The generator matrix

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

generates a code of length 70 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 63.

Homogenous weight enumerator: w(x)=1x^0+244x^63+1320x^64+2544x^65+4046x^66+5562x^67+7215x^68+7814x^69+8384x^70+7998x^71+7376x^72+5202x^73+3640x^74+2154x^75+1167x^76+504x^77+172x^78+86x^79+65x^80+10x^81+22x^82+4x^83+6x^85

The gray image is a code over GF(2) with n=560, k=16 and d=252.
This code was found by Heurico 1.16 in 40.3 seconds.