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 1 X X X X X X X X X X X X X 1 X 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 0 0 0 0 2X 2X 2X 2X 2X 2X 2X 2X 0 2X 0 0 2X 2X 2X 0 0 0 0 2X 0 2X 2X 2X 2X 0 0 0 2X 2X 2X 0 0 0 2X 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 2X 2X 2X 2X 2X 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2X 2X 2X 2X 0 0 0 0 0 2X 2X 2X 0 0 2X 2X 2X 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 2X 2X 2X 0 0 0 0 2X 2X 2X 2X 0 0 0 2X 2X 2X 2X 0 0 0 0 2X 2X 0 0 2X 2X 0 0 0 2X 2X 0 2X 2X 2X 2X 0 0 0
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 0 2X 2X 0 0 2X 2X 0 0 2X 2X 0 2X 0 2X 0 2X 2X 0 0 2X 2X 0 0 0 0 2X 2X 0 2X 2X 0 2X 2X 0 0 2X 2X 0 0
generates a code of length 68 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 67.
Homogenous weight enumerator: w(x)=1x^0+60x^67+175x^68+15x^72+1x^76+4x^83
The gray image is a code over GF(2) with n=544, k=8 and d=268.
This code was found by Heurico 1.16 in 37.7 seconds.