We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x1/5 when x is large enough (depending on m). 2010 Mathematics subject classification: primary 11N25; secondary 11A51.

The toric ideals of 3 × 3 transportation polytopes Trc are quadratically generated. The only exception is the Birkhoff polytope B3. If Trc is not a multiple of B3, these ideals even have square-free quadratic initial ideals. This class contains all smooth 3 × 3 transportation polytopes.

The notion of quadratic congruences was introduced in the recently appeared paper [1]. In this note we present another, somewhat more conceptual proof of several results from loc. cit. Our method allows to refine the notion and to generalize the results quoted. Here we deal only with the quadratic congruences for Cohen Eisenstein series. A similar phenomena exists for cusp forms of half-integra...

The focus in this work is on quadratic residue codes over the ring F2+vF2. We define these codes in terms of their idempotent generators and show that these codes share the properties analogous to that of quadratic residue codes over finite fields. We study Euclidean and Hermitian self-dual families of codes as extended quadratic residue codes over F2 + vF2. Further, we obtain two optimal self-...

We present a method for maximumlikelihood decoding of the quadratic residue code This method is based on projecting the code onto a subcode with an acyclic Tanner graph and representing the set of coset leaders by a trellis diagram This results in a two level coset decoding which can be considered a systematic generalization of the Wagner rule We show that unlike the Golay code the code does no...

An efficient maximum-likelihood soft-decision decoding algorithm for linear block codes using a generalized Dijkstra’s algorithm was proposed by Han, Hartmann, and Chen. In this correspondence we prove that this algorithm is efficient for most practical communication systems where the probability of error is less than 10 3 by finding an upper bound of the computational effort of the algorithm. ...

Three approaches are currently used for devising identity-based encryption schemes. They build on pairings, quadratic residues (QR), and lattices, respectively. Among them, the QR-based scheme proposed by Cocks in 2001 is notable in that it works in standard RSA groups: its security relies on the standard quadratic residuosity assumption. But is has also a number of deficiencies, some of them h...

Quadratic residue codes are some of the brightest gems of coding theory. Their facets reeect many topics: quadratic residues and reciprocity; multiply transitive and sporadic simple groups; lattices; group representations; and block designs and Hadamard matrices. They provide the only perfect codes for prime-power sized alphabets correcting more than one error, and some are prototypes of extrem...

An efficient table lookup decoding algorithm (TLDA) is presented to decode up to five possible errors in a binary systematic (47,24,11) quadratic residue (QR) code. The main idea of the TLDA is based on the weight of syndrome, the syndrome decoder together with a reduced-size lookup table (RSLT), and the shift-search method given by Reed et al. Thus, the size of the lookup table and computation...

Traditional sound diffusers are quasi-random phase gratings attached to reflecting surfaces whose purpose is augment the spatiotemporal incoherence of acoustic field scattered from reflective surfaces. This configuration allows one cover a large surface by periodically tiling diffuser unit cells area while reducing undesirable specular reflection for incident plane waves. However, periodic arra...