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

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+60x^82+407x^84+28x^86+7x^88+6x^90+1x^108+2x^114

The gray image is a code over GF(2) with n=672, k=9 and d=328.
This code was found by Heurico 1.16 in 0.657 seconds.