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On Rational and Periodic Power Series and on Sequential and Polycyclic Error-Correcting Codes

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On Rational and Periodic Power Series and on Sequential and Polycyclic Error-Correcting Codes On Rational and Periodic Power Series and on Sequential and Polycyclic Error-Correcting Codes A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Benigno Rafael Parra Avila November 2009 © Benigno Rafael Parra Avila. All Rights Reserved. 2 This dissertation titled On Rational and Periodic Power Series and on Sequential and Polycyclic Error-Correcting Codes by BENIGNO RAFAEL PARRA AVILA has been approved for the Department of Mathematics and the College of Arts and Sciences by Sergio R. Lopez-Permouth´ Professor of Mathematics Benjamin M. Ogles Dean, College of Arts and Sciences 3 Abstract PARRA AVILA, BENIGNO RAFAEL, Ph.D., November 2009, Mathematics On Rational and Periodic Power Series and on Sequential and Polycyclic Error-Correcting Codes (74 pp.) Director of Dissertation: Sergio R. Lopez-Permouth´ Let R be a commutative ring with identity. A power series f 2 R[[x]] with (eventually) periodic coefficients is rational. We show that the converse holds if and only if R is an integral extension over Zm for some positive integer m. Let F be a field; we prove the equivalence between two notions of rationality in F[[x1;:::; xn]], and hence in F((x1;:::; xn)), and thus extend Kronecker’s criterion for rationality from F[[x]] to the multivariable setting. We introduce the notion of sequential code, a natural generalization of cyclic and even constacyclic codes, over a (not necessarily finite) field and explore fundamental properties of sequential codes as well as connections with periodicity of sequences and with the related notion of linear recurrence sequences. A truncation of a cyclic code over F is both left and right sequential (bisequential). We prove that the converse holds if and only if F is algebraic over Fp for some prime p. We also show that all sequential codes may be obtained by a simple and explicit construction. Using this construction, we get examples of sequential codes which reach certain optimal bonds that cannot be attained by cyclic codes. Finally, we introduce the notion of polycyclic codes which is another generalization of constacyclicity. We establish a duality between sequentiality and polycyclicity. In particular, it is shown that a code C is sequential and polycyclic if and only if C and its dual C ? are both sequential if and only if C and its dual C ? are both polycyclic. 4 Furthermore, these equivalent statements characterize the family of constacyclic codes. Approved: Sergio R. Lopez-Permouth´ Professor of Mathematics 5 To my wife Iglemia and our children: Madelaine and Rafael. To my parents: Nelly and Benigno. To my brother Jos´eand my sister Jacqueline. To the memory of my grandmother Cleotilde Avila. 6 Acknowledgements I would like to express my sincere gratitude to my advisor Sergio R. Lopez-Permouth´ for his excellent guidance and patience during the realization of this work as well as for his support and encouragement since I came to Athens. I thank my wife and children, and my parents and my siblings for their support and patiently understanding along all of this project. I would like to thank my friends Rene´ and Hilcia, and Federico and his family for their support, inspiration, and encouragement in this chapter of my life, as well as my friends Add, Damiano, and Chairat for their comradery. I am also thankful to all families and friends in Athens who welcomed me and helped me care for my children while I was single-parenting in Athens. I would like to thank Dr Martin Mohlenkamp for his support, help and friendship. I also thank Dr Xiang-Dong Hou, from Florida State University, for the opportunity to collaborate with him, and my friend and classmate Steve Szabo, who has been a great source of interesting discussions and interactions. I thank my fellow graduate students Jeremy Moore and Ryan Schwiebert for many enlightening conversations and discussions. I thank my first Algebra teacher Rafael Fernandez,´ (R.I.P.), for introducing me to what has become my area of expertise. I would like to thank the Department of Mathematics, and the Center of Ring Theory and Applications for their support. Last but not least, I would like to thank the Universidad Nacional Abierta (UNA), the Oficina de Planificacion´ del Sector Universitario and Fundacion´ Gran Mariscal de Ayacucho for their support. From UNA (my university) in Venezuela, I especially thank Maruja Romero Yepez,´ Jose´ Ramon´ Ortiz, Zobeida Ramos, Mario Marino,˜ and Manuel Castro Pereira for their support, motivation, and friendship. 