Hirschfeld, JamesAlharbi, Abdulaziz2023-11-212023-11-212023-11-09https://hdl.handle.net/20.500.14154/69746Coding theory is a fascinating area that has its roots in the early 20th century when telegraph communication was prevalent. Back then, Morse code was the primary mode of transmit- ting messages, but it was prone to errors due to noise and other factors. To address this problem, mathematicians formulated error-correcting codes that were created to detect and rectify errors in telegraph message transmissions. These codes were developed based on con- cepts derived from pure mathematics and highlighted the potency of algebraic principles, for more details see [2] and [3]. A redundancy code, called also an error-correcting code, is a systematic method of representing data in a manner that allows for the detection and rectifi- cation of errors. By adding extra bits to introduce redundancy, the code enhances the data’s resilience. In case some bits experience corruption during transmission, the original data can still be reconstructed or recovered. This approach safeguards the integrity and reliabil- ity of the transmitted information, preventing and ensuring the accuracy of the received data. The main objective of this theory is to achieve efficient and dependable communica- tion, even in challenging environments. Efficiency entails minimizing the time and effort required for information transfer. On the other hand, reliability demands that the received data closely resembles the transmitted data. Coding theory aims to actualize these limits by constructing models primarily using algebraic techniques. This approach allows to bridge the gap between theoretical bounds and practical implementation. There are many types of error-correcting codes, but some of the most commonly used ones are the Hamming code, the Reed-Solomon code, and the convolutional code. These codes differ in their design and performance characteristics, and the choice of code depends on the specific application and the trade-offs between factors such as error rate, data rate, and complexity, see [4] and [1]. The Hamming code is a simple, single-error-correcting code that adds parity bits to the data to detect and correct one-bit errors. The Reed-Solomon code is a more powerful code that can correct multiple errors and has been used in many applications, including CDs and DVDs. Convolutional codes are a type of error-correcting code that is often used in wireless communication networks. In modern times, coding theory emphasizes the development and analysis of error- correcting codes, which play a vital role in ensuring the accurate and reliable transmission 1 of information over noisy communication channels. However, communication channels are often characterized by noise and susceptibility to errors, underscoring the importance of re- liable information transmission. Error-correcting codes are used in various applications such as deep space communication and the design of registration numbers, (see [5] and [1] for more information). The development of this theory has led to the creation of related areas such as cryp- tography, network coding, and compressed sensing. In particular, error-correcting codes are crucial to modern communication systems. This thesis explores the fundamental principles of coding theory and their practical applications to real-world communication systems. The focus will be on designing and analyzing error-correcting codes, including linear codes, cyclic codes, and Hamming codes. By understanding these codes, we can improve the efficiency and reliability of communication systems, which is essential in our interconnected world. For instance, in order to ensure the fast and reliable transmission of binary data (a stream of 0s and 1s ) over a noisy communication channel, it is essential to address the issue of errors that may occur. This process is similar to encrypting the data, and the same concept can also be applied to information storage systems, where the storage medium is treated as a channel. Figure 1.1 depicts a general digital communication system, (see [5] for more details). Message Codeword Received vector Decoded message Message source Encoder Channel Noise Figure 1.1: General digital communication system. Example 1.1. Suppose we examine a scenario in which we aim to transmit only two mes- sages, ’YES’ and ’NO’, using the following encoding: YES=1, NO=0. If an an error occurs, such as transmitting a 1 and receiving a 0, it can go unnoticed or undetected. So, here is some redundancy: YES=11, NO=00. If message 11 is transmitted and message 01 is received instead, an error is detected but left uncorrected. To address this issue, the introduction of additional redundancy can be employed: YES = 1111, NO = 0000. If the received message is 0100 and assuming that only one error occurred during transmis- sion, we can infer that the original message sent was 0000, corresponding to ’NO’.43encodingtheoryThesis