High rates of Fibonacci polynomials coding theory

2014 ◽  
Vol 06 (04) ◽  
pp. 1450053
Author(s):  
Bandhu Prasad
Keyword(s):  
Coding Theory ◽  
Fibonacci Polynomials ◽  
New Class ◽  
Matrix Formula ◽  
Fibonacci Polynomial ◽  
Square Matrix

In this paper, a new class of square matrix [Formula: see text] of order pm is introduced, where (p = 3, 4, 5, …), (m = 1, 2, 3, …) and for integers n, x ≥ 1. Fibonacci polynomial coding and decoding methods are followed from [Formula: see text] matrix and high code rates are obtained.

Fine rings: A new class of simple rings

2016 ◽  
Vol 15 (09) ◽  
pp. 1650173 ◽  
Author(s):  
G. Cǎlugǎreanu ◽  
T. Y. Lam
Keyword(s):  
Division Ring ◽  
Nilpotent Element ◽  
Nonzero Element ◽  
Proper Class ◽  
Invertible Matrix ◽  
Matrix Rings ◽  
Artinian Rings ◽  
New Class ◽  
Nilpotent Matrix ◽  
Square Matrix

A nonzero ring is said to be fine if every nonzero element in it is a sum of a unit and a nilpotent element. We show that fine rings form a proper class of simple rings, and they include properly the class of all simple artinian rings. One of the main results in this paper is that matrix rings over fine rings are always fine rings. This implies, in particular, that any nonzero (square) matrix over a division ring is the sum of an invertible matrix and a nilpotent matrix.

scholarly journals Some New Identities for the Generalized Fibonacci Polynomials by the Q(x) Matrix

2021 ◽  
Vol 13 (2) ◽  
pp. 21
Author(s):  
Chung-Chuan Chen ◽  
Lin-Ling Huang
Keyword(s):  
Fibonacci Polynomials ◽  
New Approach ◽  
Lucas Polynomials ◽  
Type Formula ◽  
Type Identity ◽  
Fibonacci Polynomial

We obtain some new identities for the generalized Fibonacci polynomial by a new approach, namely, the Q(x) matrix. These identities including the Cassini type identity and Honsberger type formula can be applied to some polynomial sequences such as Fibonacci polynomials, Lucas polynomials, Pell polynomials, Pell-Lucas polynomials and so on, which generalize the previous results in references.

scholarly journals A new class of Hadamard matrices

1967 ◽  
Vol 8 (1) ◽  
pp. 59-62 ◽  
Author(s):  
E. Spence
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Identity Matrix ◽  
New Class ◽  
Image Position ◽  
Square Matrix

A Hadamard matrixHis an orthogonal square matrix of ordermall the entries of which are either + 1 or - 1; i. e.whereH′denotes the transpose ofHandImis the identity matrix of orderm. For such a matrix to exist it is necessary [1] thatIt has been conjectured, but not yet proved, that this condition is also sufficient. However, many values ofmhave been found for which a Hadamard matrix of ordermcan be constructed. The following is a list of suchm(pdenotes an odd prime).

Coding theory on Lucas p numbers

2016 ◽  
Vol 08 (04) ◽  
pp. 1650074
Author(s):  
Bandhu Prasad
Keyword(s):  
Error Detection ◽  
Coding Theory ◽  
Matrix Elements ◽  
Error Detection And Correction ◽  
Matrix Formula ◽  
Code Matrix

In [K. Kuhapatanakul, The Lucas [Formula: see text]-matrix, Internat. J. Math. Ed. Sci. Tech. (2015), http://dx.doi.org/10.1080/0020739X.2015.1026612], Kuhapatanakul introduced Lucas [Formula: see text] matrix, [Formula: see text] whose elements are Lucas [Formula: see text] numbers. In this paper, we developed a new coding and decoding method followed from Lucas [Formula: see text] matrix, [Formula: see text]. We established the relations among the code matrix elements, error detection and correction for this coding theory. Correction ability of this method is [Formula: see text]% for [Formula: see text] and for [Formula: see text], the correction ability is [Formula: see text]%. In general, correction ability of this method increases as [Formula: see text] increases.

scholarly journals A Cryptographic System Based on a New Class of Binary Error-Correcting Codes

2019 ◽  
Vol 73 (1) ◽  
pp. 83-96
Author(s):  
Pál Dömösi ◽  
Carolin Hannusch ◽  
Géza Horváth
Keyword(s):  
Coding Theory ◽  
Error Correcting Code ◽  
Discrete Math ◽  
Error Correcting Codes ◽  
Key Generation ◽  
New Class ◽  
Algebraic Coding Theory ◽  
Encryption And Decryption ◽  
Construction Of Self ◽  
Cryptographic System

