The generator matrix

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

generates a code of length 13 over Z2[X]/(X^2) who´s minimum homogenous weight is 2.

Homogenous weight enumerator: w(x)=1x^0+13x^2+78x^4+286x^6+715x^8+1287x^10+1716x^12+8192x^13+1716x^14+1287x^16+715x^18+286x^20+78x^22+13x^24+1x^26

The gray image is a linear code over GF(2) with n=26, k=14 and d=2.
This code was found by Heurico 1.16 in 0.349 seconds.