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

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

Homogenous weight enumerator: w(x)=1x^0+114x^84+9x^88+2x^92+1x^96+1x^104

The gray image is a code over GF(2) with n=672, k=7 and d=336.
This code was found by Heurico 1.16 in 0.391 seconds.