Us8850298b2 combined koettervardy and chase decoding of. The objectives of this paper are to introduce a qpolynomial approach to the study of cyclic codes, lay the foundations of this approach, and construct new oneswith this approach. For this reason, almost all block codes used in practice are in fact cyclic codes. In this paper, we focus on distanceoptimal cyclic codes that arise from this construction. Cyclic codes are an interesting subclass of linear codes and have been used in consumer electronics, data transmission technologies, broadcast systems, and computer applications due to their efficient encoding and decoding algorithms. Mido spacetime codes from associative and nonassociative cyclic algebras. The second meeting of the series slovenian conference on graph theory was held at lake bled in 1991, following by the subsequent meetings at the same location in 1995, 1999, 2003, 2007, and 2011. In this work, we present an alternative qpolynomial approach to cyclic and quasicyclic codes.
Saba zerefa a computational approach to intrinsic linkedness in complete graphs mentor. Three classes of optimal ternary cyclic codes and the. The properties of the q polynomial codes and the fundamental relations between q polynomial codes and generator polynomials are proposed. Coding theory and applications cyclic codes enes pasalic. A q polynomial approach to cyclic codes was introduced by ding and ling dl. New developments in q polynomial codes springerlink. A qpolynomial approach to cyclic codes was introduced by ding and ling dl. Recently, a qpolynomial approach to the construction and analysis of cyclic codes over gfq was given by ding and ling 2.
They are prominently used in consumer electronics, data t. This at once gives the distance enumerators of large classes of errorcorrecting codes for which many special cases have been obtained previously using different methods, for example results for extended binary cyclic codes obtained by berlekamp and kasami, recent results for q ary cyclic codes obtained by li, and many results for q ary. Circuit portion 524 is provided for completion of the computation of the pitch subframe ten lpc coefficients. Quadratic and symmetric bilinear forms over finite fields and. The properties of the qpolynomial codes and the fundamental relations between qpolynomial codes and. This paper is concerned with new developments in cyclic codes from q polynomials. Let c be an n, k, dq cyclic code with generator polynomial g, then c is. A qpolynomial approach to cyclic codes sciencedirect 24 days ago the construction of a class of qpolynomial codes in section 5, which are almost optimal when q 2. Algorithms and theory of computation handbook, second edition, volume 1. This book stresses the connection between, and the applications of, design theory to graphs and codes. Complete weight distributions of two classes of cyclic codes. Jan, 2016 complete weight distribution can be used to study authentication codes and the walsh transform of monomial functions over finite fields.
Boyle, latham the standard model of particle physics from noncommutative geometry. In this paper, we focus on distanceoptimal cyclic codes that. Cyclic codes are prominently used in electronics and communication engineering. Also, the hamming weight distribution of a code can be obtained from its complete weight distribution. An introduction to linear and cyclic codes lixpolytechnique. Cyclic codes are an important class of linear codes for which the encoding and decoding can be efficiently. Algebraic soft decoding of reedsolomon codes us7159168b2 en 20021107. The objectives of this paper are to introduce a q polynomial approach to the study of cyclic codes, lay the foundations of this approach, and construct new oneswith this. Talks at conferences an analogue of tdesigns in the association schemes of alternating bilinear forms, algebraic combinatorics and group theory, osaka city university academic extension center, january 1985, pp. Bonferttaylor, petra a datadriven approach towards instruction in an introductory programming class.
The objectives of this paper are to introduce a qpolynomial approach to the study of cyclic codes, lay the foundations of this approach, and construct new oneswith this. Fundamental theory of this approach will be developed, and will be employed to construct a new family of. Ranksynd a prng based on rank metric philippegaborit 1,adrienhauteville 1. More in particular, spectral graph theory studies the relation between graph properties and the spectrum of the adjacency matrix or laplace matrix. Fundamental theory of this approach will be developed, and will be employed to construct some families of optimal and almost optimal codes in this paper. A qpolynomial approach to constacyclic codes sciencedirect.
We explicitly present the complete weight enumerators of the. The main tool for studying subsets of q and s is the beautiful theory of association schemes. The basic idea of design theory is to approximate a given space m by a good finite subset x of m. In particular the two schemes are dual to each other and, for odd q, they are isomorphic hence the association. Us20140006896a1 combined koettervardy and chase decoding. Iterative decoding with likelihood weighting us71976b2 en 20031204.
The construction of a class of qpolynomial codes in section 5, which are almost optimal when q 2. Pdf cyclic codes from cyclotomic sequences of order four. Circuit portion 520 is duplicated not shown for computing the coefficients of the q polynomial. This text is aimed at the abstract or modern algebra course taken by junior and senior math majors and many secondary math education majors. The growth of the conference was stable, from 30 participants at the 1991 meeting to almost 300 participants at the. Over gf2 the cyclic polynomial of degree 6 can be factored as. The objective of this paper is to develop another approach the qpolynomial approach. Hasseweil bound for additive cyclic codes,9th international workshop on coding and cryptography wcc 2015, paris, france, april 17, 2015. Although the qpolynomial approach for constacyclic codes is a direct generalization of ding et al. Beginning with a brief introduction to design theory and the necessary background, the book also provides relevant topics for discussion from the theory of graphs and codes. Han cyclic codes 16 encoding of cyclic codes encodingprocess.
