In this chapter we introduce our task: communicating a digital message without error (or with as few errors as possible) despite an imperfect communications medium. Figure 1.1 shows a typical communications system. In this text we will assume that our source is producing binary data, but it could equally be an analog source followed by analog-to-digital conversion. Through the early 1940s, engi...

of the Dissertation Reed-Solomon Codes and the Deep Hole Problem (DRAFT) By Matt Keti Doctor of Philosophy in Mathematics University of California, Irvine, 2015 Professor Professor Daqing Wan, Chair In many types of modern communication, a message is transmitted over a noisy medium. When this is done, there is a chance that the message will be corrupted. An error-correcting code adds redundant ...

In this paper, we present a novel class of forward error correcting codes constructed using the discrete Walsh transform. They are a class of double-error correcting codes defined on the field of real numbers. An iterative decoding algorithm for Walsh transform codes is developed and implemented. The error correcting performance of Walsh transform codes over an AWGN channel is evaluated. Select...

The previous chapter defined some properties of linear block codes and discussed two examples of linear block codes (rectangular parity and the Hamming code), but the approaches presented for decoding them were specific to those codes. Here, we will describe a general strategy for encoding and decoding linear block codes. The decoding procedure we describe is syndrome decoding, which uses the s...

A parallel concatenated coding scheme consists of two simple systematic constituent encoders linked by an interleaver. The input bits to the first encoder are scrambled by the interleaver before entering the second encoder. The codeword of the parallel concatenated code consists of the input bits followed by the parity check bits of both encoders. In this paper we propose a method to evaluate t...

Orthogonal Frequency Division Multiplexing (OFDM) has gained a lot of popularity over the years. Due to its popularity, OFDM has been adopted as a standard in cellular technology and Wireless Local Area Network (WLAN) communication systems. To improve the bit error rate (BER) performance, forward error correction (FEC) codes are often utilized to protect signals against unknown interference and...

Multi carrier code division multiple access (MC-CDMA), a strong contender for future mobile communication is a combination of orthogonal frequency division multiplexing (OFDM) and code division multiple access (CDMA). Turbo coding technique can be used for improving the performance of MC-CDMA system in terms of bit error rate (BER). The objective of this work is to form a coding scheme that red...

We study the relation between locally testable and locally decodable codes. Locally testable codes (LTCs) are error-correcting codes for which membership of a given word in the code can be tested probabilistically by examining it in very few locations. Locally decodable codes (LDCs) allow to recover each message entry with high probability by reading only a few entries of a slightly corrupted c...

In high-speed satellite-to-ground laser communications, it has been shown that the Markov-based erasure channel model is well suited for the downlink experiment results. In that model, since the average erasure rate is relatively high, a low-rate error correction code is needed. In this paper, we focus on the application of low rate systematic Luby transform (SLT) code and show that the perform...

Given an error-correcting code over strings of length n and an arbitrary input string also of length n, the list decoding problem is that of finding all codewords within a specified Hamming distance from the input string. We present an improved list decoding algorithm for decoding Reed-Solomon codes. The list decoding problem for Reed-Solomon codes reduces to the following “curve-fitting” probl...