decoding algorithms for algebraic geometric codes over rings

DECODING ALGORITHMS FOR ALGEBRAIC GEOMETRIC CODES OVER RINGS by Katherine Bartley A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy Major: Mathematics Under the Supervision of Professor Judy Walker Lincoln, Nebraska May, 2006 UMI Number: 3208054 3208054 2006 UMI Microform Copyright All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 by ProQuest Information and Learning Company. DECODING ALGORITHMS FOR ALGEBRAIC GEOMETRIC CODES OVER RINGS Katherine Bartley, Ph.D. University of Nebraska, 2006 Advisor: Judy Walker Algebraic geometric codes over rings were defined and studied in the late 1990’s by Walker, but no decoding algorithm was given. In this dissertation, we present three decoding algorithms for algebraic geometric codes over rings. The first algorithm presented is a modification of the basic algorithm for algebraic geometric codes over fields, and decodes with respect to the Hamming weight. The second algorithm presented is a modification of the Guruswami-Sudan algorithm, a list decoding algorithm for one-point algebraic geometric codes over fields. This algorithm also decodes with respect to the Hamming weight. Finally, we show how the Koetter-Vardy algorithm, a soft-decision decoding algorithm, can be used to decode one-point algebraic geometric codes over rings of the form Z/p r Z, where p is a prime, with respect to the squared Euclidean weight. ACKNOWLEDGEMENTS This dissertation would have not been possible without the support and friendship of my advisor, Judy Walker. She has provided tremendous help and encouragement over the last five years. I would also like to thank the other memb ers of my committee, Tom Marley, Lance Perez, Mark Walker and Roger Wiegand for their help and advice throughout my graduate education. In particular, I would like to thank Mark Walker for all the time he sp ent helping me understand algebraic geometry. I would also like to thank Ralf Ko etter for his help with writing Chapter 5. I would not have considered graduate school if it were not for the encouragement of my undergraduate professors. I would like to thank Vanessa Job, she introduce d me to research and the field of coding theory. I would also like to thank Judy Green and Elsa Schaefer. They helped prepare me for graduate school and gave me endless advice. Several of the graduate students have become my family here at UNL. It is partly through their support and help that I have succeeded here at UNL. In particular, I would like to thank Suanne Au, Daniel Buettner, Jennifer Everson, Pari Ford, Matt Koetz, Ed Loeb and Melissa Luckas. I will miss them greatly. Finally, I would like to thank my family, my parents John and Joan Bartley and sister Carolyn Bartley. They listened to my endless complaints and encouraged me to stick with it when I could not see the light at the end of the tunnel. Contents 1 Introduction 1 2 Background Information 7 2.1 Codes over Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Algebraic Geometric Codes over Finite Fields . . . . . . . . . . . . . 9 2.3 Curves over Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Algebraic Geometric Codes over Rings . . . . . . . . . . . . . . . . . 24 3 The Basic Decoding Algorithm for Algebraic Geometric Codes over Rings 29 3.1 The Generalized Basic Decoding Algorithm . . . . . . . . . . . . . . . 29 4 The Guruswami - Sudan Algorithm 41 4.1 Decoding Generalized Reed-Solomon Codes over a Local Artinian Ring 42 4.2 The Guruswami-Sudan Algorithm for One-Point Codes over Local Ar- tinian Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5 Decoding Algebraic Geometric Codes over Rings with Respect to the Squared Euclidean Weight 62 5.1 Squared Euclidean Weight . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2 The Koetter-Vardy Algorithm . . . . . . . . . . . . . . . . . . . . . . 64 6 A Further Application of the Guruswami-Sudan Algorithm for Code s over Rings 81 Bibliography 83 1 Chapter 1 Introduction Every communications channel contains noise which can cause errors to occur. The noise in a channel can come from several different sources. For example, rain or solar flares can cause noise in a satellite link; a scratch on a CD can be thought of as noise for that channel. Error-correcting codes are used when transmitting data across a channel to help ensure a reliable link. More specifically, through the use of error- correcting codes, one can find and correct the errors that occur during transmission as long as the number of errors does not surpass a certain bound. Let A be a commutative ring. A code C of length n over A is a subset of A n . The elements of C are called codewords. If C is a submodule of A n , then C is called a linear code. Although many of the codes used today, such as Reed-Solomon codes, are defined over finite fields, codes