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

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+136x^78+152x^79+308x^80+248x^81+576x^82+240x^83+874x^84+240x^85+507x^86+248x^87+254x^88+152x^89+98x^90+23x^92+23x^94+10x^96+4x^98+1x^100+1x^144

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 1.33 seconds.