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

generates a code of length 71 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 67.

Homogenous weight enumerator: w(x)=1x^0+80x^67+215x^68+288x^69+336x^70+256x^71+598x^72+32x^73+48x^75+129x^76+64x^77+1x^136

The gray image is a code over GF(2) with n=568, k=11 and d=268.
This code was found by Heurico 1.16 in 0.421 seconds.