The generator matrix

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

generates a code of length 84 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+520x^77+1746x^78+3246x^79+4230x^80+5940x^81+6452x^82+7578x^83+7104x^84+7292x^85+6486x^86+5704x^87+3631x^88+2684x^89+1556x^90+678x^91+306x^92+204x^93+72x^94+54x^95+32x^96+16x^97+4x^99

The gray image is a code over GF(2) with n=672, k=16 and d=308.
This code was found by Heurico 1.16 in 49 seconds.