Abstract Algebra: An Introduction (3rd Edition)

Chapter 16

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In order to prove that set is code, first prove ... more

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8 ; Hamming weight of an element x is the number ... more

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1 ; The hamming distance between and is the ... more

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Message WordCode Word000000010111101000111111 ; ... more

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a-This code doesn't detect and correct any error.b... more

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Parity-check code : The parity check is performed ... more

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Parity-check code : The parity check is performed ... more

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Function denotes hamming distance between and ... more

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The given elements are elements of . The sum of ... more

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The sum of the element in is equal to zero. The ... more

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For "correct t-words", all messages are of Hamming... more

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For "corrects t errors", all messages are of ... more

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Consider two codewords with . If the codeword is... more

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Yes. the examples: 000000, 111100, 001111, 110011... more

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is the smallest possible in (3,1) code that ... more

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Here, the standard generator matrix for the ... more

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Let be a   standard generating matrix and be a ... more

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Assume that has a basis then every codeword in ... more

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Inverse decoding improves the accuracy of results... more

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; It is a matrix with the top 3 rows as and ... more

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; It is a matrix with the top 2 rows as and ... more

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By corollary, a code that can correct every single... more

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; The parity check matrix is and the syndrome of... more

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Here, is a chart of the syndromes and the coset ... more

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The array is similar with some reshuffling of the ... more

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To simplify the solution let , . .Let be the ... more

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The standard array of decoding is formed by adding... more

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Let the be the code with parity check matrix .... more

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The decoding follows the procedure to check the ... more

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Consider and . This will give a set and .To show... more

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A (7,4) Hemming code is given by C The parity ... more

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Consider a codeword such that , From the ... more

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If the general form of a quadratic is with , then... more

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Let where . Since, is a field, where as .... more

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Let be the set of all nonzero polynomials in the ... more

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Since and , then and , . So, . ; more

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If , then in . So, has the order of with a code... more

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There exists a binary BCH code for any positive ... more

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