The generator matrix

 1  0  1  1  1  0  1  1  0  1  1  0  1  1  0  1  1  0  1  1  1  1  1  1  1  1
 0  1  1  0  1  1  0  1  1  0  1  1  0 X+1  1  0 X+1  1  0 X+1  X X+1 X+1  0  0  0
 0  0  X  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  X  0  X  X  X  X  X  0
 0  0  0  X  0  0  0  0  0  0  0  0  0  0  0  0  0  X  X  X  0  0  0  X  X  X
 0  0  0  0  X  0  0  0  0  0  0  0  0  0  X  X  X  X  0  X  X  X  0  0  0  X
 0  0  0  0  0  X  0  0  0  0  0  0  0  X  X  X  0  0  0  0  X  X  X  0  0  0
 0  0  0  0  0  0  X  0  0  0  0  0  0  X  0  X  X  X  X  0  X  0  0  X  0  0
 0  0  0  0  0  0  0  X  0  0  0  0  0  X  X  0  X  0  X  X  X  0  X  X  0  0
 0  0  0  0  0  0  0  0  X  0  0  0  X  X  X  X  X  X  0  0  0  0  0  0  0  0
 0  0  0  0  0  0  0  0  0  X  0  0  X  0  0  X  X  0  X  0  0  0  X  X  0  0
 0  0  0  0  0  0  0  0  0  0  X  0  X  0  X  0  X  0  0  X  X  0  X  0  X  0
 0  0  0  0  0  0  0  0  0  0  0  X  X  X  0  X  0  0  X  X  X  X  0  X  X  0

generates a code of length 26 over Z2[X]/(X^2) who´s minimum homogenous weight is 14.

Homogenous weight enumerator: w(x)=1x^0+52x^14+182x^16+64x^17+254x^18+192x^19+242x^20+640x^21+704x^22+1408x^23+1696x^24+1792x^25+2058x^26+1792x^27+1438x^28+1408x^29+740x^30+640x^31+409x^32+192x^33+242x^34+64x^35+110x^36+40x^38+16x^40+6x^42+2x^44

The gray image is a linear code over GF(2) with n=52, k=14 and d=14.
This code was found by Heurico 1.16 in 5.08 seconds.