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Verhoeff Algorithm
The Verhoeff algorithm[1] is a checksum formula for error detection developed by the Dutch mathematician Jacobus Verhoeff and was first published in 1969.[2][3] It was the first decimal check digit algorithm which detects all single-digit errors, and all transposition errors involving two adjacent digits,[4] which was at the time thought impossible with such a code. Verhoeff had the goal of finding a decimal code—one where the check digit is a single decimal digit—which detected all single-digit errors and all transpositions of adjacent digits. At the time, supposed proofs of the nonexistence[5] of these codes made base-11 codes popular, for example in the ISBN check digit. His goals were also practical, and he based the evaluation of different codes on live data from the Dutch postal system, using a weighted points system for different kinds of error
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Checksum
A checksum is a small-sized datum derived from a block of digital data for the purpose of detecting errors that may have been introduced during its transmission or storage. By themselves, checksums are often used to verify data integrity but are not relied upon to verify data authenticity. The procedure which generates this checksum is called a checksum function or checksum algorithm. Depending on its design goals, a good checksum algorithm will usually output a significantly different value, even for small changes made to the input
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Error Detection
In information theory and coding theory with applications in computer science and telecommunication, error detection and correction or error control are techniques that enable reliable delivery of digital data over unreliable communication channels. Many communication channels are subject to channel noise, and thus errors may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases. Any error-correcting code can be Any error-correcting code can be used for error detection. A code with minimum Hamming distance, d, can detect up to d − 1 errors in a code word
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Dihedral Group
In mathematics, a dihedral group is the group of symmetries of a regular polygon,[1][2] which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, Dn or Dihn refers to the symmetries of the n-gon, a group of order 2n
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Lookup Table

In computer science, a lookup table is an array that replaces runtime computation with a simpler array indexing operation. The savings in processing time can be significant, because retrieving a value from memory is often faster than carrying out an "expensive" computation or input/output operation.[1] The tables may be precalculated and stored in static program storage, calculated (or "pre-fetched") as part of a program's initialization phase (memoization), or even stored in hardware in application-specific platforms. Lookup tables are also used extensively to validate input values by matching against a list of valid (or invalid) items in an array and, in some programming languages, may include pointer functions (or offsets to labels) to process the matching input
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Damm Algorithm
In error detection, the Damm algorithm is a check digit algorithm that detects all single-digit errors and all adjacent transposition errors. It was presented by H. Michael Damm in 2004.[1] The Damm algorithm is similar to the Verhoeff algorithm. It too will detect all occurrences of the two most frequently appearing types of transcription errors, namely altering one single digit, and transposing two adjacent digits (including the transposition of the trailing check digit and the preceding digit).[1][2] But the Damm algorithm has the benefit that it makes do without the dedicatedly constructed permutations and its position specific powers being inherent in the Verhoeff scheme
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Commutative

In mathematics, a binary operation is commutative if changing the order of the operands does not change tIn mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, "3 − 5 ≠ 5 − 3"); such operations are not commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed
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Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.[1] Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set {1,2,3}, namely: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters
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Doi (identifier)

A digital object identifier (DOI) is a persistent identifier or handle used to identify objects uniquely, standardized by the International Organization for Standardization (ISO).[1] An implementation of the Handle System,[2][3] DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports, data sets, and official publications. However, they also have been used to identify other types of information resources, such as commercial videos. A DOI aims to be "resolvable", usually to some form of access to the information object to which the DOI refers. This is achieved by binding the DOI to metadata about the object, such as a URL, indicating where the object can be found. Thus, by being actionable and interoperable, a DOI differs from identifiers such as ISBNs and ISRCs which aim only to identify their referents uniquely
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Verhoeff Algorithm
The Verhoeff algorithm[1] is a checksum formula for error detection developed by the Dutch mathematician Jacobus Verhoeff and was first published in 1969.[2][3] It was the first decimal check digit algorithm which detects all single-digit errors, and all transposition errors involving two adjacent digits,[4] which was at the time thought impossible with such a code. Verhoeff had the goal of finding a decimal code—one where the check digit is a single decimal digit—which detected all single-digit errors and all transpositions of adjacent digits. At the time, supposed proofs of the nonexistence[5] of these codes made base-11 codes popular, for example in the ISBN check digit. His goals were also practical, and he based the evaluation of different codes on live data from the Dutch postal system, using a weighted points system for different kinds of error
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