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

generates a code of length 44 over Z3[X]/(X^3) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+200x^84+174x^85+282x^86+324x^87+330x^88+336x^89+294x^90+126x^91+22x^93+6x^94+24x^95+24x^96+12x^97+6x^98+24x^99+2x^117

The gray image is a linear code over GF(3) with n=396, k=7 and d=252.
This code was found by Heurico 1.16 in 0.519 seconds.