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Topic: Hamming code

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Error correcting code

Channel coding

Gray code

Hagelbarger code

Binary Golay code

Hamming distance

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Richard Hamming

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	Hamming Codes
		The fundamental principal embraced by Hamming codes is
		Hamming codes, as mentioned before, are capable of correcting one error or detecting two errors but not capable of doing both simultaneously.
		A Hamming code word is generated by multiplying the data bits by a generator matrix G using modulo-2 arithmetic.
	www2.rad.com /networks/1994/err_con/hamming.htm (777 words)

	Hamming (7,4) code
		Hamming, who had the use of the computer on weekends, was really frustrated by constantly having to restart his computations, so he devised the first error-correcting codes.
		Hamming noted that such a code must be able to detect n possible locations of errors, plus the possibility that there are no errors.
		He also invented the notion of perfect codes: a code for which there is an integer q such that the q-spheres c entered at the words are disjoint, and every vertex is in one of these q-spheres.
	www.cs.mcgill.ca /~smroso/hamming.html (1879 words)

	Applying Hamming Code to blocks of data
		Hamming code affords a straightforward way to protect a block of data against single bit errors.
		This is because the Hamming distance between 2 code points (the data and check bits combined are a “code point”) is enough that single bit errors can be corrected, but double bit errors appear to be a single bit error on a different code point.
		The term Hamming distance can be understood to refer to the number of bits that have to change to move from one code point to another.
	asics.chuckbenz.com /HammingCode.html (1338 words)

	Hamming code - Wikipedia, the free encyclopedia
		A code with this ability to reconstruct the original message in the presence of errors is known as an error-correcting code.
		Hamming also noticed the problems with flipping two or more bits, and described this as the "distance" (it is now called the Hamming distance, after him).
		Hamming was interested in two problems at once; increasing the distance as much as possible, while at the same time increasing the information rate as much as possible.
	en.wikipedia.org /wiki/Hamming_code (2508 words)

	Richard Hamming - Wikipedia, the free encyclopedia
		His contributions include the Hamming code (which makes use of a Hamming matrix), the Hamming window (described in section 5.8 of his book Digital Filters), Hamming numbers, Sphere-packing (or hamming bound) and the Hamming distance.
		The Richard W. Hamming Medal is an award given annually by IEEE for 'exceptional contributions to information sciences, systems and technology'.
		Hamming tries to extract general lessons -- both personal and technical -- to aid one in having a successful technical career by telling stories from his own experiences.
	en.wikipedia.org /wiki/Richard_Hamming (668 words)

	High speed Hamming code circuit and method for the correction of error bursts - Patent 4276647
		Hamming codes have the advantage of being able to detect two errors in a data field, and also to correct a one bit error.
		When Hamming codes are used in conjunction with other codes (parity checks, check sums, etc.) the detection of any number of error bits, or the corrosion of one error bit per record is obtainable.
		This invention enables a Hamming code system to correct up to a maximum of n bits for each data field by dividing the total data field into a plurality of 2.sup.m bit words where each word is defined as comprising every n.sup.th bit.
	www.freepatentsonline.com /4276647.html (4831 words)

	Hamming code - TheBestLinks.com - Hamming coding, ASCII, Algorithm, Bell Labs, ... (Site not responding. Last check: )
		Hamming also noticed the problems with flipping two or more bits, and described this as the "distance".
		Hamming Code adds three additional bits to every four data bits of the message.
		The clever part of Hamming Code is that the wrong parity bits encode the position of the incorrect data bit.
	www.thebestlinks.com /Hamming_coding.html (1172 words)

	Hamming code - Wikipedia, the free encyclopedia (Site not responding. Last check: )
		A code with this ability to reconstruct the original message in the presence of errors is known as an error-correcting code.
		Hamming was interested in two problems at once; increasing the distance as much as possible, while at the same time increasing the information rate as much as possible.
		Hamming codes can be used together with an extra parity bit, to allow for the detection of two-bit errors, without preventing single-bit errors from being corrected.
	www.kean.edu /~asetoode/home/crc2/hamming.htm (2097 words)

	The Hamming metric
		This (as we will see) measures the ability for the code to correct errors which might be introduced in transmission.
		In other words, the minimum distance of a linear code is the length of its shortest non-zero vector.
		Figure 3.1 graphs the capacity for a binary code.
	web.usna.navy.mil /~wdj/book/node121.html (617 words)

