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  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  2 X^2+X  0 X^2+X+2 X^2 X+2 X^2 X+2 X^2+2  X X^2 X+2 X+3 X+3 X+3  2 X^2+X+1  0
 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 X^2+2  2 X^2  0  2 X^2+2 X^2  0 X^2+2  2  0 X^2  2  0 X^2 X^2  2  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+270x^68+128x^69+224x^70+128x^71+269x^72+2x^76+1x^96+1x^104

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 0.25 seconds.