The generator matrix

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

generates a code of length 4 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+52x^2+96x^3+214x^4+96x^5+52x^6+1x^8

The gray image is a code over GF(2) with n=16, k=9 and d=4.
This code was found by Heurico 1.16 in 0.000132 seconds.