	The Laws of Cryptography: Hamming Code
		Codes that correct errors are essential to modern civilization and are used in devices from modems to planetary satellites.
		The Hamming code has been used for computer RAM, and is a good choice for randomly occurring errors.
		The Hamming code can accommodate any number of data bits, but it is interesting to list the maximum size for each number of check bits.
	www.cs.utsa.edu /~wagner/laws/hamming.html (937 words)

	9. Matrix Hamming Codes
		Hamming distance between two code words, A and B say, is the weight of their sum, which sum is another code word.
		One convenient form of a code (but not the only form and not even the most convenient from) is to have the first bits of the code word state the message exactly, and then tack on extra “check bits” that provide the redundancy needed to allow error correction.
		Find a code with the same code words that is in IZ form, then find the error, and the original message in terms of the basis for which the matrix takes IZ form; then take the same linear combination of those basis vectors as the message is of them.
	www-math.mit.edu /18.310/matrix_hamming_codes.html (2710 words)

	Error Control Codes (Site not responding. Last check: )
		A code is the set of all codewords of a given length that are constructed by adding a specified number of check digits in a specified way to a specified number of data bits.
		If a code has a minimum Hamming distance of 2, then at least two bits of any codeword would have to be transmitted in error or changed in memory in order to yield another codeword.
		Furthermore, the overhead due to Hamming coding decreases as the size of the data word increases: it is only 2% for a 512-byte data block, which requires only 10 check bits according to Hamming's scheme.
	www.augustana.ab.ca /~jmohr/courses/common/csc350/notes/error_codes.html (1040 words)

	Hamming Code (Site not responding. Last check: )
		Additionally, the use of Hamming code allows the erroneous bit to be identified.
		Hamming code has 11-bits comprising 7-bits of an ASCII code and an additional 4-bits which are parity bits.
		A 7-bit ASCII character may be represented by an 11-bit Hamming code in which the additional 4 parity bits take the positions 1, 2, 4 and 8 in the binary sequence and are 0 or 1 according to specified criteria which allow not only a single error to be detected, but also identified.
	www.vuw.ac.nz /scps-students/tech102/lectures/week2/ham.htm (346 words)

	Hamming-Grey Codes for Fingerprinting Descriptors
		Hamming "distance" is defined as the number of bits that are different between two binary fingerprints, [A^B] (where "[x]" is the number of bits set in "x", and "^" is the bitwise XOR operator).
		A set of Hamming codes are a set of code words that have atleast a specified number of bits different between any two non-identical codes.
		Grey codes themselves cannot be used directly to encode similarity as several pairs of codes (in addition to neighboring values) have a Hamming distance of 1, i.e.
	www.daylight.com /meetings/emug98/Sayle/codes.html (2337 words)

	Hamming - Hamming codes (Site not responding. Last check: )
		The fundamental principal embraced by Hamming codes is parity.
		Hamming code is a type of binary code that is used to detect and correct data transmission errors.
		A binary Hamming code H_r of length n==2^r-1 (with r>=2) is a linear code with parity-check matrix H whose columns consist of all nonzero binary vectors of
	spicy.mywebsiteinfo.com /?q=spicy-hamming (1040 words)

	Richard Hamming from FOLDOC (Site not responding. Last check: )
		Richard Hamming received his B.S. from the University of Chicago in 1937, his M.A. from the University of Nebraska in 1939, and his Ph.D. in mathematics from the University of Illinois at Urbana-Champaign in 1942.
		Hamming's fundamental paper on error-detecting and error-correcting codes ("Hamming codes") appeared in 1950.
		Although best known for error-correcting codes, Hamming was primarily a numerical analyst, working on integrating differential equations and the Hamming spectral window used for smoothing data before Fourier analysis.
	wombat.doc.ic.ac.uk /foldoc/foldoc.cgi?Richard+Hamming (316 words)

	Hamming Code (Site not responding. Last check: )
		Hamming codes, invented by R Hamming of Bell Labs, are used to generate FEC (Forward Error Correction) data.
		The codes are complied using a matrix, in which information and parity bits are combined in order to provide error checking and correction at the receiver.
		The basic code generates three parity bits for every four information bits, where as extended hamming codes increase the size of the matrix.
	www.mpirical.com /companion/Transmission/Hamming_Code.htm (67 words)

	Hamming code - Wikipedia, the free encyclopedia (Site not responding. Last check: )
		In telecommunication, a Hamming code is an error-detecting and error-correcting code, used in data transmission, that can (a) detect all single- and double-bit errors and (b) correct all single-bit errors.
		It was named after its inventor, Richard Hamming.
		Let us examine the Hamming (7, 4) code, in which n=7 and k=4.
	www.encyclopedia-online.info /Hamming_code (268 words)

