Introduction to Coding Theory

Abstract

Error-correcting codes constitute one of the key ingredients in achieving the high degree of reliability required in modern data transmission and storage systems. This 2006 book introduces the reader to the theoretical foundations of error-correcting codes, with an emphasis on Reed-Solomon codes and their derivative codes. After reviewing linear codes and finite fields, the author describes Reed-Solomon codes and various decoding algorithms. Cyclic codes are presented, as are MDS codes, graph codes, and codes in the Lee metric. Concatenated, trellis, and convolutional codes are also discussed in detail. Homework exercises introduce additional concepts such as Reed-Muller codes, and burst error correction. The end-of-chapter notes often deal with algorithmic issues, such as the time complexity of computational problems. While mathematical rigor is maintained, the text is designed to be accessible to a broad readership, including students of computer science, electrical engineering, and mathematics, from senior-undergraduate to graduate level.

TCP-TFEC: TCP Congestion Control based on Redundancy Setting Method for FEC over Wireless LAN

“…FEC uses the Reed-Solomon code because it is strong against burst error. The Reed-Solomon code is one of block code [21]. In the Reed-Solomon code, information is divided into consecutive bits of M information (referred to as a symbol), and the information is encoded and decoded using these symbols.…”

The Minimum Distance of Graph Codes

“…In the bipartite case also −n is an eigenvalue of A, and the corresponding eigenvector has 1s in the first half of the positions and -1 in the rest. It is known [8] that for a connected graph −n ≤ λ i ≤ n where λ i is any eigenvalue and that the second largest eigenvalue λ is closely related to the expansion properties of the graph. Large random graphs and known families with good expansion properties have λ = 2 √ n − 1 [7].…”

“…We follow the standard line of proof by defining a vector v as a modified indicator vector for the sets S and T , and then apply a well-known result (see e.g. [8], Lemma 13.6)…”

“…The recent textbook by Roth [8] contains a thorough presentation of the subject. In this paper we consider some classes of graph codes and in particular codes from bipartite expander graphs based on finite geometries.…”

PEDS: A Parallel Error Detection Scheme for TCAM Devices

“…Now, since the minimum distance ofC is 2t+1, all vectors in B M of Hamming weight t or less must have distinct syndromes. Furthermore, there are families of codes, such as BCH codes (and Hamming codes as a special case), for which the inverse mapping from syndromes to vectors of Hamming weight ≤ t can be computed efficiently [16,, [18,Chapter 7].…”