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

generates a code of length 84 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 82.

Homogenous weight enumerator: w(x)=1x^0+24x^82+200x^84+24x^86+2x^88+1x^96+4x^100

The gray image is a code over GF(2) with n=672, k=8 and d=328.
This code was found by Heurico 1.16 in 0.641 seconds.