	A Venn Diagram of the Hamming Code
		One subtracts 2 from 4096 to account for the empty set and the universal set, just as was the case for the Hamming code.
		Recall that with the Venn diagram of the Hamming code, there was a central circle for the number 8.
		In this proposed diagram, there should be 23 regions, all rotationally congruent to one another, and one central region which has 23-fold rotational symmetry, thus accounting for all 24 dimensions of the ambient vector space.
	homepages.wmich.edu /~drichter/hammingvenn.htm (506 words)

	Tables of Binary Lexicodes
		Notice that the Hamming code for 15 is the XOR of the Hamming codes for 1, 2, 4, 8, and that 1+2+4+8=15.
		One more thing: given the basis for a set with Hamming distance d (where d is odd), you can always extend that to a basis for for a set with even Hamming distance d+1 by adding one extra check bit to every value which is the XOR of all other bits in that value.
		These codes can be produced by grouping the data bits into bytes (that is, sets of 8 bits), then placing the check bits for all 256 combinations of those 8 bits in a 256-term array (one array per byte).
	burtleburtle.net /bob/math/lexicode.html (1191 words)

	Wired News: Passage: Richard W. Hamming, 82
		Richard Wesley Hamming, the mathematician responsible for devising an error-correcting protocol that was one of the key events in the development of modern computing, died Wednesday of a heart attack in Monterey, California.
		Hamming was a pioneer in the computing field, running the computing center for the Manhattan Project during World War II.
		Hamming was affiliated with Bell Labs through 1976, when he was named to a chair of computer science at the Naval Postgraduate School in Monterey.
	www.wired.com /news/technology/0,1282,9579,00.html (530 words)

	Hamming (7,4) Code Discussion and Implementation
		Traditional Hamming codes are (7, 4) codes, encoding four bits of data into seven bit blocks (a Hamming code word).
		Two Hamming codes with different parity bits or parity bits in a different bit position are considered equivalent.
		Suppose the Hamming code defined by the matrix G in the example above is being used and the code word 1011011 is received.
	michael.dipperstein.com /hamming/index.html (2977 words)

	Hamming Codes (Site not responding. Last check: )
		Example of a Hamming Matrix, whose columns are the binary representation of the numbers 1, 2, 3, 4, 5, and 6.
		These particular linear codes have a parity check matrix H (often the codes are developed through a parity matrix), in which the columns of this matrix consist of all nonzero binary vectors of length r (each vector used only once). In order to decode Hamming Codes, we use a simple syndrome decoding method.
		Where the codeword received is multiplied by the Hamming Matrix and the result is the binary representation of where the error occurred.
	www.math.uri.edu /~thoma/teaching/mth391_fall2004/hams.htm (231 words)

	EE4253 Hamming Code
		For example, given a valid Hamming codeword, a change in bit 3 changes three bits (1,2,3) such that the new codeword is a distance (d=3) from the initial word.
		The clever arrangement of the Hamming codewords ensures that this is the case for every valid codeword in the set.
		The Hamming code shown here (L=7,D=3) is useful because it is easy to generate and to check this particular set of codewords.
	www.ee.unb.ca /tervo/ee4253/hamming.htm (1094 words)

	PDC Live - Hamming Code
		This code is used extensively to protect the fields that have meaning to the Teletext system itself.
		In the table, the hammed byte value is used to index the row (most significant 4 bits) and column (least significant four bits), and the decoded nibble is read out of the table.
		The more efficient 24/18 code is based on the same principle of interleaved check bits and an overall parity bit as the simpler 8/4 code.
	pdc.ro.nu /hamming.html (886 words)

	What is Hamming code? - A Word Definition From the Webopedia Computer Dictionary
		Hamming code will detect any double errors but can only correct a single error.
		This method of error correction is best suited for situations in which randomly occurring errors are likely, not for errors that come in bursts.
		Richard Hamming, a theorist with Bell Telephone Laboratories in the 1940s, developed the Hamming code method of error correction in 1949.
	www.webopedia.com /TERM/H/Hamming_code.html (233 words)

	Constructing a Hamming Code
		It was Hamming's idea to use the sequence of parity checks to produce a syndrome.
		For now, let's assume that the minimum distance of the (7,4) Hamming code is 3.
		Hamming codes can be constructed in every dimension, but as we've already seen only those with n=2
	www.mdstud.chalmers.se /~md7sharo/coding/main/node32.html (572 words)

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