2-bit Error Detected
Contents |
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with ecc multiple bit error detected us Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics Stack Exchange is a question and ecc multiple bit error detected in memory module answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute: Sign up Here's how
Single Bit Event Upset Error Detected
it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Detect double error using Hamming code. up vote 3 down vote favorite I have a sequence of bits $$ 111011011110 $$
Hamming Code 2 Bit Error Correction
and need to detect two errors(without correction) using Hamming codes. Hamming codes contain a control bit in each $2^n$ position. Hence I should put this control bits in their positions. $$ 0010110011011110 $$ I've found a simple explanation of how to count the code for a sequence of bits. It says that each control bit responds for the following bits using these rules: First control bit responds for $2^n$ position and each following bit through $2^n$ . So the first bit responds for secded example the first, third, fifth and etc. bits. The second control bit responds for 2nd, 3rd, 6th, 7th, 10th, 11th and etc. bits. Third control bit(which is on the 4th position) responds for 4th, 5th, 6th, 7th, 12th, 13th etc. bits. And so on. The value of each of the controls bits is counted as a modulo sum of the bits, which this control bit responds for. Here is an illustration of what I mean: Assuming this rule is right, the last 16th bit(after control bits addition) is not under the responsibility of any of the control bits. So the question is: How can I detect double error(only detect, not correct) for the given sequence of bits using the Hamming code? combinatorics discrete-mathematics coding-theory share|cite|improve this question edited Apr 18 '13 at 2:29 Wolphram jonny 2741528 asked Apr 17 '13 at 10:52 RomanKapitonov 12315 Your diagram seems to indicate a new codeword starts at position $16$ since the pattern repeats,. Thus, the first $15$ bits include $4$ parity bits (using the nomenclature that is standard in coding theory will help get better answers), and so you have what is called a $[15,11]$ Hamming code. There is no way of detecting that two errors with this code. In order to detect two errors, you need to modify your scheme so that the $16$th bit is a parity check on _all_ $15$ previous bits (including the parity bits at positions $1,2,4,8$, and the next codeword starts at position $17$. &nda
article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (March 2013) (Learn how and when to remove this template message) This article may be too technical
Error Detection And Correction Using Hamming Code Example
for most readers to understand. Please help improve this article to make it understandable to non-experts, without double error detection hamming code example removing the technical details. The talk page may contain suggestions. (February 2016) (Learn how and when to remove this template message) (Learn how and single bit error correction when to remove this template message) Binary Hamming Codes The Hamming(7,4)-code (with r = 3) Named after Richard W. Hamming Classification Type Linear block code Block length 2r − 1 where r ≥ 2 Message length 2r − r − 1 http://math.stackexchange.com/questions/364324/detect-double-error-using-hamming-code 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 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 https://en.wikipedia.org/wiki/Hamming_code 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. The parity-check matrix has the property that any two columns are pairwise linearly independent. Due to the limited redundancy that Hamming codes add to the data, they can only detect and correct errors when the error rate is low. This is the case in computer memory (ECC memory), where bit errors are extremely rare and Hamming codes are widely used. In this context, an extended Hamming code having one extra parity bit is often used. Extended Hamming codes achieve a Hamming distance of fo
consecutive bit errors Error rate due to noise: The likelihood of errors due to noise depends on the transmission http://www.mathcs.emory.edu/~cheung/Courses/455/Syllabus/2-physical/errors.html medium Twisted pairs is pretty "noisy" and has an error rate of about 1 bit in every few 1000 bits transmitted. Optical fibers are very "noiseless" and can have 1 bit in several 1,000,000 bits may be in error.... Why detect errors Garbage in, garbage out: processing corrupted intermediate data will always result in bit error incorrect final outcome The sooner we can detect an error, the less computation resources we will waste Transmission errors can be detected at various levels Question we study here: How can we tell if there are errors at the physical (signal) level ? Detecting errors at physical level (signals): error detection and correction codes bit error detected Principe to detect/correct errors in a "sequence of bits": Embed extra bits into the transmitted message to conforms to a certain (mathematical) property The receiver checks for this property in the received sequence The mathematical theory that deal with this subject is called coding theory. Commonly used error detection/correction schemes Commonly used schemes: Error detection schemes: Parity checks Cyclic Redundancy Check (CRC) Error Correction Schemes: Hamming's code (can correct 1 bit error) Solomon-Reed's code (can correct 2 bit error)
One-dimensional Parity Scheme Even and Odd Parity: Even parity: Add one extra bit to the message so that the total number of 1's in the message is always even Odd parity: Add one extra bit to the message so that the total number of 1's in the message is always odd Examples: Transmitted data (unprotected): 1111000 1010101 1111111 Even parity: 11110000 10101010 11111111 Odd parity: 11110001 10101011 11111110 Error