The generator matrix

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

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+150x^78+794x^79+1110x^80+1992x^81+1696x^82+2224x^83+1612x^84+1876x^85+1247x^86+1452x^87+774x^88+704x^89+288x^90+188x^91+108x^92+80x^93+30x^94+30x^95+15x^96+4x^97+1x^98+4x^100+3x^102+1x^106

The gray image is a code over GF(2) with n=672, k=14 and d=312.
This code was found by Heurico 1.16 in 4.11 seconds.