RP-181: Formulation of Standard Quadratic Congruence of Even Composite Modulus modulo an Even Prime Raised to the Power n

The paper presented here, is a standard quadratic congruence of composite modulus, studied rigorously and found the formulation incomplete. It was partially formulated by the earlier mathematicians. The present authors realised that the earlier formulation need a completion and a reformulation of the solutions is done along with two more results. The author considered the problem for reformulation, studied and reformulated the solutions completely. A partial formulation is found in a books of Number Theory by Zuckerman at el. There the formulation is only for an odd positive integer but nothing is said about even positive integer. The authors have provided a complete formulation of the said quadratic congruence and presented here. Keywords: Composite Modulus, Quadratic Congruence, Reformulation.

RP-179: Formulation of Solutions of Standard Bi-Quadratic Congruence Modulo a POSITIVE INTEGER MULTIPLE to nth Power of Four

In this paper, the author has formulated the solutions of the standard bi-quadratic congruence of an even composite modulus modulo a positive integer multiple to nth power of four. First time a formula is established for the solutions. No literature is available for the current congruence. The author analysed the formulation of solutions in two different cases. In the first case of analysis, the congruence has the formulation which gives exactly eight incongruence solutions while in the second case of the analysis, the congruence has a different formulation of solutions and gives thirty-two incongruent solutions. A very simple and easy formulation to find all the solutions is presented here. Formulation is the merit of the paper.

Performance Analysis of a Novel 2-D Code in the Network Access Segment

In this paper, the performance of one-coincidence frequency hopping code/quadratic congruence code (OCFHC/QCC) is analysed and compared with the existing codes. The impact of variation in code weight, code length and number of wavelengths on error probability is evaluated. It is observed that for the same code weight, code length and number of available wavelengths, error probability of OCFHC/QCC is better than the existing codes under certain conditions. The large cardinality, simple design methodology and good performance suggest its use in local area network, multi-code keying and multi-rate multimedia applications.

On the number of incongruent solutions to a quadratic congruence over algebraic integers

Over the ring of algebraic integers [Formula: see text] of a number field [Formula: see text], the quadratic congruence [Formula: see text] modulo a nonzero ideal [Formula: see text] is considered. We prove explicit formulas for [Formula: see text] and [Formula: see text], the number of incongruent solutions [Formula: see text] and the number of incongruent solutions [Formula: see text] with [Formula: see text] coprime to [Formula: see text], respectively. If [Formula: see text] is contained in a prime ideal [Formula: see text] containing the rational prime [Formula: see text], it is assumed that [Formula: see text] is unramified over [Formula: see text]. Moreover, some interesting identities for exponential sums are proved.

Nature-Inspired-Based Modified Multi-Objective BB-BC Algorithm to Find Near-OGRs for Optical WDM Systems and Its Performance Comparison

Multi-objective nature-inspired-based approaches are powerful optimizing algorithms to solve the multiple objectives in NP-complete engineering design problems. This chapter proposes a nature-inspired-based modified multi-objective big bang-big crunch (M-MOBB-BC) optimization algorithm to find the Optimal Golomb rulers (OGRs) in a reasonable timeframe. The OGRs have their important application as channel-allocation algorithm that allow suppression of the four-wave mixing crosstalk in optical wavelength division multiplexing systems. The presented simulation results conclude that the proposed hybrid algorithm is superior to the existing conventional classical algorithms, namely extended quadratic congruence and search algorithm and nature-inspired-based algorithms, namely genetic algorithms, biogeography-based optimization, and simple BB-BC optimization algorithm to find near-OGRs in terms of ruler length, total occupied optical channel bandwidth, bandwidth expansion factor, computation time, computational complexity, and non-parametric statistical tests.