The generator matrix

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

generates a code of length 71 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+435x^68+288x^69+456x^70+64x^71+290x^72+64x^73+176x^74+64x^75+145x^76+32x^77+24x^78+4x^80+2x^84+1x^88+2x^92

The gray image is a code over GF(2) with n=568, k=11 and d=272.
This code was found by Heurico 1.16 in 29.5 seconds.