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	Decoding the Golay code by hand (Site not responding. Last check: 2007-10-30)
		The [24,12,8] extended binary Golay code is a well-known and remarkable combinatorial object.
		The automorphism group of the Golay code (the group of permutations of the coordinates that send codewords to codewords) is the famous Mathieu group M24 -- well known as one of the first examples of a sporadic simple group.
		She used a characterization of the Golay code as the unique code which projects onto the [6,3,4]--GF(4) hexacode and satisfies certain parity conditions.
	www.math.uic.edu /~fields/DecodingGolayHTML/introduction.html (475 words)

	PlanetMath: binary Golay code
		is the quadratic residue code of length 23.
		The Golay Code is used to define the Leech Lattice, one of the most efficient sphere-packings known to date.
		This is version 1 of binary Golay code, born on 2004-06-04.
	planetmath.org /encyclopedia/BinaryGolayCode.html (195 words)

	Linear code - Wikipedia, the free encyclopedia
		A linear code of length n and rank k is a linear subspace C with dimension k of the vector space
		Codes in general are often denoted by the letter C.
		A linear code of length n, rank k (ie having k codewords in its basis and k rows in its generating matrix) and minimum Hamming weight d is referred to as an (n,k,d) code.
	en.wikipedia.org /wiki/Linear_code (654 words)

	Error Detection and Correction Encyclopedia @ SwiftRecovery.net (Swift Recovery) (Site not responding. Last check: 2007-10-30)
		The effectiveness of the coding scheme is measured in terms of the Reed-Muller code, which is the difference of the Fountain codes levels of the uncoded and coded systems required to reach the same Memory errors and SECDED levels.
		Iterative Viterbi Decoding tells us that whatever the probability of error in transmission or storage, it is possible to construct error-correcting codes in which the likelihood of failure is arbitrarily low, although this requires adding increasing amounts of Golay code to the original, which might not be practical when the error probability is very high.
		Other block error-correcting codes, such as Golay (24,12,8) code transform a chunk of bits into a (longer) chunk of bits in such a way that errors up to some threshold in each block can be detected and corrected.
	www.swiftrecovery.net /encyclopedia/Error_detection_and_correction (2340 words)

	Reference.com/Encyclopedia/Binary Golay code
		The binary Golay code, along with the ternary Golay code, has a particularly deep and interesting connection to the theory of finite sporadic groups in mathematics.
		The other, the perfect binary Golay code, has codewords of length 23 and is obtained from the extended binary Golay code by deleting one coordinate position (conversely, the extended binary Golay code is obtained from the perfect binary Golay code by adding a parity bit).
		The perfect binary Golay code is a perfect code; that is, the spheres of radius 3 around code words form a partition of the vector space.
	www.reference.com /browse/wiki/Binary_Golay_code (907 words)

	PlanetMath: even code
		A binary code is called even if the Hamming weight of all its codewords is even.
		An even code which is not doubly-even is said to be strictly even.
		This is version 1 of even code, born on 2005-05-12.
	planetmath.org /encyclopedia/EvenCode.html (121 words)

	Free Online Encyclopaedia
		The effectiveness of the coding scheme is measured in terms of the Coding gain, which is the difference of the SNR levels of the uncoded and coded systems required to reach the same BER levels.
		Information theory tells us that whatever the probability of error in transmission or storage, it is possible to construct error-correcting codes in which the likelihood of failure is arbitrarily low, although this requires adding increasing amounts of redundant data to the original, which might not be practical when the error probability is very high.
		Other block error-correcting codes, such as Reed-Solomon codes, transform a chunk of bits into a (longer) chunk of bits in such a way that errors up to some threshold in each block can be detected and corrected.
	www.prohits.org /wiki/Error_detection (2554 words)

	Soft Decoding Techniques for Codes and Lattices, Including the Golay Code and the Leech Lattice - Conway, Sloane ... (Site not responding. Last check: 2007-10-30)
		26 the enumeration of self-dual codes (context) - Conway, Pless - 1980
		10 the classification and enumeration of self-dual codes (context) - Pless, Sloane - 1975
		1 An algorithm for soft-decision decoding of the Golay and oth..
	citeseer.ist.psu.edu /conway86soft.html (1343 words)

	Citations: Lexicographic Codes: Error Correcting Codes from Game Theory - Conway, Sloane (ResearchIndex) (Site not responding. Last check: 2007-10-30)
		Lexicodes have surprisingly good encoding parameters and include, among other famous optimal codes, the Hamming codes, the binary Golay code, and certain quadratic residue codes [5, 10] Several authors [3, 5, 9] have proved that lexicodes are linear, and comparison with optimal linear codes of....
		This paper is concerned with various classes of lexicographic codes, that is codes that are defined by a greedy algorithm; each successive code word is selected as the first word not prohibitively near to earlier codewords.
		Binary lexicodes include, among other famous optimal codes, the Hamming codes, the Golay code, and certain quadratic residue codes [4, 8] Several authors [2, 4] have proved that lexicodes are always linear.
	citeseer.ist.psu.edu /context/76450/0 (1980 words)

