The generator matrix

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

generates a code of length 69 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+66x^68+160x^69+9x^70+9x^72+6x^74+1x^78+4x^84

The gray image is a code over GF(2) with n=552, k=8 and d=272.
This code was found by Heurico 1.16 in 0.406 seconds.