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

generates a code of length 84 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 81.

Homogenous weight enumerator: w(x)=1x^0+56x^81+234x^82+162x^83+224x^84+100x^85+120x^86+52x^87+47x^88+12x^89+13x^90+2x^91+1x^130

The gray image is a code over GF(2) with n=672, k=10 and d=324.
This code was found by Heurico 1.16 in 0.422 seconds.