The generator matrix

 1  0  0  1
 0  1  0  1
 0  0  1  1
 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+76x^2+192x^3+486x^4+192x^5+76x^6+1x^8

The gray image is a code over GF(2) with n=16, k=10 and d=4.
This code was found by Heurico 1.16 in 0.000261 seconds.