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

generates a code of length 68 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 63.

Homogenous weight enumerator: w(x)=1x^0+212x^63+238x^64+426x^65+386x^66+570x^67+646x^68+528x^69+312x^70+356x^71+121x^72+126x^73+46x^74+54x^75+28x^76+24x^77+8x^78+8x^79+4x^80+1x^84+1x^108

The gray image is a code over GF(2) with n=544, k=12 and d=252.
This code was found by Heurico 1.16 in 102 seconds.