The generator matrix

 1  0  1  1  1  1  1 2X^2+X  1  1  1 2X^2  1 2X^2+X  1  1  1  X  1  1  1  1 2X  1  1  X  1 X^2+X  1  1  1  1  1  0  1  1  1  X  1  1  1  1  1  1
 0  1  1  2 2X^2+2X+1 2X^2 2X^2+2  1 2X+1 2X^2+X 2X^2+X+2  1 2X^2+1  1 2X^2+X 2X^2+X+2 2X+2  1  1  X 2X+2 2X^2+2X+1  1  0 2X^2+2X+2  1  X  1 X+1 2X+1 2X^2+X 2X X+1  1  1 X+1 2X^2+X  X 2X^2+1 2X^2+2 X^2+2 2X X^2+2X  X
 0  0 2X  0  0 2X^2+X 2X^2+X 2X^2  X 2X^2+2X X^2+2X 2X^2+X X^2 2X X^2  0  0  X X^2+X 2X^2+X 2X X^2+2X 2X 2X^2+2X 2X^2+X  X  X 2X 2X  X 2X^2+2X 2X 2X^2 2X^2 X^2+X X^2 X^2 2X^2+2X 2X^2+2X  X 2X  X  0 X^2+X
 0  0  0 X^2  0 2X^2  0 X^2 2X^2  0 X^2  0  0 X^2 2X^2 2X^2 2X^2 2X^2 2X^2  0 2X^2  0 X^2  0 2X^2 X^2  0 2X^2  0 2X^2 2X^2 X^2 X^2 2X^2  0 2X^2 X^2 X^2 X^2  0 X^2  0  0  0
 0  0  0  0 2X^2  0  0  0  0  0  0 2X^2 X^2 2X^2 X^2 X^2 2X^2 2X^2 X^2 2X^2 X^2 2X^2  0 X^2 2X^2 2X^2 X^2 2X^2 X^2 2X^2 X^2 2X^2 2X^2  0 X^2 2X^2  0  0  0 2X^2 2X^2  0 2X^2 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+140x^78+114x^79+402x^80+1104x^81+1170x^82+1770x^83+3396x^84+4158x^85+5454x^86+6786x^87+7506x^88+7830x^89+7068x^90+5352x^91+3162x^92+2138x^93+504x^94+288x^95+320x^96+120x^97+48x^98+152x^99+30x^100+28x^102+8x^105

The gray image is a linear code over GF(3) with n=396, k=10 and d=234.
This code was found by Heurico 1.16 in 5.97 seconds.