The generator matrix

 1  0  0  1  1  1  1  1  1  1
 0  1  0  1  0  0  0  0  0  0
 0  0  1  1  0  0  0  0  0  0
 0  0  0  X  0  0  0  0  0  0
 0  0  0  0  X  0  0  0  0  0
 0  0  0  0  0  X  0  0  0  0
 0  0  0  0  0  0  X  0  0  0
 0  0  0  0  0  0  0  X  0  0
 0  0  0  0  0  0  0  0  X  0
 0  0  0  0  0  0  0  0  0  X

generates a code of length 10 over Z2[X]/(X^2) who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+22x^2+141x^4+456x^6+1650x^8+3652x^10+1650x^12+456x^14+141x^16+22x^18+1x^20

The gray image is a linear code over GF(2) with n=20, k=13 and d=2.
This code was found by Heurico 1.16 in 0.0601 seconds.