The generator matrix

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

generates a code of length 70 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+278x^68+464x^70+279x^72+1x^100+1x^108

The gray image is a code over GF(2) with n=560, k=10 and d=272.
This code was found by Heurico 1.16 in 0.218 seconds.