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Elias Omega Coding
Elias ω coding or Elias omega coding is a universal code encoding the positive integers developed by Peter Elias. Like Elias gamma coding and Elias delta coding, it works by prefixing the positive integer with a representation of its order of magnitude in a universal code. Unlike those other two codes, however, Elias omega recursively encodes that prefix; thus, they are sometimes known as recursive Elias codes. Omega coding is used in applications where the largest encoded value is not known ahead of time, or to compress data in which small values are much more frequent than large values. To encode a positive integer ''N'': #Place a "0" at the end of the code. #If ''N'' = 1, stop; encoding is complete. #Prepend the binary representation of ''N'' to the beginning of the code. This will be at least two bits, the first bit of which is a 1. #Let ''N'' equal the number of bits just prepended, minus one. #Return to Step 2 to prepend the encoding of the new ''N''. To decode an E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Code (data Compression)
In data compression, a universal code for integers is a prefix code that maps the positive integers onto binary codewords, with the additional property that whatever the true probability distribution on integers, as long as the distribution is monotonic (i.e., ''p''(''i'') ≥ ''p''(''i'' + 1) for all positive ''i''), the expected lengths of the codewords are within a constant factor of the expected lengths that the optimal code for that probability distribution would have assigned. A universal code is ''asymptotically optimal'' if the ratio between actual and optimal expected lengths is bounded by a function of the information entropy of the code that, in addition to being bounded, approaches 1 as entropy approaches infinity. In general, most prefix codes for integers assign longer codewords to larger integers. Such a code can be used to efficiently communicate a message drawn from a set of possible messages, by simply ordering the set of messages b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Elias
Peter Elias (November 23, 1923 – December 7, 2001) was a pioneer in the field of information theory. Born in New Brunswick, New Jersey, he was a member of the Massachusetts Institute of Technology faculty from 1953 to 1991. In 1955, Elias introduced convolutional codes as an alternative to block codes. He also established the binary erasure channel and proposed list decoding of error-correcting codes as an alternative to unique decoding. Career Peter Elias was a member of the Massachusetts Institute of Technology faculty from 1953 to 1991. From 1957 until 1966, he served as one of three founding editors of Information and Control. Awards Elias received the Claude E. Shannon Award of the IEEE Information Theory Society (1977); the Golden Jubilee Award for Technological Innovation of the IEEE Information Theory Society (1998); and the IEEE Richard W. Hamming Medal (2002). Family background Peter Elias was born on November 23, 1923, in New Brunswick, New Jersey. His mother ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elias Gamma Coding
Elias γ code or Elias gamma code is a universal code encoding positive integers developed by Peter Elias. It is used most commonly when coding integers whose upper-bound cannot be determined beforehand. Encoding To code a number ''x'' ≥ 1: # Let N = \lfloor \log_2 x \rfloor be the highest power of 2 it contains, so 2''N'' ≤ ''x'' < 2''N''+1. # Write out ''N'' zero bits, then # Append the binary form of ''x'', an ''N''+1-bit binary number. An equivalent way to express the same process: # Encode ''N'' in unary; that is, as ''N'' zeroes followed by a one. # Append the remaining ''N'' binary digits of ''x'' to this representation of ''N''. To represent a number x, Elias gamma (γ) uses 2 \lfloor \log_2(x) \rfloor + 1 bits. The code begins (the implied probability distribution for the code is added for clarity): Decoding To decode an Elias gamma-coded integer: #Read and count 0s from the stream until you reach the first 1. Call this count of zeroes ''N''. #Consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elias Delta Coding
Elias δ code or Elias delta code is a universal code encoding the positive integers developed by Peter Elias. Encoding To code a number ''X'' ≥ 1: # Let ''N'' = ⌊log2 ''X''⌋; be the highest power of 2 in ''X'', so 2''N'' ≤ ''X'' < 2''N''+1. # Let ''L'' = ⌊log2 ''N''+1⌋ be the highest power of 2 in ''N''+1, so 2''L'' ≤ ''N''+1 < 2''L''+1. # Write ''L'' zeros, followed by # the ''L''+1-bit binary representation of ''N''+1, followed by # all but the leading bit (i.e. the last ''N'' bits) of ''X''. An equivalent way to express the same process: #Separate ''X'' into the highest power of 2 it contains (2''N'') and the remaining ''N'' binary digits. #Encode ''N''+1 with . #Append the remaining ''N'' binary digits to this representation of ''N''+1. To re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Of Magnitude
