The generator matrix

 1  0  0  1  1
 0  1  0  0  2
 0  0  1  0  0
 0  0  0  X  0
 0  0  0  0  X

generates a code of length 5 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 1.

Homogenous weight enumerator: w(x)=1x^0+18x^1+125x^2+504x^3+1666x^4+3564x^5+1666x^6+504x^7+125x^8+18x^9+1x^10

The gray image is a code over GF(2) with n=20, k=13 and d=2.
This code was found by Heurico 1.16 in 0.00929 seconds.