In a seminal paper, Robin proved that the Riemann hypothesis holds if and only if the inequality σ ( n ) = ∑ d | n d > e γ n log ⁡ log ⁡ n \sigma (n)=\sum _{d|n} d > e^\gamma n \log \log n holds for all integers greater than 5040, where γ \gamma is the Euler-Mascheroni constant. In this article, we prove new results on putative violations of Robin’s criterion.

In [1], Dalzell proved that $\pi = \frac{{22}}{7} - \int_0^1 {\frac{{{t^4}{{(1 - t)}^4}}}{{1 + {t^2}}}}$ . He then used this equation to derive a new series converging to π. In [2], Backhouse studied the general case of integrals of the form $\int_0^1 {\frac{{{t^m}{{(1 - t)}^m}}}{{1 + {t^2}}}dt}$ and derived conditions on m and n so that they could be used to evaluate π. As a sequel, he derived accurate rational approximations of π. This work was extended in [3] where new rational approximations of π are obtained. Some related integrals of the forms $\int_0^1 {\frac{{{t^m}{{(1 - t)}^m}}}{{1 + {t^2}}}P(t)\,dt}$ and $\int_0^1 {\frac{{{t^m}{{(1 - t)}^m}}}{{\sqrt {1 - {t^2}} }}P(t)dt}$ with P(t) being of polynomial form are also investigated. In [4] the author gives more new approximations and new series for the case m = n = 4k. In [5] new series for π are obtained with the integral $\int_0^a {\frac{{{t^{12m}}{{(a - t)}^{12m}}}}{{1 + {t^2}}}dt}$ where $a = 2 - \sqrt 3$ . The general problem of improving the convergence speed of the arctan series by transformation of the argument has also been considered in [6, 7]. In the present work the author considers an alternative form for the denominators in integrals. As a result, new series are obtained for multiples of π by some algebraic numbers.

In a recent study, Li et al. proposed an algorithm for the decoding of binary quadratic residue codes with tiny memory requirements. In this Letter, this algorithm is modified in order to dramatically improve the decoding speed. The case of the (89, 45, 17) binary quadratic residue code is used to illustrate the new algorithm.

Belief propagation and variants are the usual decoding algorithms for low-density parity check codes. The successive overrelaxations are applied to increase the convergence speed of the decoding process. Performances are reported and improvements over the reference algorithms are observed. This technique is also compatible with the high-speed parallel group decoding involved in modern standards such as DVB-S2.

Recently K. Gaitanas gave a formula which matches with the counting prime function for an infinite set of values of its argument. In this note, we give a construction of an infinite number of such formulae.

In this letter, a new framework for the construction of low density parity check codes is exposed. Parameters of the new codes are estimated, performance simulations are reported and compared with some classical constructions.

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We consider the least number x for which a change of sign of p(x)- li(x) occurs. First, we consider modifications of Lehman's method that enable us to obtain better estimates of some error terms. Second, we establish a new smaller upper bound for the first x for which the difference is positive. Third, we use numerical computations to improve the final result.

The Nordstrom-Robinson code is a binary nonlinear code of length 16 with 28 = 256 codewords. Its minimum distance which is equal to 6 is greater than that of any binary linear code of length 16 and encoding rate 1/2. For this reason, the Nordstrom-Robinson code has attracted interests in practical applications of the communication area. In this paper, we present two new algorithms for soft decoding of this code.

We study the Mertens conjecture and revisit and improve the original constructivedisproof of Janos Pintz. We obtain a new lower bound for the minimal counterexample and new numerical results for this conjecture.

At present, the main challenge for hardware implementation turbo decoders is to achieve the high data rates required by current and future communication system standards. In order to address this challenge, a low complexity radix-16 SISO decoder for the Max-Log- MAP algorithm is proposed in this paper. Based on the elimination of parallel paths in the radix-16 trellis diagram, architectural solutions to reduce the hardware complexity of the different blocks of a SISO decoder are detailed. Moreover, two complementary techniques are introduced order to overcome BER/FER performance degradation when turbo decoders based on the proposed SISO decoder are considered. Thus, a penalty lower than 0.05dB is observed for a 8 state binary turbo code with respect to a traditional radix-2 turbo decoder for 6 decoding iterations.

In this article, we report on computations that led to the discovery of a new Lehmer pair of zeros for the Riemann ζ \zeta function. Given this new close pair of zeros, we improve the known lower bound for de Bruijn-Newman constant Λ \Lambda . The Riemann hypothesis is equivalent to the assertion Λ ≤ 0 \Lambda \leq 0 . In this article, we establish that in fact we have Λ > − 1.14541 × 10 − 11 \Lambda > -1.14541 \times 10^{-11} . This new bound confirms the belief that if the Riemann hypothesis is true, it is barely true.

Turbocodes have been introduced as a new encoding technique for error correction over noisy channels. Their performance is close to theoretical limits. In this paper, we describe the use of the cross-entropy principle in the turbocode domain and we show that it is possible to use it to obtain an additional gain in terms of error correction performance.

In the area of digital communications, turbocodes provide a flexible way to design error-correcting codes which are decodable near the theoretical limits. Classical constructions use optimal fixed convolutional codes. However it has been proven that, in some cases, time varying convolutional codes outperform fixed codes in terms of minimum distance. In this paper, we discuss the use of these codes in the design of turbocodes.

Une multitude de facteurs liés à aux activités humaines (télévision, satellite, radio, stations radar…) perturbe les nombreuses télécommunications (liaisons wifi, téléphones portables, liaisons satellite) qui peuplent de nos jours notre environnement. Ce bruit important peut conduire à des erreurs de transmission de données. Les codes correcteurs sont une forme de codage basée sur la redondance de l’information et destinés à corriger ces erreurs. L’intégration des turbocodes dans ces systèmes est venue bousculer la théorie précédemment établie. La correction d’erreur peut se pratiquer maintenant à des niveaux de bruit jusqu’alors inaccessibles, avec en plus une structure de décodeurs grandement simplifiée.

AbstractThis article is devoted to describing heuristics to improve the efficiency of Cannon's algorithm. As an application, the method is used to derive a new presentation for the Lyons sporadic group.

In this article, we study the problem of changes of sign of pi(x) - li(x). We provide three improvements. First, we give better esimates of error term for Lehman's theorem. Second, we rigorously prove the positivity of this difference for a region formerly conjectured by Patrick Demichel. Third, we improve the estimates for regions of positivity by using number theoretic results.

Permutation design is a crucial point for turbocodes in terms of error correction performance. Many permutation design models have been proposed to deal with this point. However, most of them are defined in terms of several free parameters that have to be correctly chosen. The selection of these parameters generally leads to a systematic search in a wide domain of parameters value. In this article, we focus on the ARP model and describe an algorithm which avoids combinatorial enumeration.