The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+3x^68+56x^69+3x^70+1x^74

The gray image is a code over GF(2) with n=544, k=6 and d=272.
This code was found by Heurico 1.16 in 0.203 seconds.