3 Error Correction
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citations to reliable sources. Unsourced material may be challenged and removed. (August 2008) (Learn how and error detection and correction using hamming code example when to remove this template message) In information theory and error correction and detection coding theory with applications in computer science and telecommunication, error detection and correction or error error correction code control are techniques that enable reliable delivery of digital data over unreliable communication channels. Many communication channels are subject to channel noise, and thus errors error correction techniques may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases. Contents 1 Definitions 2 History 3 Introduction 4 Implementation 5 Error detection schemes 5.1 Repetition codes 5.2 Parity bits
Reed Solomon Error Correction
5.3 Checksums 5.4 Cyclic redundancy checks (CRCs) 5.5 Cryptographic hash functions 5.6 Error-correcting codes 6 Error correction 6.1 Automatic repeat request (ARQ) 6.2 Error-correcting code 6.3 Hybrid schemes 7 Applications 7.1 Internet 7.2 Deep-space telecommunications 7.3 Satellite broadcasting (DVB) 7.4 Data storage 7.5 Error-correcting memory 8 See also 9 References 10 Further reading 11 External links Definitions[edit] The general definitions of the terms are as follows: Error detection is the detection of errors caused by noise or other impairments during transmission from the transmitter to the receiver. Error correction is the detection of errors and reconstruction of the original, error-free data. History[edit] The modern development of error-correcting codes in 1947 is due to Richard W. Hamming.[1] A description of Hamming's code appeared in Claude Shannon's A Mathematical Theory of Communication[2] and was quickly generalized by Marcel J. E. Golay.[3] Introduction[edit] The general idea for achieving error d
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Error Correcting Code Example
how and when to remove this template message) This article may be too technical error correcting codes pdf for most readers to understand. Please help improve this article to make it understandable to non-experts, without removing the technical details. error correcting codes lecture notes The talk page may contain suggestions. (February 2016) (Learn how and when to remove this template message) (Learn how and when to remove this template message) Binary Hamming Codes The Hamming(7,4)-code (with r = https://en.wikipedia.org/wiki/Error_detection_and_correction 3) Named after Richard W. Hamming Classification Type Linear block code Block length 2r − 1 where r ≥ 2 Message length 2r − r − 1 Rate 1 − r/(2r − 1) Distance 3 Alphabet size 2 Notation [2r − 1, 2r − r − 1, 3]2-code Properties perfect code v t e In telecommunication, Hamming codes are a family of linear error-correcting codes that generalize the https://en.wikipedia.org/wiki/Hamming_code Hamming(7,4)-code, and were invented by Richard Hamming in 1950. Hamming codes can detect up to two-bit errors or correct one-bit errors without detection of uncorrected errors. By contrast, the simple parity code cannot correct errors, and can detect only an odd number of bits in error. Hamming codes are perfect codes, that is, they achieve the highest possible rate for codes with their block length and minimum distance of three.[1] In mathematical terms, Hamming codes are a class of binary linear codes. For each integer r ≥ 2 there is a code with block length n = 2r − 1 and message length k = 2r − r − 1. Hence the rate of Hamming codes is R = k / n = 1 − r / (2r − 1), which is the highest possible for codes with minimum distance of three (i.e., the minimal number of bit changes needed to go from any code word to any other code word is three) and block length 2r − 1. The parity-check matrix of a Hamming code is constructed by listing all columns of length r that are non-zero, which means that the dual code of the Hamming code is the punctured Hadamard code.
