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  X  1  1  X  X  1  1
 0 X^2  0  0  0  0  0  0  0  0 X^2 2X^2 2X^2 X^2 X^2 X^2  0 2X^2 2X^2 X^2 X^2 X^2  0 X^2 2X^2 X^2 2X^2  0 X^2 X^2 2X^2 X^2 X^2  0  0 X^2 2X^2 X^2 2X^2 2X^2 X^2 X^2 2X^2  0
 0  0 X^2  0  0  0  0 X^2 2X^2 2X^2 2X^2  0  0 2X^2 X^2 2X^2 X^2  0 X^2 X^2 X^2  0 2X^2 2X^2 2X^2  0 2X^2 X^2 2X^2 2X^2 2X^2 X^2 X^2 X^2 2X^2 X^2 2X^2 X^2 X^2 2X^2 2X^2 X^2  0 X^2
 0  0  0 X^2  0  0 X^2 2X^2  0 2X^2  0  0 2X^2 X^2 X^2 2X^2  0 X^2 2X^2 2X^2 2X^2 X^2 X^2  0 X^2 2X^2  0 X^2  0 X^2  0 X^2 2X^2 2X^2 2X^2 X^2  0  0  0 2X^2  0  0 X^2 X^2
 0  0  0  0 X^2  0 2X^2 2X^2 X^2  0 2X^2 2X^2 2X^2  0 2X^2 2X^2  0 2X^2 2X^2 X^2  0  0 2X^2  0 2X^2 2X^2 X^2 2X^2  0 X^2 X^2  0 2X^2  0 2X^2  0  0 X^2 2X^2 X^2  0 X^2  0 X^2
 0  0  0  0  0 X^2 2X^2 2X^2 2X^2 2X^2 2X^2 2X^2 X^2 2X^2 X^2 X^2 2X^2 X^2  0 2X^2 2X^2  0  0  0 2X^2 X^2  0 2X^2 2X^2 X^2 2X^2 2X^2  0 X^2 X^2 X^2  0 X^2 2X^2 2X^2 X^2 2X^2 X^2 X^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+154x^78+108x^81+54x^82+324x^85+184x^87+5022x^88+84x^90+432x^91+100x^96+36x^99+38x^105+14x^108+8x^114+2x^123

The gray image is a linear code over GF(3) with n=396, k=8 and d=234.
This code was found by Heurico 1.16 in 5.8 seconds.