The generator matrix

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

generates a code of length 70 over Z4[X]/(X^2+2X+2) 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 0.265 seconds.