The generator matrix

 1  0  0  0  1
 0  1  0  0 X+1
 0  0  1  0  1
 0  0  0  1  1
 0  0  0  0 X^2

generates a code of length 5 over Z2[X]/(X^3) who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+125x^2+480x^3+2050x^4+2880x^5+2050x^6+480x^7+125x^8+1x^10

The gray image is a linear code over GF(2) with n=20, k=13 and d=4.
As d=4 is an upper bound for linear (20,13,2)-codes, this code is optimal over Z2[X]/(X^3) for dimension 13.
This code was found by Heurico 1.16 in 0.0128 seconds.