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

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+54x^81+18x^82+50x^84+108x^85+52x^87+1674x^88+28x^90+144x^91+14x^93+14x^96+10x^99+8x^102+4x^105+6x^108+2x^123

The gray image is a linear code over GF(3) with n=396, k=7 and d=243.
This code was found by Heurico 1.16 in 0.0511 seconds.