Studying the locator polynomials of minimum weight codewords of BCH codes

Primitive binary cyclic codes of length n=2/sup m/ are considered. A BCH code with designed distance delta is denoted B(n, delta ). A BCH code is always a narrow-sense BCH code. A codeword is identified with its locator polynomial, whose coefficients are the symmetric functions of the locators. The...

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Bibliographic Details
Published in:IEEE transactions on information theory 1992-05, Vol.38 (3), p.960-973
Main Authors: Augot, D., Charpin, P., Sendrier, N.
Format: Article
Language:English
Subjects:
Applied sciences
Coding, codes
Computer Science
Cryptography and Security
Exact sciences and technology
Galois fields
Information Theory
Information, signal and communications theory
Mathematics
Signal and communications theory
Telecommunications and information theory
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Summary:Primitive binary cyclic codes of length n=2/sup m/ are considered. A BCH code with designed distance delta is denoted B(n, delta ). A BCH code is always a narrow-sense BCH code. A codeword is identified with its locator polynomial, whose coefficients are the symmetric functions of the locators. The definition of the code by its zeros-set involves some properties for the power sums of the locators. Moreover, the symmetric functions and the power sums of the locators are related to Newton's identities. An algebraic point of view is presented in order to prove or disprove the existence of words of a given weight in a code. The principal result is the true minimum distance of some BCH codes of length 255 and 511. which were not known. The minimum weight codewords of the codes B(n2/sup h/-1) are studied. It is proved that the set of the minimum weight codewords of the BCH code B(n,2/sup m-2/-1) equals the set of the minimum weight codewords of the punctured Reed-Muller code of length n and order 2, for any m.< >
ISSN:0018-9448
1557-9654
DOI:10.1109/18.135638