On Hadamard $ 2 $-$ (51, 25, 12) $ and $ 2 $-$ (59, 29, 14) $ designs

Abstract

<p>In this paper, we classified Hadamard $ 2 $-$ (51, 25, 12) $ designs having a non-abelian automorphism group of order 10, i.e., $ {\rm{Frob}}_{10} \cong Z_5:Z_2 $, where a subgroup isomorphic to $ Z_2 $ fixed setwise all the $ Z_5 $-orbits of the sets of points and blocks. Furthermore, we classified $ 2 $-$ (59, 29, 14) $ designs having a non-abelian automorphism group of order 14, i.e., $ {\rm{Frob}}_{14} \cong Z_7:Z_2 $, where a subgroup isomorphic to $ Z_2 $ fixed setwise all the $ Z_7 $-orbits of the sets of points and blocks. We also showed that there was no Hadamard $ 2 $-$ (59, 29, 14) $ design with automorphism group $ {\rm{Frob}}_{21} \cong Z_7:Z_3 $, where a subgroup of order 3 fixed setwise all the $ Z_7 $-orbits of points and blocks. Additionally, we used Hadamard $ 2 $-$ (59, 29, 14) $ designs obtained to construct ternary linear codes and linear codes over the finite field of order 5. We constructed ternary self-dual codes and self-dual codes over the finite field of order 5 from the corresponding Hadamard matrices of order 60.</p>