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  X  X  1  1  1  1  1  1  1  1  1  1  X  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^2  X  X
 0 X^2+2  0 X^2  0  0 X^2 X^2+2  0  0 X^2 X^2+2  0  0 X^2 X^2+2  0  0 X^2 X^2+2  0  0 X^2 X^2 X^2+2  2  2  2 X^2  2 X^2+2 X^2+2  2  2  0  0  0  2  0 X^2+2 X^2 X^2+2 X^2  2 X^2+2  2 X^2  2 X^2+2  2 X^2  2 X^2+2  2 X^2  2 X^2+2  2 X^2  2 X^2+2  2 X^2  2  2 X^2+2 X^2 X^2 X^2+2 X^2+2 X^2+2
 0  0 X^2+2 X^2  0 X^2+2 X^2  0  0 X^2+2 X^2  0  0 X^2+2 X^2  0  2 X^2 X^2+2  2  2 X^2 X^2+2 X^2+2  2 X^2 X^2  0 X^2+2 X^2  2  2  0 X^2  2 X^2  2 X^2 X^2 X^2+2  2 X^2+2  2  2 X^2+2 X^2+2  2  2 X^2+2 X^2+2  2  2 X^2 X^2  0  0 X^2 X^2+2  0  2 X^2 X^2  0  0 X^2+2 X^2  0 X^2  0  0  2
 0  0  0  2  0  0  2  0  2  2  0  2  2  2  0  2  2  2  2  2  2  2  2  0  0  0  0  0  0  0  2  0  0  0  0  0  0  2  0  0  0  0  0  2  2  2  2  2  2  2  2  0  2  2  2  2  0  0  0  0  2  2  2  2  0  0  0  0  2  0  2
 0  0  0  0  2  2  2  2  2  2  0  2  0  0  2  0  0  0  0  0  2  2  2  2  2  0  2  2  0  0  2  0  0  2  2  2  0  2  0  0  0  2  2  2  2  0  0  0  0  2  2  0  0  2  0  2  2  0  2  2  2  0  2  0  2  0  0  0  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+70x^68+304x^70+256x^71+334x^72+48x^74+10x^76+1x^128

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 23.6 seconds.