	[No title]
		The automorphism group of the binary (ternary) Golay code of length 24 (12) is the Mathieu group M_{24} (M_{12}) which is one of the 26 sporadic finite simple groups.
		Besides the costruction of good codes, we demonstrate a hand decoding of the binary Reed-Muller code of length 32 and dimension 16 by projecting it onto the binary Hamming code of length 8 and dimension 4 over GF(4).
		For lengths 36 and 38, we construct more codes than were previously known and for length 58 we construct several new codes which have previously unknown weight enumerators.
	www.math.louisville.edu /~jlkim/abs_the.html (880 words)

	Binary Golay Code (Site not responding. Last check: 2007-10-30)
		One is called extended binary Golay code which is an error-correcting code that encodes 12 bits of data in a 24-bit word in such a way that any triple-bit error can be corrected and any quadruple-bit error can be detected.
		The other is called perfect binary Golay Code which has odewords of length 23 and is obtained from the extended binary Golay code by deleting one coordinate position.
		Conversely, the extended binary Golay code can be obtained from the perfect binary Golay Code by adding a parity bit.
	www.javvin.com /telecomglossary/BinaryGolayCode.html (99 words)

	Decoding the Golay code by hand: Conclusions (Site not responding. Last check: 2007-10-30)
		Thus an algorithm (essentially Polya counting) computing the minimum distance of the codes we have considered is possible for codes having dimensions much higher than would be practical for generic codes.
		M12 (the automorphism group of the extended ternary Golay code of length 12) is constucted from the group of a dodecahedron with certain additional "twists".
		Since this group and the extended binary Golay code are intimately related, it seems natural (in view of our construction) to consider whether there are natural geometric generators for M24 that can be viewed via the dodecahedron.
	www.southernct.edu /~fields/DecodingGolay/conclusions.html (873 words)

	CLASSICAL and QUANTUM INFORMATION THEORY
		C24 is a binary Golay code [24,12,8] is a code of length 24, dimension 12, and minimal distance 8 over the binary field F2.
		The Nordstrom-Robinson code is the binary image of the octacode under the Gray map.
		A common fundamental structure causes quantum-error-correcting codes to be based on GF(4), the hexacode H6 to be related to the Golay codes and Leech lattice, and an RNA code to be based on 4 nucleotides UGAC, taken 3 at a time.
	www.valdostamuseum.org /hamsmith/info.html (6749 words)

	Decoding the Golay code by hand: Conclusions
		Thus an algorithm (essentially Polya counting) computing the minimum distance of the codes we have considered is possible for codes having dimensions much higher than would be practical for generic codes.
		M12 (the automorphism group of the extended ternary Golay code of length 12) is constucted from the group of a dodecahedron with certain additional "twists".
		Since this group and the extended binary Golay code are intimately related, it seems natural (in view of our construction) to consider whether there are natural geometric generators for M24 that can be viewed via the dodecahedron.
	www2.math.uic.edu /~fields/DecodingGolayHTML/conclusions.html (873 words)

	Even code at AllExperts
		A binary code is called an even code if the Hamming weight of all its codewords is even.
		Furthermore, a binary code is called doubly-even if the Hamming weight of all its codewords is divisible by 4.
		Examples of doubly-even codes are the extended binary Hamming code of block length 8 and the extended binary Golay code of block length 24.
	en.allexperts.com /e/e/ev/even_code.htm (193 words)

	Amazon.com: "binary cyclic code": Key Phrase page (Site not responding. Last check: 2007-10-30)
		It is proved that in a binary cyclic code which is invariant under a certain group of permutations, the weights of all the code words cannot be divisible by...
		Such a code does exist; it is the binary Golay code.
		We shall introduce the binary Golay code as a binary cyclic code in Chapter 5.
	www.amazon.com /phrase/binary-cyclic-code (560 words)

	Binary Golay code - Wikipedia, the free encyclopedia
		It is possible to send standard 8-bit bytes using this standard Golay code via using 8-to-12 modulation.
		Other bit allocation schemes may be used to allow 8-bit data to share bandwidth with 4-bit telemetry.
		These modes may be listed as (0,7), (1,6), (2,5), and (3,4), where the first number indicates the number of errors which may be corrected, and the second the number of errors which will be detected by each mode.
	en.wikipedia.org /wiki/Binary_Golay_code (925 words)

	Sphere Packing
		This is achieved by having the distributions of dimensions which are allowed correspond to the codewords in a binary Golay code including parity bit which have exactly eight ones in them.
		This is achieved by having the pattern of signs for the coefficients of absolute value 1 correspond to the pattern of 0 and 1 bits in a codeword in the binary Golay code without the parity bit.
		To keep points of the second and third kind far enough away from each other, there are other restrictions on which representation of the Golay code can be used in each case, so this description is not a full construction of the Leech lattice.
	www.quadibloc.com /math/pakint.htm (2129 words)

