The generator matrix

 1  0  0  0  1  1  1 X+2  1  1
 0  1  0  0  X X^2+1  1  1 X+1  0
 0  0  1  0 X+1  1  X X+1  3 X^2
 0  0  0  1  1  X X+1 X^2+1 X^2+3 X^2

generates a code of length 10 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 6.

Homogenous weight enumerator: w(x)=1x^0+96x^6+938x^7+3971x^8+15024x^9+25421x^10+15140x^11+3930x^12+896x^13+114x^14+2x^15+2x^16+1x^18

The gray image is a code over GF(2) with n=80, k=16 and d=24.
This code was found by Heurico 1.16 in 2.42 seconds.