The generator matrix
1 0 0 0 1 1 1 1 1 1 1 X 1
0 1 0 0 1 0 0 1 1 0 0 0 0
0 0 1 0 1 0 1 0 1 0 0 0 0
0 0 0 1 1 1 0 0 1 0 0 0 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+103x^4+347x^6+128x^7+1514x^8+1280x^9+2565x^10+3968x^11+3662x^12+5632x^13+3662x^14+3968x^15+2565x^16+1280x^17+1514x^18+128x^19+347x^20+103x^22+1x^26
The gray image is a linear code over GF(2) with n=26, k=15 and d=4.
This code was found by Heurico 1.16 in 1.63 seconds.