However, certain classes of linear codes have tractable decoding algorithms:. A cyclic code is a linear code where a rotation of any codeword is another codeword.
Since all cyclic codes are linear codes, any linear combination of codewords is still a codeword. Cyclic codes are closely connected to polynomials, and we can write a codeword as coefficients of a polynomial: for example, is.
A cyclic code is then the set of polynomial multiples of a generator polynomial :. Furthermore, must have some additional properties:. For example, consider a binary cyclic code of block length 3, with generator polynomial.
Then the codewords are:. Cyclic codes are very good at correcting burst errors, where errors are assumed to occur consecutively, instead of distributed evenly across the entire word. However, cyclic codes do not always have high distance.
Some special classes of cyclic codes do have high distance, and are more suitable for general error correction:. What is a code? Linear Codes A linear code is a code where a linear combination of codewords is another codeword. A generator matrix for this code is: If we add up any linear combination of the rows of G remember to do arithmetic modulo 10 , we get a codeword.
In our case, a parity-check matrix is: Using the parity-check matrix, we can easily detect errors, but correcting to the nearest codeword is still NP-hard for general linear codes. However, certain classes of linear codes have tractable decoding algorithms: Hamming Codes : class of single-error correcting codes with distance 3. Binary Golay Code : block length 24 and distance 8, so it can correct 3 errors.
Shannon's colleague Richard Hamming had been laboring on error-correction for early computers even before Shannon's paper, and he made some of the first breakthroughs of coding theory. Although these notes shall discuss these areas as mathematical subjects, it must always be remembered that the primary motivation for such work comes from its practical engineering applications.
Mathematical beauty can not be the sole gauge of worth. Throughout this manuscript, one should concentrate on the algebra of coding theory, but keep in mind the fundamental bounds of information theory and the practical desires of engineering.
Jonathan I. Hall Homepage No information is available for this author.
Follow Book Categories. Cancel Delete. Reed-Solomon codes are used in audio CDs and mobile telephony and are especially good at correcting burst errors e.
To reach the planned 3 hours per week, lectures will only be held on the dates shown in the table below. Tutorials are held every week. Exercises and problem sheets will be available on Uniworx.
For you to have the best experience on Lulu. You must be logged in to post a review. Terms and Conditions: Jonathan I. Please log in. In the context of the simple repeat code, this may not appear effective. The Viterbi algorithm is the optimum algorithm used to decode convolutional codes.
Date: Feb. Suche Search Site only in current section.
"Notes on Coding Theory" -- J.I. Hall. These notes are not intended for broad distribution. If you want to use them in any way, please contact me. These notes were written over a period of years as part of an advanced under- .. Here we shall concentrate on the algebra of coding theory.