The generator matrix

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

generates a code of length 87 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 83.

Homogenous weight enumerator: w(x)=1x^0+204x^83+638x^84+708x^85+644x^86+496x^87+352x^88+300x^89+218x^90+120x^91+131x^92+76x^93+98x^94+52x^95+26x^96+28x^97+1x^100+3x^104

The gray image is a code over GF(2) with n=696, k=12 and d=332.
This code was found by Heurico 1.16 in 0.468 seconds.