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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] |
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] |
Sign Bit
In computer science, the sign bit is a bit in a signed number representation that indicates the sign of a number. Although only signed numeric data types have a sign bit, it is invariably located in the most significant bit position, so the term may be used interchangeably with "most significant bit" in some contexts. Almost always, if the sign bit is 0, the number is non-negative (positive or zero). If the sign bit is 1 then the number is negative, although formats other than two's complement integers allow a signed zero: distinct "positive zero" and "negative zero" representations, the latter of which does not correspond to the mathematical concept of a negative number. In the two's complement representation, the sign bit has the weight where is the number of bits. In the ones' complement representation, the most negative value is , but there are two representations of zero, one for each value of the sign bit. In a sign-and-magnitude representation of numbers, the value o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numeral Systems
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. The same sequence of symbols may represent different numbers in different numeral systems. For example, "11" represents the number ''eleven'' in the decimal numeral system (used in common life), the number ''three'' in the binary numeral system (used in computers), and the number ''two'' in the unary numeral system (e.g. used in tallying scores). The number the numeral represents is called its value. Not all number systems can represent all numbers that are considered in the modern days; for example, Roman numerals have no zero. Ideally, a numeral system will: *Represent a useful set of numbers (e.g. all integers, or rational numbers) *Give every number represented a unique representation (or at least a standard representation) *Reflect the algebraic and arithme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elsevier
Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', the '' Current Opinion'' series, the online citation database Scopus, the SciVal tool for measuring research performance, the ClinicalKey search engine for clinicians, and the ClinicalPath evidence-based cancer care service. Elsevier's products and services also include digital tools for data management, instruction, research analytics and assessment. Elsevier is part of the RELX Group (known until 2015 as Reed Elsevier), a publicly traded company. According to RELX reports, in 2021 Elsevier published more than 600,000 articles annually in over 2,700 journals; as of 2018 its archives contained over 17 million documents and 40,000 e-books, with over one billion annual downloads. Researchers have criticized Elsevier for its high profit marg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IEEE Transactions On Information Theory
''IEEE Transactions on Information Theory'' is a monthly peer-reviewed scientific journal published by the IEEE Information Theory Society. It covers information theory and the mathematics of communications. It was established in 1953 as ''IRE Transactions on Information Theory''. The editor-in-chief is Muriel Médard (Massachusetts Institute of Technology). As of 2007, the journal allows the posting of preprints on arXiv. According to Jack van Lint, it is the leading research journal in the whole field of coding theory. A 2006 study using the PageRank network analysis algorithm found that, among hundreds of computer science-related journals, ''IEEE Transactions on Information Theory'' had the highest ranking and was thus deemed the most prestigious. '' ACM Computing Surveys'', with the highest impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Posit (number Format)
Unums (''universal numbers'') are a family of formats and arithmetic, similar to floating point, proposed by John L. Gustafson in 2015. They are designed as an alternative to the ubiquitous IEEE 754 floating-point standard. The latest version (known as posits) can be used as a drop-in replacement for programs that do not depend on specific features of IEEE 754. Type I Unum The first version of unums, formally known as Type I unum, was introduced in Gustafson's book ''The End of Error'' as a superset of the IEEE-754 floating-point format. The defining features of the Type I unum format are: * a variable-width storage format for both the significand and exponent, and * a ''u-bit'', which determines whether the unum corresponds to an exact number (''u'' = 0), or an interval between consecutive exact unums (''u'' = 1). In this way, the unums cover the entire extended real number line ˆ’∞,+∞ For computation with the format, Gustafson proposed using inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Golomb Coding
Golomb coding is a lossless data compression method using a family of data compression codes invented by Solomon W. Golomb in the 1960s. Alphabets following a geometric distribution will have a Golomb code as an optimal prefix code, making Golomb coding highly suitable for situations in which the occurrence of small values in the input stream is significantly more likely than large values. Rice coding Rice coding (invented by Robert F. Rice) denotes using a subset of the family of Golomb codes to produce a simpler (but possibly suboptimal) prefix code. Rice used this set of codes in an adaptive coding scheme; "Rice coding" can refer either to that adaptive scheme or to using that subset of Golomb codes. Whereas a Golomb code has a tunable parameter that can be any positive integer value, Rice codes are those in which the tunable parameter is a power of two. This makes Rice codes convenient for use on a computer since multiplication and division by 2 can be implemented more e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential-Golomb Coding
An exponential-Golomb code (or just Exp-Golomb code) is a type of universal code. To encode any nonnegative integer ''x'' using the exp-Golomb code: # Write down ''x''+1 in binary # Count the bits written, subtract one, and write that number of starting zero bits preceding the previous bit string. The first few values of the code are: 0 ⇒ 1 ⇒ 1 1 ⇒ 10 ⇒ 010 2 ⇒ 11 ⇒ 011 3 ⇒ 100 ⇒ 00100 4 ⇒ 101 ⇒ 00101 5 ⇒ 110 ⇒ 00110 6 ⇒ 111 ⇒ 00111 7 ⇒ 1000 ⇒ 0001000 8 ⇒ 1001 ⇒ 0001001 ... In the above examples, consider the case 3. For 3, x+1 = 3 + 1 = 4. 4 in binary is '100'. '100' has 3 bits, and 3-1 = 2. Hence add 2 zeros before '100', which is '00100' Similarly, consider 8. '8 + 1' in binary is '1001'. '1001' has 4 bits, and 4-1 is 3. Hence add 3 zeros before 1001, which is '0001001'. This is identical to the Elias gamma code of ''x''+1, allowing it to encode 0. Extension to negative numbers Exp-Golomb coding is used in the H.2 ... [...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] |
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 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] |
