The generator matrix

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

generates a code of length 13 over Z2[X]/(X^2) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+78x^4+715x^8+3764x^12+2048x^14+1287x^16+286x^20+13x^24

The gray image is a linear code over GF(2) with n=26, k=13 and d=4.
This code was found by Heurico 1.16 in 0.372 seconds.