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

generates a code of length 55 over Z3[X]/(X^2) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+88x^84+298x^87+536x^90+742x^93+42x^95+1020x^96+294x^98+1202x^99+1302x^101+1530x^102+4074x^104+1794x^105+8106x^107+1998x^108+10626x^110+2178x^111+9330x^113+2034x^114+4560x^116+1892x^117+1032x^119+1620x^120+1092x^123+796x^126+444x^129+222x^132+122x^135+38x^138+20x^141+14x^144+2x^147

The gray image is a linear code over GF(3) with n=165, k=10 and d=84.
This code was found by Heurico 1.16 in 75.2 seconds.