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  0  1  1  X  1  1  1  1  1  1  X  0  1  0  1  1  1  1  1  1
 0  X  0  0 2X 2X^2+X 2X^2+2X  X 2X 2X^2+X 2X^2  0 2X 2X^2 2X^2+X 2X 2X^2+X X^2+2X X^2+X 2X^2+2X X^2+X X^2+X X^2+X 2X^2 2X^2 X^2 2X^2 2X 2X^2 2X  X  X 2X^2+2X 2X^2+2X 2X  X X^2  X X^2 X^2+2X X^2+2X 2X^2+2X 2X^2 2X^2
 0  0  X 2X  0 X^2+2X X^2+X  X X^2+2X 2X^2+2X  X X^2+2X 2X X^2 X^2 X^2 X^2+X X^2+X X^2+X  X 2X^2+2X  0 X^2+X 2X^2+X  X 2X^2 2X^2+2X X^2+2X 2X^2 X^2 X^2 2X^2+2X X^2+2X 2X^2 X^2 2X  0 2X^2 X^2+2X 2X^2+X 2X^2  X 2X^2 2X^2
 0  0  0 X^2  0  0 2X^2  0  0 X^2 2X^2 2X^2 2X^2 X^2 2X^2  0 2X^2 2X^2  0 X^2 2X^2 X^2 X^2  0  0  0 2X^2 X^2 2X^2 2X^2 X^2 2X^2 X^2  0 2X^2  0  0 2X^2  0 X^2 X^2  0 2X^2  0
 0  0  0  0 X^2 2X^2  0 X^2 2X^2  0 2X^2 2X^2 2X^2  0  0 2X^2  0 2X^2 2X^2 X^2  0  0 2X^2 X^2 2X^2 2X^2 X^2 X^2 2X^2  0 2X^2 2X^2 X^2  0 X^2 X^2 X^2 X^2  0 2X^2 X^2 X^2  0 X^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+106x^78+84x^79+186x^80+488x^81+402x^82+246x^83+1170x^84+930x^85+1698x^86+2646x^87+2958x^88+3180x^89+2700x^90+1116x^91+288x^92+480x^93+204x^94+192x^95+264x^96+90x^97+36x^98+128x^99+48x^100+6x^101+24x^102+8x^105+2x^108+2x^117

The gray image is a linear code over GF(3) with n=396, k=9 and d=234.
This code was found by Heurico 1.16 in 1.27 seconds.