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

generates a code of length 44 over Z3[X]/(X^3) who´s minimum homogenous weight is 83.

Homogenous weight enumerator: w(x)=1x^0+126x^83+40x^84+480x^86+104x^87+486x^88+708x^89+78x^90+54x^92+12x^93+78x^95+4x^96+12x^98+2x^102+2x^126

The gray image is a linear code over GF(3) with n=396, k=7 and d=249.
This code was found by Heurico 1.16 in 0.0463 seconds.