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

generates a code of length 84 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+70x^78+236x^79+305x^80+312x^81+426x^82+484x^83+513x^84+572x^85+406x^86+256x^87+193x^88+84x^89+94x^90+56x^91+22x^92+40x^93+8x^94+8x^95+5x^96+4x^98+1x^140

The gray image is a code over GF(2) with n=672, k=12 and d=312.
This code was found by Heurico 1.16 in 1.02 seconds.