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

generates a code of length 71 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 70.

Homogenous weight enumerator: w(x)=1x^0+28x^70+186x^71+31x^72+4x^75+3x^78+2x^79+1x^86

The gray image is a code over GF(2) with n=568, k=8 and d=280.
This code was found by Heurico 1.16 in 0.125 seconds.