Abstract In this paper we introduce a new cryptographic system which is based on the idea of encryption due to [McEliece, R. J. A public-key cryptosystem based on algebraic coding theory, DSN Progress Report. 44, 1978, 114–116]. We use the McEliece encryption system with a new linear error-correcting code, which was constructed in [Hannusch, C.—Lakatos, P.: Construction of self-dual binary 22k, 22k−1, 2k-codes, Algebra and Discrete Math. 21 (2016), no. 1, 59–68]. We show how encryption and decryption work within this cryptosystem and we give the parameters for key generation. Further, we explain why this cryptosystem is a promising post-quantum candidate.

scholarly journals Restrictions and Generalizations on Comma-Free Codes

10.37236/114 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Alexander L. Churchill
Keyword(s):  
Word Length ◽  
Coding Theory ◽  
Correspondence Problem ◽  
Major Difficulty ◽  
Alphabet Size ◽  
New Class ◽  
Np Complete

A significant sector of coding theory is that of comma-free coding; that is, codes which can be received without the need of a letter used for word separation. The major difficulty is in finding bounds on the maximum number of comma-free words which can inhabit a dictionary. We introduce a new class called a self-reflective comma-free dictionary and prove a series of bounds on the size of such a dictionary based upon word length and alphabet size. We also introduce other new classes such as self-swappable comma-free codes and comma-free codes in q dimensions and prove preliminary bounds for these classes. Finally, we discuss the implications and applications of combining these original concepts, including their implications for the NP-complete Post Correspondence Problem.

scholarly journals Construction of a new class of generating functions of binary products of some special numbers and polynomials

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 1001-1013
Author(s):  
Souhila Boughaba ◽  
Ali Boussayoud ◽  
Serkan Araci ◽  
Mohamed Kerada ◽  
Mehmet Acikgoz
Keyword(s):  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Fibonacci Polynomials ◽  
Pell Numbers ◽  
New Class ◽  
Special Numbers And Polynomials ◽  
Symmetric Properties

In this paper, we derive some new symmetric properties of k-Fibonacci numbers by making use of symmetrizing operator. We also give some new generating functions for the products of some special numbers such as k-Fibonacci numbers, k-Pell numbers, Jacobsthal numbers, Fibonacci polynomials and Chebyshev polynomials.

CODING THEORY ON FIBONACCI n-STEP NUMBERS

2014 ◽  
Vol 06 (02) ◽  
pp. 1450017 ◽  
Author(s):  
MANJUSRI BASU ◽  
MONOJIT DAS
Keyword(s):  
Coding Theory ◽  
Square Matrix

We consider the series of Fibonacci n-step numbers and a class of square matrix of order n based on Fibonacci n-step numbers with determinant +1 or -1. Thereby, we introduce a new coding theory called Fibonacci n-step coding theory and establish generalized relation among the code elements for all values of n. For n = 2, the correct ability of the method is 93.33%; for n = 3, the correct ability of the method is 99.80% and for n = 4, the correct ability of the method is 99.998%. In general, the correct ability of the method increases as n increases.

On Exchange QB∞-Rings

2007 ◽  
Vol 14 (04) ◽  
pp. 613-623 ◽  
Author(s):  
Huanyin Chen
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Exchange Ring ◽  
New Class ◽  
Necessary And Sufficient ◽  
Invertible Matrices ◽  
Diagonal Reduction ◽  
Square Matrix

In this paper, we introduce a new class of rings, the QB∞-rings. We investigate necessary and sufficient conditions under which an exchange ring is a QB∞-ring. The modules over an exchange QB∞-ring are studied. Also, we prove that every regular square matrix over an exchange QB∞-ring admits a diagonal reduction by pseudo-invertible matrices.

scholarly journals On Period of the Sequence of Fibonacci Polynomials Modulo

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
İnci Gültekin ◽  
Yasemin Taşyurdu
Keyword(s):  
Cyclic Group ◽  
Fibonacci Polynomials ◽  
Fibonacci Polynomial ◽  
Fibonacci Sequences

It is shown that the sequence obtained by reducing modulo coefficient and exponent of each Fibonacci polynomials term is periodic. Also if is prime, then sequences of Fibonacci polynomial are compared with Wall numbers of Fibonacci sequences according to modulo . It is found that order of cyclic group generated with matrix is equal to the period of these sequences.

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