	Sphere Packing
		This is achieved by having the distributions of dimensions which are allowed correspond to the codewords in a binary Golay code including parity bit which have exactly eight ones in them.
		This is achieved by having the pattern of signs for the coefficients of absolute value 1 correspond to the pattern of 0 and 1 bits in a codeword in the binary Golay code without the parity bit.
		To keep points of the second and third kind far enough away from each other, there are other restrictions on which representation of the Golay code can be used in each case, so this description is not a full construction of the Leech lattice.
	www.hypermaths.org /quadibloc/math/pakint.htm (2129 words)

	New Quasi-cyclic and Quasi-twisted Codes and an Optimal Family of Polynomial Codes \| Department of Mathematics
		It is an important problem in coding theory to explicitly construct efficient codes which have sufficient structure to facilitate encoding and decoding.
		A class of codes known as quasi-cyclic (QC) codes have been shown to be promising toward solving this problem.
		One of these codes, obtained by lifting the generator polynomial of the famous binary Golay code, turns out to have the best parameters among all known binary codes.
	www.math.ohio-state.edu /node/164 (378 words)

	How to Make the Mathieu Group M(24)
		Thus, you saw it coming, codewords correspond to subsets of the 24 vertices of the polyhedron X. The action of the Mathieu group on the Golay code is not transitive.
		The purpose here is to show that one may use the polyhedron X for computations inside the Golay code.
		Nevertheless, the Golay code is spanned by the disc octads and the tetrahedral dodecads.
	homepages.wmich.edu /~drichter/mathieu.htm (1988 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.
		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).
		If the original code took up n bits, the new code takes up n words, and for all bit positions i, the ith bit position of those n words form an instance of the original n-bit code.
	burtleburtle.net /bob/math/lexicode.html (1191 words)

	PPI no. 4, 1999
		Principles of iterative decoding of LDC codes are formulated and an iterative algorithm for the decoding of homogeneous LDC codes is described.
		This algorithm is close to the decoding algorithm of binary BCH codes with distance 6.
		Since a [12,6,6] quaternary code yields a cascade description of a binary extended [24,12,8] Golay code, this description provides a new method for decoding binary Golay codes.
	www.engr.umd.edu /~abarg/ppi/contents/4-99-abstracts.html (580 words)

	The Golay Code on the Dodecadodecahedron
		This is the story about the uniform polyhedron known as the "dodecadodecahedron" and its close relationship with the extended binary linear Golay code of length 24.
		The extended binary linear Golay code of length 24 is merely a particular 12-dimensional subspace of F
		spans the Golay code is the facial octad.
	homepages.wmich.edu /~drichter/golay.htm (1979 words)

	Springer Online Reference Works
		From a purely mathematical point of view, the Golay codes are the most interesting codes constructed as yet (1996).
		One thus obtains the binary and the ternary Golay code.
		He is known for his work on infrared spectroscopy and the invention of the capillary column, but to mathematicians mainly for his discovery of the two Golay codes.
	eom.springer.de /G/g110160.htm (339 words)

	Error-correcting codes in MAPLE
		We give a program which returns all the codewords in a code and all the codewords in its dual code, provided the prime p and the generator matrix are given.
		The dual code of this Hamming (7,4) code is a simplex (7,3) code.
		A BCH code is a cyclic code which included the binary Hamming codes as a special case.
	web.usna.navy.mil /~wdj/codes.htm (814 words)

	Contents
		In fact, for coding theory and cryptography, many systems use such large parameters that working practical ``by hand'' is virtually impossible and computers are not just useful but required.
		Cyclic codes are a very broad class of codes which includes, for example Reed-Solomon codes, the codes used on todays CDs.
		Other codes, such as the Goppa codes and the low density parity check codes, do not need group theory but were too advanced to fit naturally into chapter 3.
	web.usna.navy.mil /~wdj/book/node1.html (878 words)

	Binary Golay code - Gurupedia
		error-correcting code which encodes 12 bits of data in a 24-bit word in such a way that any triple-bit error can be corrected and any quadruple-bit error can be detected.
		In mathematical terms, the binary Golay code consists of a 12-dimensional subspace W of the space V=F
		In the binary Golay code, all code words have order 0, 8, 12, 16, or 24.
	www.gurupedia.com /b/bi/binary_golay_code.htm (231 words)

	[GUAVA] 4 Generating Codes
		Then functions that generate cyclic codes from a prescribed set of roots of the generator polynomial are described, including the BCH codes (see RootsCode, BCHCode, ReedSolomonCode and QRCode).
		is a subset of the roots of the code.
		This function returns the same toric code as in J. Hansen, Toric surfaces and error-correcting codes, except that the polytope can be more general This is a truncated RS code.
	www-gap.dcs.st-and.ac.uk /oldsite/pkg/guava/htm/CHAP004.htm (3010 words)

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