The generator matrix

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

generates a code of length 70 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 66.

Homogenous weight enumerator: w(x)=1x^0+48x^66+742x^67+604x^68+782x^69+458x^70+472x^71+209x^72+280x^73+160x^74+162x^75+32x^76+106x^77+20x^78+16x^79+1x^82+2x^84+1x^90

The gray image is a code over GF(2) with n=560, k=12 and d=264.
This code was found by Heurico 1.16 in 0.282 seconds.