The generator matrix

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

generates a code of length 71 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+264x^69+140x^70+256x^71+94x^72+232x^73+18x^74+16x^75+1x^80+1x^90+1x^106

The gray image is a code over GF(2) with n=568, k=10 and d=276.
This code was found by Heurico 1.16 in 1.45 seconds.