The generator matrix

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

generates a code of length 70 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 67.

Homogenous weight enumerator: w(x)=1x^0+380x^67+336x^68+304x^69+144x^70+248x^71+234x^72+300x^73+46x^74+32x^75+4x^76+4x^77+12x^79+2x^90+1x^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 56 seconds.