The generator matrix

 1  1  1  1  X
 0  X  0  0 X^2+X
 0  0  X  0  X
 0  0  0  X  X
 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+45x^2+338x^4+256x^5+338x^6+45x^8+1x^10

The gray image is a linear code over GF(2) with n=20, k=10 and d=4.
This code was found by Heurico 1.16 in 0.000482 seconds.