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

generates a code of length 87 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 86.

Homogenous weight enumerator: w(x)=1x^0+84x^86+128x^87+14x^88+24x^90+4x^102+1x^112

The gray image is a code over GF(2) with n=696, k=8 and d=344.
This code was found by Heurico 1.16 in 0.547 seconds.