On Extended Codes of Linear Codes Over Finite Fields

The Hong Kong University of Science and Technology
Department of Computer Science and Engineering


PhD Thesis Defence


Title: "On Extended Codes of Linear Codes Over Finite Fields"

By

Miss Tingfang CHEN


Abstract:

The construction of linear codes over finite fields with good parameters is 
one of the most interesting and fundamental topics in coding theory. This 
problem is of particular interest especially if the parameters of the code 
are "optimal" in a well-known sense. One method to construct new codes from 
known linear codes is to extend them, and one of the classical ways of 
extending an $[n,k,d]$ linear code is to add an overall parity-check bit. The 
main goal of investigating extended codes is to generate codes with new 
parameters, weight distributions, and new properties, which may get codes 
with some interesting applications. The study of the extended codes also 
helps to verify some properties of the original linear codes.

However, the extended codes of linear codes have not yet been well studied. 
The standardly extended codes of certain families of cyclic codes have been 
investigated to some extent, but little is known about their parameters or 
properties. There are a very small number of references and results on the 
extended codes, and the properties of the extended codes of most linear codes 
are open in general, requiring further investigation.

In this thesis, we introduce the descriptions of standardly and nonstandardly 
extended codes of linear codes. Some general theorems and basic results about 
the standardly extended codes are explored. We investigate the extended codes 
of some classical linear codes, including cyclic codes, nonbinary Hamming 
codes and projective two-weight codes. Several families of distance-optimal 
or dimension-optimal linear codes are obtained with this extending technique, 
and the parameters of certain extended codes of many families of linear codes 
are also settled.


Date:                   Monday, 19 May 2025

Time:                   2:30pm - 4:30pm

Venue:                  Room 2128B
                        Lift 19

Chairman:               Dr. Ivan Chi Ho IP (MATH)

Committee Members:      Prof. Cunsheng DING (Supervisor)
                        Dr. Dimitris PAPADOPOULOS
                        Prof. Ke YI
                        Dr. Maosheng XIONG (MATH)
                        Prof. Qing XIANG (SUSTech)