This paper explores the role of group theory in providing a proof for the Law of Quadratic Reciprocity, which states that for distinct odd primes pand q, q is a quadratic residue mod p if and only if p is a quadratic residue mod q, unless p and q are both congruent to 3 mod 4. The Law of Quadratic Reciprocity is an important result in number theory; it provides us with a simple method to determ...

In this paper we study elementary approaches to classical theorems on representations of primes of the form ax2 + by2, in particular the two squares theorem. While most approaches make use of quadratic residues, we study a route initiated by Liouville, and simplified by Heath-Brown and Zagier.

We point out that off-shell four-dimensional spacetime-supersymmetry implies strange hermiticity properties for the N=1 RNS superstring. However, these hermiticity properties become natural when the N=1 superstring is embedded into an N=2 superstring.

(where χ (mod q) is a Dirichlet character) arise naturally in many classical problems of analytic number theory, from estimating the least quadratic nonresidue (mod p) to bounding L-functions. Recall that for any character χ (mod q) , |Sχ(x)| is trivially bounded above by φ(q). A folklore conjecture (which is a consequence of the Generalized Riemann Hypothesis) predicts that for nonprincipal ch...

We construct a class of codes of length such that the minimum distance outside of a certain subcode is, up to a constant factor, bounded below by the square root of , a well-known property of quadratic residue codes. The construction, using the group algebra of an Abelian group and a special partition or splitting of the group, yields quadratic residue codes, duadic codes, and their generalizat...

despite of beneficial effects of antibiotics in treatments of infectious diseases, antibiotic residues in meat, milk, eggs and another products can cause serious side effects for human health. it is clear that, antibiotic residues in food stuff, cause bacterial resistance, allergic reactions, toxicity, carcinogenic effects and change of natural micro flora of intestine in consumers. so, the aim...

In this paper a new algebraic decoding approach for (17,9,5) Quadratic Residue Code is proposed by using the Inverse Free Berlekamp-MasseyAlgorithm i.e. IFBM algorithm. By using an efficient algorithm an unknown syndrome are also developed in this paper. With the help of unknown syndromes, we achieve the alternative consecutive syndromes which are needed for the application of the Berlekamp-Mas...

Let C be a binary extremal self-dual doubly-even code of length n ≥ 48. If such a code has an automorphism σ of prime order p ≥ 5 then the number of fixed points in the permutation action on the coordinates is bounded by the number of p-cycles. It turns out that large primes p, i.e. n − p small, occur extremely rarely in Aut(C). Examples are the extended quadratic residue codes. We prove that d...

The aim of this work is to compare in terms of performance, area and power dissipation, the implementations of complex FIR filters based on the following number representations: traditional two’s complement (TCS), Quadratic Residue Number System (QRNS) and radix-2j Redundant Complex Number Systems (RCNS)). The resulting implementations, designed to work at the same clock rate, show that filters...