The generator matrix

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

generates a code of length 86 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+488x^77+1598x^78+3908x^79+6538x^80+10318x^81+15107x^82+21132x^83+25852x^84+30616x^85+30617x^86+30902x^87+26578x^88+21594x^89+14972x^90+9950x^91+5950x^92+3252x^93+1447x^94+784x^95+276x^96+132x^97+59x^98+26x^99+16x^100+10x^101+8x^102+2x^103+5x^104+4x^105+2x^109

The gray image is a code over GF(2) with n=688, k=18 and d=308.
This code was found by Heurico 1.16 in 740 seconds.