Self-dual 2-quasi-cyclic Codes And Dihedral Codes

With the development of coding theory,cyclic codes and quasi-cyclic codes have become important research topics in both theory and practice.Double circulant codes,2-quasi-cyclic codes and dihedral codes possess nice algebraic structures and are sdudied extensively.In this paper 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 precise relationship between the three kinds of codes.1.We show a necessary and sufficient condition for a 2-quasi-cyclic code being a dihedral code.2.We obtain a necessary and sufficient condition for a self-dual 2-quasi-cyclic code being a dihedral code(if char F =2),or a consta-dihedral code(if char F ?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.3.We show a necessary and sufficient condition that the three classes(the selfdual double circulant codes,the self-dual 2-quasi-cyclic codes,and the self-dual(consta-)dihedral codes)are coincide each other.