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  X  2  X X^2  X  0  X  X X^2+2  X  X  X  0  X  X X^2+2  X  X  0 X^2+2  X  X  2  X  X  X  X X^2  0  X  X  X  X  X  X  1
 0  X  0 X^2+X+2 X^2 X^2+X X^2+2  X  0 X^2+X+2  0 X^2+X X^2  X X^2+2  X  2 X^2+X+2  2 X^2+X  2 X^2+X+2  2 X^2+X X^2+2 X+2 X^2 X+2 X^2+2 X+2 X^2 X+2 X^2+X  X X+2  X X^2+X+2  X  X X^2  X X^2+2  2 X^2+X+2  X  0  X  X X^2+X+2  X  X  X X^2+X X+2  X X^2+X X^2+X X+2 X+2  X  0  2  0 X^2+2 X^2  0 X^2+2 X^2
 0  0 X^2+2 X^2 X^2  2  2 X^2+2  2 X^2+2 X^2  0 X^2+2 X^2  0  2  2  2 X^2 X^2+2  0  0 X^2+2 X^2 X^2+2 X^2+2  2  0 X^2 X^2  0  2  0 X^2  2 X^2+2 X^2  0 X^2 X^2+2  2 X^2+2 X^2  0 X^2 X^2  2 X^2+2 X^2+2 X^2+2  2  0 X^2+2 X^2+2 X^2+2  2 X^2 X^2  0 X^2 X^2 X^2+2  0  0 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 66.

Homogenous weight enumerator: w(x)=1x^0+100x^66+124x^67+101x^68+120x^69+22x^70+4x^71+24x^72+6x^73+5x^74+1x^76+1x^80+2x^81+1x^82

The gray image is a code over GF(2) with n=544, k=9 and d=264.
This code was found by Heurico 1.16 in 0.296 seconds.