this tutorial page. 6.02 Practice Problems: Error Correcting Codes Problem . For each of the following sets of codewords, please give the appropriate (n,k,d) designation where n is http://web.mit.edu/6.02/www/f2012/handouts/tutprobs/ecc.html number of bits in each codeword, k is the number of message bits http://www.computing.dcu.ie/~humphrys/Notes/Networks/data.hamming.html transmitted by each code word and d is the minimum Hamming distance between codewords. Also give the code rate. {111, 100, 001, 010} n=3, k=2 (there are 4 codewords), d = 2. The code rate is 2/3. {00000, 01111, 10100, 11011} n=5, k=2 (there are 4 codewords), d = 2. The code rate is 2/5. {00000} error correction A bit of a trick question: n=5, k=0, d = undefined. The code rate is 0 -- since there's only one codeword the receiver can already predict what it will receive, so no useful information is transferred. Problem . Suppose management has decided to use 20-bit data blocks in the company's new (n,20,3) error correcting code. What's the minimum value of n that will permit the code to be error correcting code used for single bit error correction? n and k=20 must satisfy the constraint that n + 1 ≤ 2n-k. A little trial-and-error search finds n=25. Problem . The Registrar has asked for an encoding of class year ("Freshman", "Sophomore", "Junior", "Senior") that will allow single error correction. Please give an appropriate 5-bit binary encoding for each of the four years. We want a (5,2,3) block code. For such a code, 00000 is a codeword by definition. Every other codeword must have weight at least 3, and 00111 is an obvious choice (or any permutation thereof). We now need only two more codewords and each must have at least three ones, and must also have a Hamming distance of 3 from the second codeword above. A little bit of trial and error shows that 11011 and 11100 work. So: 00000, 00111, 11011, 11100 should satisfy the Registrar Problem . For any block code with minimum Hamming distance at least 2t + 1 between code words, show that: An (n, k) block code can represent in its parity bits at most 2n-k patterns, and these must cover all the error cases we wish to correct, as well as the one case with no errors. When the minimum Hamming distan
limit for minimum number of check bits to do 1-bit error-correction. Bits of codeword are numbered: bit 1, bit 2, ..., bit n. Check bits are inserted at positions 1,2,4,8,.. (all powers of 2). The rest are the m data bits. Each check bit checks (as parity bit) a number of data bits. Each check bit checks a different collection of data bits. Check bits only check data, not other check bits. Hamming Codes used in: Wireless comms, e.g. Fixed wireless broadband. High error rate. Need correction not detection. Any number can be written as sum of powers of 2 First note every number can be written in base 2 as a sum of powers of 2 multiplied by 0 or 1. i.e. As a simple sum of powers of 2. Number is sum of these: 1 2 4 8 16 Number: 1 x 2 x 3 x x 4 x 5 x x 6 x x 7 x x x 8 x 9 x x 10 x x 11 x x x 12 x x 13 x x x 14 x x x 15 x x x x 16 x 17 x x 18 x x 19 x x x 20 x x 21 x x x 22 x x x 23 x x x x 24 x x 25 x x x 26 x x x 27 x x x x 28 x x x 29 x x x x 30 x x x x 31 x x x x x ... Scheme for check bits Now here is our scheme for which bits each check bit checks: Checked by check bit: 1 2 4 8 16 Bit: 1 (not applicable - this is a check bit) 2 (n/a) 3 x x 4 (n/a) 5 x x 6 x x 7 x x x 8 (n/a) 9 x x 10 x x 11 x x x 12 x x 13 x x x 14 x x x 15 x x x x 16 (n/a) 17 x x 18 x x 19 x x x 20 x x 21 x x x 22 x x x 23 x x x x 24 x x 25 x x x 26 x x x 27 x x x x 28 x x x 29 x x x x 30 x x x x 31 x x x x x 32 (n/a) ... Check bit records odd or even parity of all the bits it covers, so any one-bit error in the data will lead to error in the check bit. Assume one-bit error: If any data bit bad, then multiple check bits will be bad (never just one check bit). Calculating the Hamming Code (check bits do even parity here) How it works 21 (as sum of powers of 2) = 1 + 4 + 16 Bit 21 is checked by check bits 1, 4 and 16. No other bit is checked by exactly these 3 check bits. If assume one-bit error, then if exactly these 3 check bits are bad, then we know that data bit 21 was bad and no other. Assume one-bit error: Error in a data bit: Will cause multiple errors in check bits. Will cause errors in exactly the check bits that correspond to the powers of 2 that the bit number can be written as a sum of. Error in a check bit: Will affect nothing except that check bit. i.e. One bad check bit (not multiple bad check bits as above). Hamming Code example for 3-bit data Consider standard encoding of numbers 0 to 7: 000 001 010 011 100 101 110 111 (bits 1 to 3). Encode this such that a 1 bit error can be detected and corrected. Add check bits: cc0c00 cc0c01 cc0c10 cc0c11 cc1c00 cc1c01 cc1c10 cc1c11 (n