The generator matrix

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

generates a code of length 86 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+173x^80+708x^81+1460x^82+1526x^83+2226x^84+1618x^85+2126x^86+1362x^87+1593x^88+1148x^89+1068x^90+506x^91+390x^92+180x^93+136x^94+94x^95+19x^96+24x^97+6x^98+14x^100+2x^101+3x^102+1x^110

The gray image is a code over GF(2) with n=688, k=14 and d=320.
This code was found by Heurico 1.16 in 4 seconds.