Group 10 - Linear Block Codes Linear Block Codes By William...

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Unformatted text preview: Linear Block Codes Linear Block Codes By William Gecosky What are Linear Block Codes used What are Linear Block Codes used for? Linear block codes are used in error correction and detection schemes. Linear block codes allow for more efficient encoding and decoding algorithms. Linear block codes are an important aspect of code theory. The elements in the binary linear block codes are called codewords History of Linear Block Codes History of Linear Block Codes The first error correction code was invented by Richard W. Hamming, a theorist in Bell Telephone laboratories in the late forties. Hamming discovered a solution that would allow a computer to overcome an input error, and restore the original input without having the program restart. The codes he'd invented, known as Hamming codes, were the first non­trivial linear block codes. What makes a code a block code? What makes a code a block code? In block coding, the binary information stream is segmented into message block of fixed length. The length of the input segment is denoted by k. Therefore, there are distinct input messages. The encoder, according to certain rules, transforms the input segment of length k into output segment of fixed length n. Therefore, there are distinct output code words. Such code is often called a (n, k) code. Linear Block Codes Applications Linear Block Codes Applications Linear codes are applied in methods of transmitting symbols (e.g., bits) on a communications channel so that, if errors occur in the communication, some errors can be detected by the recipient of a message block. Linear block code have numerous applications in error correction and detection. Linear block codes are used in many cryptosystems. Types of Linear Block Codes Types of Linear Block Codes Cyclic Codes Repetition Codes Parity Codes Polynomial Codes Reed Solomon Codes Algabraic Geometric Codes Reed­Muller Codes Perfect Codes Linear Block Codes Symbol Linear Block Codes Symbol Alphabets Linear block codes are summarized by their symbol alphabets (e.g. binary or ternary) and parameters (n,m, dmin ) where, I. n is the length of the codeword, in symbols, m is the number of source symbols that will be used for encoding at once, dmin is the minimum hamming distance for the code II. III. Block­Coding Features Block­Coding Features 1) 2) The diagram shown here illustrates the class of block coding techniques with included categories. BCH, Hamming, and Reed­Solomon codes are all special kinds of linear block codes. Linear Block Codes Properties Linear Block Codes Properties Linear block codes have the property of linearity, i.e the sum of any two codeword's is also a code word, and they are applied to the source bits in blocks, hence the name linear block codes. There are some block codes that are not linear, but it is difficult to prove that a code is a good one without this property Linear block codes are defined by high­density parity­check matrices. Linear Block Codes and Coding Linear Block Codes and Coding Theory Coding theory is a branch of mathematics and computer science dealing with the error­ prone process of transmitting data across noisy channels via clever means so that a large number of errors that occur can be corrected. Linear block codes are a large part of coding theory. There are two classes of codes, source coding and channel coding. Source coding attempts to compress the data from a source in order to transmit it more efficiently. This is commonly used on the Internet. Channel coding adds extra data bits, commonly called parity bits, to make the transmission of data more robust to disturbances present on the transmission channel Linear Block Codes in the Real Linear Block Codes in the Real World Compression of data on computers commonly uses linear block codes to reduce network load and make files smaller. A typical music CD uses a powerful Reed­Solomon (a type of linear block code) code to correct for scratches and dust. Cell phones also use powerful coding techniques to correct for the fading and noise of high frequency radio transmission. Data modems, telephone transmission and of course NASA all employ powerful channel coding to get the bits through. Works Cited Works Cited 1. 2. 3. 4. 5. Vera Pless, Introduction to the Theory of Error­Correcting Codes, John Wiley & Sons, Inc (1982) Audrey Terras (1999). Fourier Analysis on Finite Groups and Applications. Cambridge University Press "Linear Block Codes." Nation Master. 2006. 1 Feb 2009 <http://www.nationmaster.com/encyclopedia/Linear­block­ codes>. Copy the citation below and paste it into your document. "Coding Theory." Global Oneness. 1 Feb 2009 <http://www.experiencefestival.com/a/Coding_theory/id/19 89372>. Koren, Debby. Error Correcting Codes. Tel­Aviv University . 1 Feb 2009 <http://www2.rad.com/networks/2002/errors/>. ...
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