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

generates a code of length 70 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 65.

Homogenous weight enumerator: w(x)=1x^0+224x^65+289x^66+216x^67+360x^68+672x^69+645x^70+684x^71+363x^72+168x^73+227x^74+144x^75+11x^76+40x^77+22x^78+28x^79+1x^84+1x^126

The gray image is a code over GF(2) with n=560, k=12 and d=260.
This code was found by Heurico 1.16 in 150 seconds.