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

generates a code of length 44 over Z3[X]/(X^2) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+154x^78+118x^81+54x^82+324x^85+180x^87+648x^88+66x^90+432x^91+100x^96+42x^99+50x^105+16x^108+2x^123

The gray image is a linear code over GF(3) with n=132, k=7 and d=78.
This code was found by Heurico 1.16 in 3.82 seconds.