# Asymptotic Properties of Quasi-Group Codes

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.

• 9 publications
• 5 publications
research
07/03/2018

### The Concatenated Structure of Quasi-Abelian Codes

The decomposition of a quasi-abelian code into shorter linear codes over...
research
07/30/2020

### Quasi-Cyclic Codes

Quasi-cyclic codes form an important class of algebraic codes that inclu...
research
06/18/2019

### Asymptotic performance of metacyclic codes

A finite group with a cyclic normal subgroup N such that G/N is cyclic i...
research
08/17/2021

### Self-dual 2-quasi-abelian Codes

A kind of self-dual quasi-abelian codes of index 2 over any finite field...
research
03/10/2022

### Double Constacyclic Codes over Two Finite Commutative Chain Rings

Many kinds of codes which possess two cycle structures over two special ...
research
01/23/2021

### Automorphism Groups and Isometries for Cyclic Orbit Codes

We study orbit codes in the field extension 𝔽_q^n. First we show that th...
research
02/24/2021

### Simplified quasi-likelihood analysis for a locally asymptotically quadratic random field

The asymptotic decision theory by Le Cam and Hajek has been given a luci...