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

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

Homogenous weight enumerator: w(x)=1x^0+108x^86+10x^88+4x^92+4x^94+1x^96

The gray image is a code over GF(2) with n=688, k=7 and d=344.
This code was found by Heurico 1.16 in 0.766 seconds.