The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+213x^6+1724x^7+8208x^8+30048x^9+50711x^10+29976x^11+8268x^12+1728x^13+179x^14+12x^15+3x^16+1x^18

The gray image is a code over GF(2) with n=80, k=17 and d=24.
This code was found by Heurico 1.16 in 16.1 seconds.