Three classes of optimal ternary cyclic codes and the weight distributions of their dualsj. A new view on cyclic codes based on q polynomials has been introduced by ding and ling. In this work, we present an alternative q polynomial approach to cyclic and quasi cyclic codes. Jan 01, 2016 we consider linear cyclic codes with the locality property or locally recoverable codes lrc codes. Ieice transactions on fundamentals of electronics, communications and computer sciences vol. A qpolynomial approach to cyclic codes sciencedirect. Polynomial, determine the ones digit when the expression is evaluated 23 to the power of 502, literal equations, in single variable inequalities if. C, we consider the related polynomial representation of the dual code c.
Cyclic codes have wide applications in data storage systems and communication systems. Combined koettervardy and chase decoding of cyclic codes download pdf info publication number us20140006896a1. Crc concise encyclopedia of mathematics, second edition. Our purpose is to recall some basic aspects about linear and cyclic codes. The spherical tdesigns are those finite subsets x of the unit sphere m s n. An algebraic approach to the theory of association schemes of the coding theory 10 1973 ppoly. A family of lrc codes that generalises the classical construction of reedsolomon codes was constructed in a recent paper by tamo and barg ieee transactions on information theory, no. It is known that q m, q and s m, q carry the structure of a translation association scheme with. Decoding reedsolomon codes and related codes represented by graphs.
Quadratic and symmetric bilinear forms over finite fields. Het s w re w recently it was proven in 15 that this problem had a probabilistic. This paper is devoted to the study of the qpolynomial approach to constacyclic codes. Three classical approaches to the study and construction of cyclic codes are those based on the generator matrix, the generator polynomial and the idempotent. In proceedings of the 2012 information theory workshop itw 2012, pp. The zeroes of gx belong to extension field gfqm of gfq. We consider linear cyclic codes with the locality property or locally recoverable codes lrc codes.
The properties of cyclic codes suggest a very simple method to encode a message. An algebraic approach to the theory of association schemes of. On the explicit constructions of certain unitary tdesigns. Complete weight distribution can be used to study authentication codes and the walsh transform of monomial functions over finite fields. Cyclic codes from cyclotomic sequences of order four. The resultant final new values of p1p11 and q1q11 are provided to circuit portion 524. A new view on cyclic codes based on qpolynomials has been introduced by ding and ling. Recently, a q polynomial approach to the construction and analysis of cyclic codes over gf q was given by ding and ling 2. Recently, a qpolynomial approach to the construction and analysis of cyclic codes over gfqwas given by ding and ling 2. Hasseweil bound for additive cyclic codes,9th international workshop on coding and cryptography wcc 2015, paris, france, april. Spectra of graphs monograph february 1, 2011 springer preface algebraic graph theory is the branch of mathematics that studies graphs by using algebraic properties of associated matrices. Boyko, mariya the role of socialist competition in the soviet mathematics curriculum reform of the 1960s and 1970s.
The objective of this paper is to present another q polynomial approach to all cyclic codes over gf q. And we also calculate the minimum distance of some cyclic codes and give lower. Reference list of indexed articles, discrete mathematics. In this paper, we investigate the complete weight distributions of two classes of cyclic codes. Recently, a qpolynomial approach to the construction and analysis of cyclic codes over \gfq was given by ding and. Jordan lee stability of finite difference schemes on the diffusion equation with discontinuous coefficients mentor. Cyclic codes have been an interesting topic of both mathematics and engineering for decades. The basic idea of coding theory consists of adding some kind of redundancy. Pdf an introduction to linear and cyclic codes researchgate. This paper is concerned with new developments in cyclic codes from qpolynomials. Conference talks a new qpolynomial approach to cyclic and quasicyclic codes, codes, cryptology and curves, eindhoven, holland, march 78, 2019. Three classes of optimal ternary cyclic codes and the weight.
The objective of this paper is to present another qpolynomial approach to all cyclic codes over gfq. Now that we have a polynomial approach to describe a cyclic code. Request pdf a qpolynomial approach to cyclic codes cyclic codes have been an interesting topic of both mathematics and engineering for decades. A cyclic code has generator polynomial gx that is a divisor of every codeword. Reproducible codes and cryptographic applications paolo.
These approaches have their advantages and disadvantages in dealing with cyclic codes. As a generalization of cyclic codes, constacyclic codes is an important and interesting class of codes due to their nice algebraic structures and various applications in engineering. They have been studied for decades and a lot of progress has been made, for example, some constructions and properties of them can be found in 1,4,5,6,9,10,11,12,14,15,20,21,22,23. Linear equations in full written form using two variables and the ymx b formula, algrbra cheat, simplifying radical expressions, degree of polynomial, how to solve reciprocal equation g pe q. A qpolynomial approach to cyclic codes request pdf. Ling finite fields and their applications 20 20 1a14 the discovery of the family of almost optimal cyclic codes of section 5 demonstrates that the qpolynomial approach to cyclic codes could be promising. Universitext eberhard freitag rolf busam complex analysis 2005 springer. Cyclic codes from twoprime generalized cyclotomic sequences. Every cyclic code over gf q is a qpolynomial code one basic question concerning the qpolynomial approach is whether every cyclic code of length over gf q can be expressed as the code c i. In this chapter, we will give a general introduction to cyclic codes, discussing both the.
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