The generator matrix

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

generates a code of length 68 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 61.

Homogenous weight enumerator: w(x)=1x^0+128x^61+1287x^62+2588x^63+4206x^64+5438x^65+7025x^66+7782x^67+8505x^68+8206x^69+7393x^70+5450x^71+3697x^72+1872x^73+1191x^74+428x^75+191x^76+94x^77+32x^78+6x^79+8x^80+6x^81+2x^83

The gray image is a code over GF(2) with n=544, k=16 and d=244.
This code was found by Heurico 1.16 in 38.4 seconds.