The generator matrix

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

generates a code of length 5 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+173x^2+1728x^3+11986x^4+37760x^5+11986x^6+1728x^7+173x^8+1x^10

The gray image is a code over GF(2) with n=40, k=16 and d=8.
This code was found by Heurico 1.16 in 0.531 seconds.