7 Table of Contents Abstract . .3 Dedication . .5 Acknowledgements . .6 List of Tables . .8 List of Figures . .9 1 Preface . 10 2 Definitions and Preliminaries . 14 2.1 Sequences . 14 2.2 Power Series Ring . 20 2.3 Codes . 22 3 Rationality of Power Series . 29 3.1 Introduction . 29 3.2 Rationality in R[[x]] and Periodicity of Coefficients . 29 3.3 Two Definitions of Rationality in F[[x1;:::; xn]] . 32 3.4 Generalization of Kronecker’s Criterion to F[[x1;:::; xn]] . 40 4 Sequential Codes . 49 4.1 Introduction . 49 4.2 Sequential Codes . 49 4.3 Constructing Sequential Codes . 53 5 Polycyclic Codes . 60 5.1 Introduction . 60 5.2 Polycyclic Codes . 62 5.3 Sequential Codes . 66 References . 72 8 List of Tables 4.1 (22; 5; 12) sequential codes over F3 ........................ 59 9 List of Figures 3.1 Relations among various rings . 33 10 1 Preface We focus on two main areas of mathematics: the theories of formal power series and of error-correcting codes. These two topics have become intertwined in my thesis through a series of problems which preserve their own intrinsic interest but, at the same time, feed from one another creating a natural symbiosis. In the second chapter, background and terminology about sequences, series and codes, are introduced. The third chapter of the dissertation deals with characterizations of fields of Rational Functions as subrings of rings of Laurent Series. As an example, the following is one of the main results from that chapter. The result generalizes a classic theorem, in fact, the equivalence between (i) and (ii) is known in the literature as Kronecker’s theorem, see for instance X i Theorem 1.0.1 Let f = ai x 2 F[[x]] be a power series over a field F, and i≥0 0 1 B a a a ::: C B 0 1 2 C B C B a a a ::: C B 1 2 3 C A = B C : (1.1) B a a a ::: C B 2 3 4 C B : : : C @B : : : AC The following statements are equivalent. (i)f 2 F(x). (ii) rank A < 1: (iii) There exist m; r ≥ 0 such that 0 1 B C B am am+1 ··· C B C B C B am+1 am+2 ··· C rank B C ≤ r : B : : C B : : C B C B C @ am+l am+r+1 ··· A 11 The significance of condition (iii) is that one can characterize multivariable rational functions in a fashion similar to Kronecker’s theorem. However, the analogs of condition (ii) and (iii) are no longer equivalent in the multivariable setting and it is the one for (iii) that characterizes rationality. Our criterion is given next as Theorem 1.0.2 and in order to n state it we must introduce some notation: For i = (i1;:::; in) and j = ( j1;:::; jn) 2 N , n i i1 in i ≤ j means that it ≤ jt for all 1 ≤ t ≤ n. For i = (i1;:::; in) 2 N , define x = x1 ··· xn . If h 2 F x ;:::; x h h;:::; h n [ 1 n], define deg = (degx1 degxn ). To each -dimensional infinite X i f g n 2 array ai i2N over a field F, we associate a power series f = ai x F[[x1;:::; xn]] and n i2N a function A : Nn × Nn −! F (1.2) (i; j) 7−! ai+ j: For a fixed i 2 Nn, define n Ai : N −! F (1.3) j 7−! ai+ j: (Note that when n = 1, A is the infinite circulant matrix in (1.1) and Ai is the row of A n n with label i.) For convenience, we define ai = 0 for i 2 Z n N . Therefore, in (1.3), we may allow i to be in Zn. Theorem 1.0.2 In the above notation, f 2 F(x1;:::; xn) if and only if there exist n k = (k1;:::; kn), m = (m1;:::; mn) 2 N such that dimF (A j−k+m(1) ;:::; A j−k+m(n) ): j ≤ l < (k1 + 1) ··· (kn + 1); (1.4) where m(i) = (0;:::; 0; m + 1; 0;:::; 0) and h i is the linear span. | {z i } i While focusing on the single variable case, we discovered a kind of error-correcting block code whose description is intimately related to periodic sequences. We referred to those codes as (right) sequential codes. The notion of right sequential codes is a generalization of the concept of cyclic and even of that of constacyclic codes. 12 Interestingly, these sequential codes played an important role in the characterization of rational functions mentioned above. Furthermore, we discovered that certain families of fields can be characterized precisely by the structure of their sequential codes. To be precise, a field is such that its (two-sided) sequential codes are exactly the truncations of cyclic codes if and only if the field is a (not necessarily finite) algebraic extension of a prime field Zp. In particular bisequential codes over finite fields are always truncations of cyclic codes. In this dissertation, we establish the foundation to study sequential codes.
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