The generator matrix

 1  0  0  1  1  1  0 X^2+2 X^2+2 X^2+2  1  1  1  1  X  1 X^2+X  1  1 X^2+X+2  1 X^2+X  X  1  1  X  1  1  0  1  2  1  1  1  X  1  1  1  1  1 X^2+X X^2  1  1  1 X+2  1  1  1 X^2+X X^2+X+2  2 X^2  1  1  1  1 X^2  1  1  1  X X+2  1  X  1 X^2+2 X^2+X+2  0  1 X^2 X^2+2  1 X^2+X  0 X^2  1 X+2 X^2  1  2 X^2+2 X^2+X+2  1
 0  1  0  0 X^2+1 X^2+3  1  X  1  1  3  3 X^2+2 X^2 X^2+X+2 X^2+X  1 X+3 X^2+X  1 X^2+X+1 X^2+2  1 X^2+X+2 X^2+X  1 X^2+X+3 X^2+3 X^2+X+2 X^2+2  1 X+3  X X^2+1  1  2  1  0  0  1  1  1  3 X^2+2  X  2 X^2 X+3 X^2+3  1  0  1  1  1 X^2+X+2 X+1 X^2+X+3  1  1  2 X+2  1  1 X^2+X  1 X+2 X^2+2 X^2+X+2  1  0  1 X^2+X X^2+X  1  0  0 X^2+X+1  1 X^2  2  1  1  1  0
 0  0  1 X+1 X^2+X+1 X^2 X^2+X+1  1  X  3 X^2+3 X+2  X X^2+1  1  0  2 X^2+2  1 X+1 X^2+X+3  1 X^2+X+2 X+1  X X^2+3 X^2+1 X^2+X  1 X^2+X+2 X^2+X  X X^2+1  1 X^2+X+3  2  2 X+1  3 X+1 X^2+3 X^2 X^2+X+3  X X^2+2  1  0  0 X+2  1  1 X^2+2 X+3  X X^2+X  1 X^2+X+3 X^2+1 X^2+X+3 X^2+X+1 X+1 X^2+2 X^2+2 X^2+X X^2+X+2  1  1  1 X^2+X+1  X X^2+X  1 X^2+2 X+2  1  1 X+1  1  1 X^2+1 X^2 X^2 X+2 X^2+2
 0  0  0 X^2 X^2  0 X^2 X^2+2 X^2  2  0 X^2 X^2  0 X^2+2  2  2  2  0 X^2+2  0  2 X^2 X^2 X^2+2  0 X^2+2 X^2+2 X^2  0 X^2+2  0  2  2  0 X^2+2 X^2  2 X^2+2  0 X^2+2 X^2+2 X^2+2  2 X^2+2 X^2+2  2 X^2  2 X^2 X^2 X^2  0  0  0  2 X^2+2 X^2+2  2 X^2+2  2 X^2+2  2 X^2 X^2+2 X^2 X^2  2  2 X^2+2  0  0 X^2  0  2 X^2+2 X^2 X^2+2  2 X^2 X^2+2  2  2 X^2+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+147x^78+828x^79+1250x^80+1684x^81+1722x^82+2162x^83+1836x^84+1878x^85+1461x^86+1208x^87+768x^88+534x^89+341x^90+304x^91+104x^92+70x^93+39x^94+24x^95+9x^96+10x^97+2x^99+1x^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 3.81 seconds.