The generator matrix

 1  0  0  1  1  1  1  1  1 2X^2+X 2X  1  1  1  1  1 X^2+X X^2+X  1  1  1 X^2  1 X^2+X  1  1  1  1 2X^2+X  1  1  X X^2+X  1  1  1  1  1  1  1  1 2X^2+X 2X  1
 0  1  0  1 2X^2  1 X+2  0 2X^2+2X+1  1  1 2X^2+2X+2 2X^2+2X+2  2  2 2X^2+2X+1  1  1 X^2+2 2X^2+2X  0  1 2X^2+2X X^2+2X X^2+2X+1 2X+2 X^2+1 X^2+X X^2+2X 2X^2+2X X^2+X+2  1  1 2X^2 X+1 2X^2+X X^2+1 X^2+2X+1 X^2+2X+1 X^2+X X^2+1  1  1  0
 0  0  1  2 2X^2+2X+1  1 X+1  2 2X^2 2X^2+2 X+1 2X^2 2X^2+2X+2 X+1 2X^2+X X^2+X+2 2X^2+2X+2  1 X^2+2X+2  1 X^2+2 2X^2 2X  1 X^2+2X X^2+2 2X 2X^2+2X+2  1 X^2+X+1 X^2 X^2 2X+2 X^2+X+2 X+2 X^2+2X 2X+2  1 2X^2+2X 2X^2+X+2 X^2+2 X^2+X+1 X^2+X  0
 0  0  0 2X 2X^2 2X^2+2X 2X^2+X X^2+2X X^2 2X^2 2X^2+2X X^2+X  X 2X^2  0 X^2+X  X  0 X^2+2X X^2+X X^2+X X^2+2X 2X^2+2X 2X X^2+2X 2X^2 X^2+X X^2 2X^2+X 2X  X  X 2X^2+2X X^2+2X 2X^2+2X 2X^2+X  0 2X^2+X X^2+X 2X^2 2X^2+X 2X^2 X^2+2X 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+176x^78+300x^79+1686x^80+2704x^81+4200x^82+6066x^83+9432x^84+10914x^85+17736x^86+19754x^87+21222x^88+24702x^89+22196x^90+14610x^91+11112x^92+5538x^93+2532x^94+1284x^95+548x^96+132x^97+90x^98+146x^99+36x^100+12x^101+6x^102+6x^104+6x^105

The gray image is a linear code over GF(3) with n=396, k=11 and d=234.
This code was found by Heurico 1.16 in 40.9 seconds.