Self-dual 2-quasi-cyclic codes and dihedral codes

Authors：Fan, Yun^*; Zhang, Yuchang

Source：Finite Fields and their Applications, 2023, 85: 1001002.

DOI：10.1016/j.ffa.2022.102127

• Finite fields
• Self-dual codes
• Double circulant codes
• 2-quasi-cyclic codes
• Dihedral group codes
• Consta-dihedral codes

We characterize the structure of 2-quasi-cyclic codes over a finite field F by the so-called Goursat Lemma. With the characterization, we exhibit a necessary and sufficient condition for a 2-quasi-cyclic code being a dihedral code. And we obtain a necessary and sufficient condition for a self-dual 2-quasi-cyclic code being a dihedral code (if charF = 2), or a consta-dihedral code (if charF =? 2). As a consequence, any self-dual 2-quasi-cyclic code generated by one element must be (consta-)dihedral. In particular, any self-dual double circulant code must be (consta-)dihedral. We also obtain necessary and sufficient conditions under which the three classes (the selfdual double circulant codes, the self-dual 2-quasi-cyclic codes, and the self-dual (consta-)dihedral codes) of codes coincide with each other.

• Institution
  y