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  1  1  1  1  X  X  X  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X
 0 X^2+2  0 X^2  0  0 X^2 X^2+2  0  0 X^2 X^2+2  0  0 X^2 X^2+2  2  2 X^2+2 X^2  2  2 X^2+2 X^2  2  2 X^2+2 X^2  2  2 X^2+2 X^2  2  0 X^2+2 X^2 X^2 X^2 X^2+2 X^2  2  0 X^2 X^2  0  0  2  2 X^2+2 X^2+2 X^2+2 X^2  2  0 X^2+2 X^2+2  2  0 X^2+2 X^2  0  0  2  2  0  0  2 X^2+2 X^2+2 X^2+2  2  0 X^2 X^2  0  2 X^2+2 X^2  2  0 X^2+2 X^2+2  2 X^2+2
 0  0 X^2+2 X^2  0 X^2+2 X^2  0  2 X^2 X^2+2  2  2 X^2 X^2+2  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+2 X^2  0 X^2 X^2+2  0  2 X^2 X^2+2  0  2 X^2 X^2+2 X^2 X^2+2  0  2  2  0 X^2+2 X^2  2  0 X^2+2 X^2 X^2+2 X^2 X^2+2 X^2  2  0  0  2  2  0  0  2  2  0  0  2 X^2 X^2+2 X^2 X^2+2  0  0 X^2+2  2 X^2+2 X^2 X^2+2 X^2 X^2+2  0
 0  0  0  2  2  2  0  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  0  0  2  2  2  2  2  2  0  0  2  2  2  2  0  0  2  2  0  0  2  2  2  2  0  0  0  0  0  0  0  0  0  0  2  2  0  0  0  0  2  2  0  2  2  2  2  2  0  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+56x^82+32x^83+356x^84+32x^85+12x^86+16x^88+4x^90+1x^104+2x^116

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 18.3 seconds.