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

generates a code of length 68 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 62.

Homogenous weight enumerator: w(x)=1x^0+56x^62+94x^64+64x^66+256x^67+1152x^68+256x^69+112x^70+24x^78+32x^80+1x^128

The gray image is a code over GF(2) with n=544, k=11 and d=248.
This code was found by Heurico 1.16 in 34.2 seconds.