The generator matrix

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

generates a code of length 87 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 84.

Homogenous weight enumerator: w(x)=1x^0+174x^84+48x^85+224x^86+160x^87+220x^88+48x^89+96x^90+50x^92+2x^104+1x^128

The gray image is a code over GF(2) with n=696, k=10 and d=336.
This code was found by Heurico 1.16 in 1.06 seconds.