The generator matrix

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

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+113x^68+4x^70+4x^72+4x^74+1x^76+1x^80

The gray image is a code over GF(2) with n=544, k=7 and d=272.
This code was found by Heurico 1.16 in 0.172 seconds.