The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  2  X  2  X  0  X  0  X  X  X X^2+2  X  X X^2 X^2+2  X X^2+2  X  2  X X^2  X  X  0 X^2+2  X  X  0 X^2  X  2  X X^2  1  1  1  1
 0  X  0 X^2+X+2 X^2 X^2+X X^2+2  X  0 X^2+X+2  0 X^2+X X^2  X X^2+2  X  2 X^2+X+2  2 X^2+X  2 X^2+X+2  2 X^2+X X^2+2 X+2 X^2 X+2 X^2+2 X+2 X^2 X+2 X^2+X  X X^2+X  X X^2+X+2  X X^2+X+2  X  2 X^2  X  X X+2  X  X  X  X  X X^2+X+2  X X^2+X  X X+2  X  X  X X^2+X+2 X+2  X  X X^2+X  X X+2  X  0  0 X^2  0
 0  0 X^2+2 X^2 X^2  2  2 X^2+2  2 X^2+2 X^2  0 X^2+2 X^2  0  2  2  2 X^2 X^2+2  0  0 X^2+2 X^2 X^2+2 X^2+2  2  0 X^2 X^2  0  2  0 X^2  2 X^2+2 X^2  0 X^2+2  2 X^2 X^2 X^2  0  0  2 X^2 X^2+2 X^2+2  2  2  2 X^2  0 X^2  0 X^2+2 X^2  0  2 X^2 X^2+2 X^2+2  0 X^2+2  2  0 X^2+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+52x^68+32x^69+344x^70+32x^71+41x^72+8x^74+1x^96+1x^104

The gray image is a code over GF(2) with n=560, k=9 and d=272.
This code was found by Heurico 1.16 in 0.313 seconds.