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

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

Homogenous weight enumerator: w(x)=1x^0+113x^78+126x^79+346x^80+342x^81+430x^82+472x^83+604x^84+428x^85+425x^86+282x^87+227x^88+102x^89+91x^90+12x^91+26x^92+24x^93+24x^94+4x^95+12x^96+4x^98+1x^138

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 1.03 seconds.