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 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 0
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+55x^4+11x^6+330x^8+165x^10+462x^12+2048x^13+462x^14+165x^16+330x^18+11x^20+55x^22+1x^26
The gray image is a linear code over GF(2) with n=26, k=12 and d=4.
This code was found by Heurico 1.16 in 0.0499 seconds.