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

generates a code of length 71 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 70.

Homogenous weight enumerator: w(x)=1x^0+60x^70+128x^71+62x^72+1x^78+1x^80+2x^86+1x^94

The gray image is a code over GF(2) with n=568, k=8 and d=280.
This code was found by Heurico 1.16 in 0.328 seconds.