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 generating the coset. Moreover, for any odd 9, we exhibit optimal cubic cosets of (1 ) whose weights take on exactly five values.

On cryptographic properties of the cosets of R(1, m)

“…We have (24) We obviously deduce Note the extension of this property to , for any . Indeed, we have for such a Thus,…”

“…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 .…”

“…It has been recently proved by Canteaut that, for any odd , all almost-optimal functions of degree are three-valued. For , she has proved that there is only one weight polynomial for almost-optimal non-three-valued cosets of which are contained in ; moreover, she proves that such cosets exist [24].…”

Construction of systems of polynomial equations with exact p-Divisibility via the covering method

“…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.…”

On codes with low peak-to-average power ratio for multi-code CDMA

“…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.…”