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

generates a code of length 68 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 66.

Homogenous weight enumerator: w(x)=1x^0+54x^66+210x^67+495x^68+204x^69+57x^70+2x^71+1x^130

The gray image is a code over GF(2) with n=544, k=10 and d=264.
This code was found by Heurico 1.16 in 6.2 seconds.