An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic distributions are common in nature and considering the order of magnitude of values sampled from such a distribution can be more intuitive. When the reference value is 10, the order of magnitude can be understood as the number of digits in the base-10 representation of the value. Similarly, if the reference value is one of some powers of 2, since computers store data in a binary format, the magnitude can be understood in terms of the amount of computer memory needed to store that value. Differences in order of magnitude can be measured on a base-10 logarithmic scale in “decades” (i.e., factors of ten). Examples of numbers of different magnitudes can be found at Orders of magnitude (numbers). Definition Generally, the order of magnitude ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Data Compression
In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compression reduces bits by identifying and eliminating statistical redundancy. No information is lost in lossless compression. Lossy compression reduces bits by removing unnecessary or less important information. Typically, a device that performs data compression is referred to as an encoder, and one that performs the reversal of the process (decompression) as a decoder. The process of reducing the size of a data file is often referred to as data compression. In the context of data transmission, it is called source coding; encoding done at the source of the data before it is stored or transmitted. Source coding should not be confused with channel coding, for error detection and correction or line coding, the means for mapping data onto a signal. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal numbers'', and numbers used for ordering are called ''ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports jersey numbers). Some definitions, including the standard ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural numbers form a set. Many other number sets are built by success ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Numeral System
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one). The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation. History The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, Juan Caramuel y Lobkowitz, and Gottfried Leibniz. However, systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, China, and India. Leibniz was specifica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Code (data Compression)
In data compression, a universal code for integers is a prefix code that maps the positive integers onto binary codewords, with the additional property that whatever the true probability distribution on integers, as long as the distribution is monotonic (i.e., ''p''(''i'') ≥ ''p''(''i'' + 1) for all positive ''i''), the expected lengths of the codewords are within a constant factor of the expected lengths that the optimal code for that probability distribution would have assigned. A universal code is ''asymptotically optimal'' if the ratio between actual and optimal expected lengths is bounded by a function of the information entropy of the code that, in addition to being bounded, approaches 1 as entropy approaches infinity. In general, most prefix codes for integers assign longer codewords to larger integers. Such a code can be used to efficiently communicate a message drawn from a set of possible messages, by simply ordering the set of messages b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Googol
A googol is the large number 10100. In decimal notation, it is written as the digit 1 followed by one hundred zeroes: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000. Etymology The term was coined in 1920 by 9-year-old Milton Sirotta (1911–1981), nephew of U.S. mathematician Edward Kasner. He may have been inspired by the contemporary comic strip character Barney Google. Kasner popularized the concept in his 1940 book ''Mathematics and the Imagination''. Other names for this quantity include ''ten duotrigintillion'' on the short scale, ''ten thousand sexdecillion'' on the long scale, or ''ten sexdecilliard'' on the Peletier long scale. Size A googol has no special significance in mathematics. However, it is useful when comparing with other very large quantities such as the number of subatomic particles in the visible universe or the number of hypothetical possibilities in a chess game ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levenshtein Coding
Levenshtein coding is a universal code encoding the non-negative integers developed by Vladimir Levenshtein. Encoding The code of zero is "0"; to code a positive number: #Initialize the step count variable ''C'' to 1. #Write the binary representation of the number without the leading "1" to the beginning of the code. #Let ''M'' be the number of bits written in step 2. #If ''M'' is not 0, increment ''C'', repeat from step 2 with M as the new number. #Write ''C'' "1" bits and a "0" to the beginning of the code. The code begins: To decode a Levenshtein-coded integer: #Count the number of "1" bits until a "0" is encountered. #If the count is zero, the value is zero, otherwise #Start with a variable ''N'', set it to a value of 1 and repeat ''count minus 1'' times: #Read ''N'' bits, prepend "1", assign the resulting value to ''N'' The Levenshtein code of a positive integer is always one bit longer than the Elias omega code of that integer. However, there is a Levenshtein code fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite sets. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
