Extended codes of linear codes over finite fields

PhD Thesis Proposal Defence


Title: "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 in verifying some 
properties of the original linear codes.

However, the extended codes of linear codes are not well studied yet. 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:                   Thursday, 26 September 2024

Time:                   9:30am - 11:30am

Venue:                  Room 5501
                        Lifts 25/26

Committee Members:      Prof. Cunsheng Ding (Supervisor)
                        Dr. Dimitris Papadopoulos (Chairperson)
                        Dr. Amir Goharshady
                        Dr. Maosheng Xiong (MATH)