The generator matrix

 1  0  1  1  1 X^2+X  1  1 X^2+2  1  1 X+2  1  1  2  1  1 X^2+X+2  1  1 X^2  1  1  X  1  1  0  1  1 X^2+X  1  1 X^2+2  1  1 X+2  1  1  1  1  1  1  1  1  2 X^2+X+2 X^2  X  X  X  0  X  X  2  X  X  X  X X^2+2 X^2  1  1  1  1  1  1  1  1
 0  1 X+1 X^2+X X^2+1  1 X^2+2 X^2+X+3  1 X+2  3  1  2 X+3  1 X^2+X+2 X^2+3  1 X^2 X^2+X+1  1  X  1  1  0 X+1  1 X^2+X X^2+1  1 X^2+2 X^2+X+3  1 X+2  3  1  2 X^2+X+2 X+3 X^2+3 X^2  X X^2+X+1  1  1  1  1  1  0 X^2+X  X X^2+2 X^2+X+2  X  2 X^2 X+2  X  X  X  0  2 X^2+2 X^2 X+1 X+3 X^2+X+3 X^2+X+1

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

Homogenous weight enumerator: w(x)=1x^0+244x^68+6x^72+4x^76+1x^80

The gray image is a code over GF(2) with n=544, k=8 and d=272.
This code was found by Heurico 1.16 in 0.094 seconds.