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

generates a code of length 85 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 78.

Homogenous weight enumerator: w(x)=1x^0+516x^78+1762x^79+3304x^80+4588x^81+5581x^82+6402x^83+7255x^84+7708x^85+6852x^86+6772x^87+5189x^88+3884x^89+2734x^90+1474x^91+865x^92+320x^93+168x^94+78x^95+34x^96+24x^97+5x^98+8x^99+8x^100+4x^101

The gray image is a code over GF(2) with n=680, k=16 and d=312.
This code was found by Heurico 1.16 in 48.7 seconds.