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

generates a code of length 81 over Z3[X]/(X^3) who´s minimum homogenous weight is 144.

Homogenous weight enumerator: w(x)=1x^0+94x^144+168x^147+236x^150+30x^151+238x^153+180x^154+204x^156+540x^157+188x^159+1110x^160+13284x^162+1386x^163+198x^165+864x^166+112x^168+264x^169+104x^171+104x^174+104x^177+98x^180+68x^183+56x^186+26x^189+14x^192+4x^195+6x^198+2x^222

The gray image is a linear code over GF(3) with n=729, k=9 and d=432.
This code was found by Heurico 1.16 in 4.39 seconds.