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  1  1  1  1  1  1  1  1  1  1  1  X  X  1
 0  X X^2+2 X^2+X  0 X^2+X X^2+2 X+2  0 X^2+X X+2 X^2+2  0 X^2+X X^2+2  X  0 X^2+X X^2+2 X+2  0 X^2+X X^2+2  X  0 X^2+X X^2+2  X  0 X^2+X+2 X^2  X  2 X^2+X+2 X^2 X+2  2 X^2+X+2 X^2  X  2 X^2+X+2 X^2  X  2 X^2+X+2 X+2 X^2  2 X^2+X X^2+2 X+2 X^2+X+2 X^2 X+2  2  2 X^2+X+2 X^2 X+2  2 X^2+X+2 X^2  X  0 X^2+X X^2+2 X+2 X^2+X X^2+X  0
 0  0  2  0  0  0  2  0  0  2  2  2  0  2  2  2  2  0  0  2  2  2  0  0  2  0  0  2  2  2  0  0  2  2  2  2  2  2  2  2  0  0  0  0  0  0  0  0  2  2  0  0  0  2  0  2  0  2  2  2  0  0  0  2  0  0  0  0  2  2  0
 0  0  0  2  0  0  0  2  2  2  2  2  2  0  2  0  2  0  0  2  2  2  0  0  0  2  2  0  0  0  2  2  0  0  0  0  2  2  2  2  2  2  2  2  0  0  0  0  0  0  2  0  2  2  2  2  2  2  0  0  0  0  0  2  0  0  0  2  2  0  0
 0  0  0  0  2  2  2  2  2  0  2  0  0  2  2  0  0  0  0  0  2  2  2  2  2  2  2  2  0  0  0  0  0  2  2  0  2  2  0  0  0  0  2  2  0  0  2  2  2  0  0  0  2  2  0  0  2  0  0  2  2  2  0  2  0  0  2  2  0  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+18x^68+96x^69+109x^70+576x^71+109x^72+96x^73+17x^74+1x^78+1x^138

The gray image is a code over GF(2) with n=568, k=10 and d=272.
This code was found by Heurico 1.16 in 0.36 seconds.