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

generates a code of length 87 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 80.

Homogenous weight enumerator: w(x)=1x^0+494x^80+2082x^81+2942x^82+4920x^83+5612x^84+6548x^85+6880x^86+7402x^87+6816x^88+6922x^89+4919x^90+4336x^91+2593x^92+1554x^93+750x^94+372x^95+163x^96+132x^97+45x^98+24x^99+17x^100+10x^101+2x^103

The gray image is a code over GF(2) with n=696, k=16 and d=320.
This code was found by Heurico 1.16 in 51.7 seconds.