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  X  X  X  X  X  X  X  1  1  1  1  1  1  1  1  X  X  X  X 2X+2 2X+2 2X+2 2X+2 2X+2  X  X  X  1  1  1  1  X 2X  X 2X 2X+2 2X+2 2X 2X  0  0  0  0 2X+2
 0 2X  0  0  0 2X 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0 2X 2X  0 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0 2X 2X  0  0 2X 2X  0  0 2X  0  0 2X  0 2X 2X  0  0 2X 2X 2X  0 2X 2X 2X  0  0  0
 0  0 2X  0 2X 2X 2X  0  0  0 2X 2X 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0  0 2X 2X  0 2X 2X  0  0  0 2X 2X 2X 2X  0  0  0 2X 2X  0  0 2X 2X  0 2X  0 2X 2X  0 2X 2X  0  0  0  0 2X  0 2X 2X  0  0 2X 2X  0  0
 0  0  0 2X 2X  0 2X 2X  0 2X 2X  0  0 2X 2X  0  0 2X 2X  0  0 2X 2X  0 2X 2X  0  0  0 2X 2X  0 2X 2X  0  0 2X 2X  0 2X 2X  0  0 2X 2X  0  0  0 2X  0 2X  0 2X  0 2X  0 2X  0  0 2X 2X 2X  0  0  0 2X 2X  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+120x^68+1x^72+4x^76+1x^80+1x^88

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 23.8 seconds.