Asymptotic Properties of Quasi-Group Codes
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This is a manuscript of a chapter prepared for a book. The good codes possess large information length and large minimum distance. A class of codes is said to be asymptotically good if there exists a positive real δ such that, for any positive integer N we can find a code in the class with code length greater than N, and with both the rate and the relative minimum distance greater than δ. The linear codes over any finite field are asymptotically good. More interestingly, the (asymptotic) GV-bound is a phase transition point for the linear codes; i.e., asymptotically speaking, the parameters of most linear codes attain the GV-bound. It is a long-standing open question: whether or not the cyclic codes over a finite field (which are an important class of codes) are asymptotically good? However, from a long time ago the quasi-cyclic codes of index 2 were proved to be asymptotically good. This chapter consists of some of our studies on the asymptotic properties of several classes of quasi-group codes. We'll explain the studies in a consistent and self-contained style. We begin with the classical results on linear codes. In many cases we consider the quasi-group codes over finite abelian groups (including the cyclic case as a subcase of course), and study their asymptotic properties along two directions: (1) the order of the group (the coindex) is fixed while the index is going to infinity; (2) the index is small while the order of the group (the coindex) is going to infinity. Finally we describe the story on dihedral codes. The dihedral groups are non-abelian but near to cyclic groups (they have cyclic subgroups of index 2). The asymptotic goodness of binary dihedral codes was obtained in the beginning of this century, and extended to the general dihedral codes recently.
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