The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  2  8  8  1  0  1  2  1  1  1  1  2  2 12  4  1  8  2  2  8  4
 0  2  0  6  0  6  8  6 14  0  0 14  6 14  2  2 10  2  4 14  0  4  8 14 10  6  2 12  6  4 14 10  4 12
 0  0 12  0  0  0 12  0  0  8  4  4  4  0 12  8 12  4  0  8  4 12 12  8  8  8  4 12  4  4  4 12 12  4
 0  0  0 12  0  0  0  4  8 12  4 12  0  4  4  8 12 12  4  0  8 12 12  4  4  0  4  8  0  8  0  0  8  4
 0  0  0  0 12  0  4  4 12  4  0  4  4  8 12  4  8  0  8 12  4 12  4  8  4  8  8  4  0  8  8  8  8  0
 0  0  0  0  0  8  0  0  8  8  8  0  8  8  8  0  0  8  0  8  8  8  0  8  8  0  8  8  0  0  0  8  8  0

generates a code of length 34 over Z16 who´s minimum homogenous weight is 27.

Homogenous weight enumerator: w(x)=1x^0+78x^27+240x^28+408x^29+763x^30+1174x^31+1886x^32+2450x^33+2414x^34+2446x^35+1923x^36+1190x^37+743x^38+314x^39+165x^40+110x^41+45x^42+20x^43+9x^44+2x^45+2x^46+1x^50

The gray image is a code over GF(2) with n=272, k=14 and d=108.
This code was found by Heurico 1.16 in 4.22 seconds.