The generator matrix

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

generates a code of length 68 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 65.

Homogenous weight enumerator: w(x)=1x^0+308x^65+335x^66+412x^67+195x^68+308x^69+175x^70+172x^71+19x^72+48x^73+33x^74+24x^75+8x^76+8x^81+1x^92+1x^94

The gray image is a code over GF(2) with n=544, k=11 and d=260.
This code was found by Heurico 1.16 in 0.25 seconds.