The generator matrix
1 0 0 1 1 1 1 1 0 2X 1 1 1 0 1 X 1 X 1 1 1 1 1 0 1 1 0 1 1 1 2X 1 1 1 1 1 1 0 1 X 1 1 1 1 X 1 1 1 1 1 X 1 1 1 1 X X X 1 1 1 1 X 1 X 2X X 1 1 0 1 X 1 0 1 1 2X 2X
0 1 0 0 1 2 1 2 1 1 0 2X+1 2X+2 1 0 1 2 1 2X X+2 X+2 1 2X+1 0 X 2X+1 1 2X X 2X+2 1 2 0 2X+1 2X+2 1 0 1 X+1 0 X+2 2X+1 X+2 X 1 X+2 2X 2X+1 1 X+2 1 2X+2 1 X+1 2 1 1 1 2X 0 2X 2X 1 X+1 X 0 2X X+2 1 1 2X+1 1 2X+1 2X 2X+2 X+1 1 1
0 0 1 1 2 2 1 0 2 2X+1 2 2X 2X+1 X+2 0 X X+2 1 X+1 2X+1 X 2X+1 2 1 X+2 2X 2X+1 X+2 X+1 2X 2 1 X 2 X+2 1 X+2 X 0 1 2X X+2 1 X 2X+1 X+2 1 X X 1 0 2X X+1 X+2 X+1 0 1 2 X+2 1 2X+2 1 X+1 X+1 1 1 1 2X X+1 X 2 0 2X 1 2X 2X+2 2X 2
0 0 0 2X 0 0 0 0 0 0 0 0 2X 0 0 0 0 0 2X 2X 0 0 2X X X 2X 2X 2X 2X X 0 2X X 0 X X 2X X X 2X X 2X 2X 0 X X X X 2X X X X X 0 2X X X 0 2X X X X X 2X 0 0 X 0 X X X 0 0 0 0 2X 0 X
0 0 0 0 2X 0 0 0 0 0 X X 0 X 2X X 2X X 2X X X 2X X 2X X X 0 X 0 0 0 0 2X X 0 X X 0 2X 2X 2X 0 2X 0 X 2X X X 0 2X 0 X 0 2X 2X 0 X 2X 0 0 X X X 2X 2X 0 X X X 0 2X 2X X 2X X 2X 0 2X
0 0 0 0 0 X 0 X X 2X X 2X 2X 0 X X 2X X X X 0 2X 2X X 2X 0 0 2X 2X 0 0 0 X 0 2X 2X X 2X 0 0 2X 0 0 X 0 2X 0 0 2X 0 0 0 X X 2X 0 2X X 0 2X X X X 2X X X X 2X 0 0 X 0 0 X X X 2X 0
0 0 0 0 0 0 X X X X 2X 2X X 0 X X X 0 X 0 2X 0 X 0 0 2X X 2X 0 X X X X 2X 2X 2X 0 0 2X 0 X X X 2X 0 2X 0 0 2X X 2X 2X 0 2X 0 0 X 0 2X 0 2X 0 0 X 2X X X X X X 0 2X X 0 X 0 0 2X
generates a code of length 78 over Z3[X]/(X^2) who´s minimum homogenous weight is 137.
Homogenous weight enumerator: w(x)=1x^0+84x^137+178x^138+168x^139+366x^140+494x^141+348x^142+978x^143+1074x^144+738x^145+1818x^146+1526x^147+1134x^148+2712x^149+2154x^150+1524x^151+3444x^152+2922x^153+2196x^154+4026x^155+2858x^156+2154x^157+4320x^158+3016x^159+1974x^160+3666x^161+2364x^162+1368x^163+2682x^164+1708x^165+960x^166+1380x^167+786x^168+420x^169+546x^170+308x^171+126x^172+192x^173+122x^174+12x^175+24x^176+66x^177+6x^179+50x^180+32x^183+14x^186+4x^189+2x^192+4x^195
The gray image is a linear code over GF(3) with n=234, k=10 and d=137.
This code was found by Heurico 1.16 in 64.3 seconds.