2001
DOI: 10.1109/18.904547
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On the weight distributions of optimal cosets of the first-order Reed-Muller codes

Abstract: Abstract-We study the weight distributions of cosets of the first-order Reed-Muller code (1 ) for odd , whose minimum weight is greater than or equal to the so-called quadratic bound. Some general restrictions on the weight distribution of a coset of (1 ) are obtained by partitioning its words according to their weight divisibility. Most notably, we show that there are exactly five weight distributions for optimal cosets of ( 1 7) in (5 7) and that these distributions are related to the degree of the function … Show more

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Cited by 7 publications
(9 citation statements)
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“…We have (24) We obviously deduce Note the extension of this property to , for any . Indeed, we have for such a Thus,…”
Section: Proofmentioning
confidence: 92%
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“…We have (24) We obviously deduce Note the extension of this property to , for any . Indeed, we have for such a Thus,…”
Section: Proofmentioning
confidence: 92%
“…In the case where is odd it is not so easy to find cosets satisfying . Actually, the existence of such cosets is just proved by Canteaut in [24]; she exhibits almost-optimal cosets with five weights which are contained in . These weights are and .…”
Section: Proposition Iv3: For Anymentioning
confidence: 97%
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“…An area that has been applied frequently in coding theory and cryptography is the p-divisibility of exponential sums ( [3,4,11]). Sometimes getting an extra power of p in the p-divisibility can make a difference in the applications.…”
Section: Introductionmentioning
confidence: 99%
“…For higher non-linear orders, the problem of determining the 'flattest' possible Walsh-Hadamard spectrum is an open problem related to the determination of the covering radius of RM (1, m). We refer the reader to [3] for recent results on this problem.…”
Section: A Walsh-hadamard Transforms and Bent Functionsmentioning
confidence: 99%