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

generates a code of length 71 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 66.

Homogenous weight enumerator: w(x)=1x^0+362x^66+48x^67+509x^68+304x^69+644x^70+592x^71+505x^72+272x^73+438x^74+64x^75+219x^76+80x^78+45x^80+12x^82+1x^112

The gray image is a code over GF(2) with n=568, k=12 and d=264.
This code was found by Heurico 1.16 in 6 seconds.