The generator matrix

 1  0  0  1  1  1 2X  1  1  1 3X+2 3X  1 X+2  1 2X+2 X+2  0  1  1  1  1 2X 3X+2  2  1  1 2X  1 3X  X X+2  1  1 3X  1 2X  1 3X  1 2X+2  1  1  1  1  1  X  2 3X+2  1  1  0  1  1 2X+2 3X+2  1  1  1  1  0  2  0  1  1  1  1  1
 0  1  0 2X+2  3 2X+3  1  2 2X X+3  1 2X 3X+3  1  X  1  1 3X  X 3X+3  1  1  1 X+2  1 3X+2  X 2X+2 3X+3  1  1  1  0 3X X+2 X+1  1 3X  1  2 3X 3X+1 2X+3  3 X+3  2  1  1  2 3X+2  3 2X 2X+2 X+3  1  1 3X+2 X+1  3  X  1  1  2 X+3 2X+2  1  2  0
 0  0  1 3X+3 X+3  2 X+3 3X  3 2X+3 2X+2  1  2 X+3  X 3X  3  1 2X+1 X+3 X+2  1  3  1 2X 2X 3X+3  1 3X+2  3  X 3X+1  3 3X  1  1 2X+3 2X+3 2X  0  1 2X+2 3X+1 2X+2 3X X+3 3X 3X+2  1 2X+2  1  1 3X+1 3X+3 3X+2  1 2X+2 X+3 3X+2  0 3X+3  3  1 2X  3 3X+1 2X  0
 0  0  0 2X 2X 2X  0 2X  0  0 2X 2X  0 2X 2X  0 2X  0  0 2X 2X  0  0  0 2X  0  0 2X  0  0  0  0 2X  0 2X 2X 2X 2X  0 2X 2X 2X  0  0 2X  0 2X 2X  0 2X 2X 2X 2X  0  0 2X  0 2X  0 2X 2X  0  0  0 2X  0  0  0

generates a code of length 68 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 63.

Homogenous weight enumerator: w(x)=1x^0+118x^63+619x^64+918x^65+1193x^66+956x^67+1321x^68+838x^69+737x^70+472x^71+380x^72+232x^73+231x^74+70x^75+41x^76+44x^77+15x^78+6x^80

The gray image is a code over GF(2) with n=544, k=13 and d=252.
This code was found by Heurico 1.16 in 0.86 seconds.