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

generates a code of length 86 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 79.

Homogenous weight enumerator: w(x)=1x^0+602x^79+1778x^80+3030x^81+4536x^82+5848x^83+6595x^84+7282x^85+7627x^86+6898x^87+6525x^88+5250x^89+3679x^90+2604x^91+1620x^92+882x^93+409x^94+172x^95+102x^96+68x^97+13x^98+4x^99+11x^100

The gray image is a code over GF(2) with n=688, k=16 and d=316.
This code was found by Heurico 1.16 in 52.4 seconds.