The generator matrix
1 0 0 0 1 1 1 0 1 1 1 X 1 X X 1 1 X 1 1 0 X 1 1 0 1 1
0 1 0 0 0 1 1 1 0 0 1 1 X+1 1 X 0 1 1 X X+1 0 0 X X 1 X 0
0 0 1 0 1 1 0 1 0 1 1 0 0 X+1 1 0 0 0 X 0 X 1 1 X X 0 0
0 0 0 1 1 0 1 1 1 0 1 1 X 0 1 0 1 X+1 X+1 0 1 X+1 X X+1 0 1 0
0 0 0 0 X 0 0 0 0 0 0 X 0 X 0 X X X 0 X X X X X X X 0
0 0 0 0 0 X 0 0 0 0 0 X 0 0 X X X X 0 0 0 0 0 X X 0 X
0 0 0 0 0 0 X 0 0 0 0 0 0 X X X 0 X 0 X X 0 0 0 X 0 0
0 0 0 0 0 0 0 X 0 0 0 0 X 0 X X 0 0 X 0 X 0 X X 0 0 X
0 0 0 0 0 0 0 0 X 0 0 0 X X X 0 X 0 0 0 0 X 0 X X X 0
0 0 0 0 0 0 0 0 0 X 0 X X 0 X 0 X 0 X X X X X 0 0 X 0
0 0 0 0 0 0 0 0 0 0 X X X X 0 0 0 0 X X 0 X 0 X 0 0 X
generates a code of length 27 over Z2[X]/(X^2) who´s minimum homogenous weight is 16.
Homogenous weight enumerator: w(x)=1x^0+68x^16+72x^17+257x^18+300x^19+603x^20+876x^21+1321x^22+1992x^23+2545x^24+3152x^25+3444x^26+3608x^27+3304x^28+3128x^29+2530x^30+2000x^31+1449x^32+904x^33+569x^34+284x^35+203x^36+60x^37+69x^38+8x^39+17x^40+2x^42+2x^44
The gray image is a linear code over GF(2) with n=54, k=15 and d=16.
This code was found by Heurico 1.16 in 21.7 seconds.