The generator matrix

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

generates a code of length 70 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+28x^69+206x^70+14x^72+4x^77+1x^78+1x^80+1x^94

The gray image is a code over GF(2) with n=560, k=8 and d=276.
This code was found by Heurico 1.16 in 22.5 seconds.