The generator matrix

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

generates a code of length 84 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+196x^79+647x^80+1076x^81+1136x^82+1004x^83+929x^84+844x^85+730x^86+564x^87+350x^88+216x^89+211x^90+112x^91+67x^92+56x^93+17x^94+28x^95+5x^96+1x^98+1x^102+1x^104

The gray image is a code over GF(2) with n=672, k=13 and d=316.
This code was found by Heurico 1.16 in 1.19 seconds.