Extended Versions of Polynomial Remainder Codes and Chinese Remainder Codes
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Abstract:
A polynomial remainder code, derived from the Chinese remainder theorem, is a class of linear block codes, where the Reed-Solomon (RS) code is a special case. In this letter, an extended version of polynomial remainder codes is introduced, where the class of doubly extended RS codes is a special case. Furthermore, the extended version of Chinese remainder codes is also presented. The erasure decoding methods for both the codes are proposed. Finally, an application of the extended polynomial remainder codes is discussed.Keywords:
Chinese remainder theorem
Reed–Solomon error correction
Polynomial code
Polynomial remainder codes are a large class of codes derived from the Chinese remainder theorem that includes Reed-Solomon codes as a special case. In this paper, we revisit these codes and study them more carefully than in previous work. We explicitly allow the code symbols to be polynomials of different degrees, which leads to two different notions of weight and distance. Algebraic decoding is studied in detail. If the moduli are not irreducible, the notion of an error locator polynomial is replaced by an error factor polynomial. We then obtain a collection of gcd-based decoding algorithms, some of which are not quite standard even when specialized to Reed-Solomon codes.
Chinese remainder theorem
Reed–Solomon error correction
Polynomial code
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A polynomial remainder code, derived from the Chinese remainder theorem, is a class of linear block codes, where the Reed-Solomon (RS) code is a special case. In this letter, an extended version of polynomial remainder codes is introduced, where the class of doubly extended RS codes is a special case. Furthermore, the extended version of Chinese remainder codes is also presented. The erasure decoding methods for both the codes are proposed. Finally, an application of the extended polynomial remainder codes is discussed.
Chinese remainder theorem
Reed–Solomon error correction
Polynomial code
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In 1986 H. Brandstrom described concatenated codes using polynomials in two indeterminates. The author discusses concatenated codes using this idea with some modifications. He describes the code as a principal idea in a quotient ring. By doing so, the author shows the differences and similarities between product codes and concatenated codes and why concatenated codes are much better than product codes. The concatenated codes considered consist of cyclic inner and outer codes. Typically, the outer code is an RS (Reed-Solomon) code and the inner code is a BCH (Bose-Chaudhuri-Hocquenguem) code. The codes for the product codes are also cyclic and typically BCH-codes.< >
BCH code
Reed–Solomon error correction
Cyclic code
Polynomial code
Code (set theory)
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We present an efficient algorithm based on the robust Chinese remainder theorem (CRT) to perform single frequency determination from multiple undersampled waveforms. The optimal estimate of common remainder in robust CRT, which plays an important role in the final frequency estimation, is first discussed. To avoid the exhausted searching in the optimal estimation, we then provide an improved algorithm with the same performance but less computation. Besides, the sufficient and necessary condition of the robust estimation was proposed. Numerical examples are also provided to verify the effectiveness of the proposed algorithm and related conclusions.
Chinese remainder theorem
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For pt. I see ibid., vol. 47, p. 197-205, (2000). It has been shown that Wang's implementation technique of residue number system to binary conversion using a new Chinese remainder theorem formulation (CRT1) is the same as the well-known CRT. It has also been shown that his approach of using mixed radix digits to reduce the hardware requirements is no different from that described by Huang earlier and needs some corrections.
Chinese remainder theorem
Residue number system
Residue (chemistry)
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The residue-to-binary conversion is the crucial step for residue arithmetic. The traditional methods are the Chinese remainder theorem (CRT) and the mixed radix conversion. This paper presents new Chinese remainder theorems I, II, and Ill for the residue-to-binary conversion, with the following detailed results. (1) The big weights in the original CRT are reduced to a matrix of numbers less than the moduli P/sub i/. (2) The new Chinese remainder theorem I is a parallel algorithm in mixed radix format. The delay is reduced from O(n) to O(logn). (3) The new Chinese remainder theorem II reduces the modulo operation from the size M to a size less than /spl radic/M. (4) The new Chinese remainder theorem II can be easily extended to the new Chinese remainder theorem III for non-prime moduli sets. (5) A summary of a long list of references on residue-to-binary conversion is also presented.
Chinese remainder theorem
Residue number system
Modulo operation
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A coupled theory in parallel with,Chinese Remainder Theorem,Chinese Complementary Remainder Theorem, is given, which helps solve the problem of common solution for linear complementation formulas of one unknown.In this paper,a conversion theorem of equal values for complementation and congruent is also given.It is a coupled rule that has research value and certain universal sense. The result is a further expansion and complement to Chinese Remainder Theorem.
Chinese remainder theorem
Complement
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Chinese remainder theorem
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Chinese remainder theorem plays an important role in number theory and algebra.A proof of Chinese remainder throrem in k[x] is given,and give the applications in proving Lagrange interpolating formula and Jordan-Chevally theory by Chinese remainder theorem in k[x].
Chinese remainder theorem
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Abstract The fourth-century Chinese mathematical text, titled Sun Tsu San Ching (Master Sun’s arithmetic manual), contains the following problem: There is an unknown number of objects. When counted in ‘threes’, the remainder is 2; when counted in ‘fives’, the remainder is 3; and when counted in ‘sevens’, the remainder is 2. How many objects are there? Sun Tsu describes a way for solving the problem and provides an answer to the question.
Chinese remainder theorem
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