The generator matrix

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

generates a code of length 71 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 65.

Homogenous weight enumerator: w(x)=1x^0+106x^65+627x^66+1174x^67+1537x^68+2052x^69+1962x^70+2156x^71+1885x^72+1702x^73+1189x^74+878x^75+497x^76+292x^77+180x^78+60x^79+42x^80+24x^81+7x^82+4x^83+6x^84+2x^86+1x^90

The gray image is a code over GF(2) with n=568, k=14 and d=260.
This code was found by Heurico 1.16 in 3.23 seconds.