The generator matrix

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

generates a code of length 69 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 67.

Homogenous weight enumerator: w(x)=1x^0+276x^67+148x^68+236x^69+78x^70+236x^71+26x^72+20x^73+1x^80+1x^86+1x^102

The gray image is a code over GF(2) with n=552, k=10 and d=268.
This code was found by Heurico 1.16 in 1.64 seconds.