The generator matrix

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

generates a code of length 70 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 67.

Homogenous weight enumerator: w(x)=1x^0+460x^67+209x^68+328x^69+96x^70+340x^71+200x^72+376x^73+12x^75+3x^76+20x^79+1x^88+2x^96

The gray image is a code over GF(2) with n=560, k=11 and d=268.
This code was found by Heurico 1.16 in 8.59 seconds.