The generator matrix

 1  1
 0 X^3+X^2

generates a code of length 2 over Z2[X]/(X^4) who´s minimum homogenous weight is 1.

Homogenous weight enumerator: w(x)=1x^0+4x^1+54x^2+4x^3+1x^4

The gray image is a linear code over GF(2) with n=16, k=6 and d=4.
As d=6 is an upper bound for linear (16,6,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 6.
This code was found by Heurico 1.16 in 3.81e-009 seconds.