The generator matrix

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

generates a code of length 29 over Z2[X]/(X^2) who´s minimum homogenous weight is 16.

Homogenous weight enumerator: w(x)=1x^0+128x^16+53x^18+262x^20+142x^22+430x^24+256x^25+390x^26+1024x^27+630x^28+1536x^29+622x^30+1024x^31+415x^32+256x^33+472x^34+130x^36+250x^38+50x^40+106x^42+2x^44+10x^46+3x^50

The gray image is a linear code over GF(2) with n=58, k=13 and d=16.
This code was found by Heurico 1.16 in 4.87 seconds.