The generator matrix

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

generates a code of length 70 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 64.

Homogenous weight enumerator: w(x)=1x^0+544x^64+1280x^65+2253x^66+2824x^67+3542x^68+4034x^69+4401x^70+3898x^71+3660x^72+2498x^73+1730x^74+1032x^75+582x^76+232x^77+110x^78+50x^79+43x^80+18x^81+17x^82+4x^83+12x^84+2x^85+1x^86

The gray image is a code over GF(2) with n=560, k=15 and d=256.
This code was found by Heurico 1.16 in 16.4 seconds.