=9 and d=237.
This code was found by Heurico 1.16 in 2.5 seconds.