## Date of Submission

3-22-2008

## Date of Award

3-22-2009

## Institute Name (Publisher)

Indian Statistical Institute

## Document Type

Doctoral Thesis

## Degree Name

Doctor of Philosophy

## Subject Name

Mathematics

## Department

Theoretical Statistics and Mathematics Unit (TSMU-Kolkata)

## Supervisor

Sikdar, K. (TSMU-Kolkata; ISI)|Barua, R. (TSMU-Kolkata; ISI)

## Abstract (Summary of the Work)

In everyday life, there arise many situations where two parties, sender and receiver, need to communicate. The channel through which they communicate is assumed to be binary symmetric, that is, it changes 0 to 1 and vice versa with equal probability. At the receiverâ€™s end, the sent message has to be recovered from the corrupted received word using some reasonable mechanism. This real life problem has attracted a lot of research in the past few decades. A solution to this problem is obtained by adding redundancy in a systematic manner to the message to construct a codeword. The collection of all codewords forms a code. Study of codes is referred to as coding theory. Reed-Solomon, Reed-Muller, BCH, Goppa, etc., are some well-studied families of codes. There are many applications which use codes. A compact disc (CD) uses Reed-Solomon codes to recover the data from scratches. Cellphones use codes to correct fading and high frequency noises. Satellite communication is another high profile area which uses codes.Coding theory is a multi-disciplinary subject drawing inputs from electronics, computer science, algebra, geometry, combinatorics, etc. The subject originated with fundamental contributions from Shannon [57] and Hamming [36]. For a fine treatment of the subject refer to [46], [8] and [66].Coding theory deals with design of codes, efficient encoding and efficient decoding algorithms. Broadly, a code is a collection of tuples. But to make the analysis easier, it is assumed that this collection has some mathematical structure. Normally, the collection is assumed to form a Fq-vector subspace of F n q and such codes are said to be linear. Study of codes possessing some nice algebraic structure is called algebraic coding theory. This thesis concentrates only on linear codes. A linear code is often represented as a triple [n, k, d]q, where n denotes the length, k the dimension (also the message block length) and d the minimum distance of the code. Here, the distance refers to the Hamming distance between two tuples of length n over Fq, which is the number of coordinates where they differ.At the senderâ€™s end, a message m is encoded using the encoding function E of the underlying code, to obtain codeword c. This codeword is sent over the noisy binary symmetric channel. The channel introduces a random noise e, so that y = c + e is received at the other end. At the receiverâ€™s end, the sent message must be recovered form y using some reasonable mechanism.Let D denote the decoding function, which satisfies D Â· E(m) = m. A Hamming sphere in F n q of radius r centered at x is the collection of all vectors having distance less than r from x. Spheres of radius greater than or equal to âŒŠ d 2 âŒ‹ may have common points, but those of radius at most âŒŠ dâˆ’1 2 âŒ‹ are disjoint. Hence, it is assumed that the channel corrupts at most âŒŠ dâˆ’1 2 âŒ‹ coordinates. If the received word y ends up in a sphere with centre c = E(m1), then y is decoded as m1. This is known as unique decoding.Associated with any linear code are two matrices G and H, which are known as the generator and parity check matrices respectively. The code C is defined using these matrices as follows:

## Control Number

ISILib-TH317

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

## DOI

http://dspace.isical.ac.in:8080/jspui/handle/10263/2146

## Recommended Citation

Das, M. Prem Laxman Dr., "Studies on Construction and List Decoding of Codes on Some Towers of Function Fields." (2009). *Doctoral Theses*. 39.

https://digitalcommons.isical.ac.in/doctoral-theses/39

## Comments

ProQuest Collection ID: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:28842815