The generator matrix

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

generates a code of length 84 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+112x^79+440x^80+360x^81+512x^82+448x^83+459x^84+458x^85+463x^86+272x^87+369x^88+106x^89+45x^90+32x^91+10x^92+2x^93+1x^94+2x^102+2x^105+1x^116+1x^122

The gray image is a code over GF(2) with n=672, k=12 and d=316.
This code was found by Heurico 1.16 in 0.75 seconds.