The generator matrix

 1  1
 0 X^3

generates a code of length 2 over Z2[X]/(X^4) who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+30x^2+1x^4

The gray image is a linear code over GF(2) with n=16, k=5 and d=8.
As d=8 is an upper bound for linear (16,5,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 5.
This code was found by Heurico 1.16 in 6.87e-008 seconds.