The generator matrix

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

generates a code of length 89 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+664x^82+1818x^83+3252x^84+4590x^85+5814x^86+6712x^87+7278x^88+7186x^89+6454x^90+6106x^91+5566x^92+4070x^93+2609x^94+1660x^95+997x^96+402x^97+196x^98+72x^99+34x^100+20x^101+31x^102+4x^105

The gray image is a code over GF(2) with n=712, k=16 and d=328.
This code was found by Heurico 1.16 in 50.9 seconds.