The generator matrix

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

generates a code of length 71 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+304x^68+344x^69+382x^70+270x^71+248x^72+168x^73+164x^74+54x^75+42x^76+28x^77+33x^78+4x^82+4x^84+1x^98+1x^100

The gray image is a code over GF(2) with n=568, k=11 and d=272.
This code was found by Heurico 1.16 in 0.297 seconds.