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

generates a code of length 44 over Z3[X]/(X^3) who´s minimum homogenous weight is 81.

Homogenous weight enumerator: w(x)=1x^0+340x^81+546x^84+972x^86+1158x^87+1944x^89+1170x^90+162x^93+156x^96+88x^99+12x^102+10x^108+2x^126

The gray image is a linear code over GF(3) with n=396, k=8 and d=243.
This code was found by Heurico 1.16 in 27.1 seconds.