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

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

Homogenous weight enumerator: w(x)=1x^0+56x^78+290x^79+150x^80+380x^81+437x^82+550x^83+563x^84+492x^85+373x^86+362x^87+121x^88+92x^89+59x^90+94x^91+29x^92+28x^93+2x^94+12x^95+4x^99+1x^142

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 seconds.