over Z/4Z received increased interest when, in 1994, Hammons, Kumar, Calderbank, Sloane and Sol´e [8] showed that certain non- linear binary codes, such as the Nordstrom-Robinson code, are nonlinear projections of linear codes over Z/4Z. Algebraic geometric codes over finite fields were defined in 1977 by V. D. Goppa [5], and have had a strong influence on the field of coding theory. For example, in 2 1982, Tsfasman, Vlˇadut¸ and Zink [29] used modular curves to prove the existence of a sequence of codes with asymptotically better parameters then any previously known sequence of codes. Algebraic geometric codes over finite fields are constructed by using a smooth, absolutely irreducible, projective curve X over a finite field F q . Given a divisor Weil D on X and a set of distinct F q -rational points P = {P 1 , . . . , P n } on X such that supp D ∩P = ∅, the algebraic geometric code C L (X, P, D) is defined by evaluating functions in the Riemann-Roch space of D at points in P, i.e, C L (X, P, D) = {(f(P 1 ), . . . , f(P n )) | f ∈ L(D)}. In the late 1990’s, Walker [33] defined algebraic geometric codes over rings, thus combining two different areas of coding theory. Given a local Artinian ring A and a smooth irreducible projective s cheme X of relative dimension one over Spec A whose closed fiber X is absolutely irreducible, a Cartier divisor on X is the analog of a Weil divisor on X, and the A-module Γ(X, O X (D)) is the analog of the Riemann-Ro ch space L(D). Lastly, a set of pairwise disjoint A-points Z = {Z 1 , . . . , Z n } on X is the analog of a set of distinct F q -rational points P = {P 1 , . . . , P n } on X. Given an A-point Z i ∈ Z, there is a non-canonical isomorphism γ i : Γ(Z i , O X (D)| Z i ) → A. Any system γ = {γ i } of these isomorphisms gives us a map α : Γ(X, O X (D)) → A n . The algebraic geometric code C L (X, Z, O X (D), γ) is the image of α. If D = mZ for an A-point Z disjoint from the points of Z and m > 2g − 2, where g is the genus of X, then C L (X, Z, O X (mZ), γ) is called a one-point algebraic geometric code. In [33], Walker proved several properties about algebraic geometric codes over rings, but did not provide any decoding algorithms, although it is be lieved that many of the 3 algorithms for decoding algebraic geometric codes over finite fields can be modified to decode algebraic geometric codes over rings. The goal of this dissertation is to modify three of the decoding algorithms for algebraic geometric codes over finite fields to make them work for algebraic geometric codes over rings. In Chapter 3, the modification of the basic algorithm for algebraic geometric codes over finite fields is given. The basic algorithm is a classical decoding algorithm and works by finding error locator functions for a received word y. Although the modi- fied basic algorithm decodes algebraic geometric codes over any finite local Artinian Gorenstein ring A, it does so with respect to the Hamming weight, whe re the Ham- ming weight of a vector x ∈ A n is defined to be w H (x) = |{i | x i = 0}|. But, for codes over Z/4Z, the more pertinent weight measure is the Lee weight, or more generally, the squared Euclidean weight for codes over rings of the form Z/p r Z. For A = Z/4Z, the Lee weight w L : Z/4Z → Z is given by w L (0) = 0, w L (1) = 1, w L (2) = 2, w L (3) = 1. The Lee weight of x ∈ (Z/4Z) n is defined to be w L (x) = n  i=1 w L (x i ). The Lee weight is a special case of the squared Euclidean weight. Let A = Z/p r Z, where p is a prime. We can identify the elements of A with the complex p r -th roots 4 of unity via the map e p r : Z/p r Z → C defined by e p r (x) = e 2πix p r . The squared Euclidean weight of x ∈ Z/p r Z, w E 2 (x), is the square of the Euclide an distance between e p r (x) and e p r (0) in the complex plane. Note that for A = Z/4Z, w L (x) = 1 2 w E 2 (x) for all x ∈ A. The motivation for decoding codes over rings with respect to the Lee weight or the squared Euclidean weight is two-fold. The first motivation comes from the Gray map. The Gray map λ : Z/4Z → F 2 2 is defined by λ(0) = (0, 0), λ(1) = (0, 1), λ(2) = (1, 1), λ(3) = (1, 0). Extending the Gray map to (Z/4Z) n , λ : ((Z/4Z) n , w L ) → (F 2n 2 , w H ) is an isometry. For a code C over Z/4Z, decoding C with resp ect to the Lee weight is equivalent to decoding λ(C) with respect to the Hamming weight. The second motivation for decoding codes over Z/p r Z with respect to the squared Euclidean weight comes from communication channels. Because of the way that data is encoded for transmission across a channel, there is a higher probability that a symbol c i of a codeword c will get changed to (c i ± 1) (mod p r ) rather than any of the other symb ols. Decoding algorithms for codes over Z/p r Z need to take this into consideration. We can achieve this goal by decoding with respect to the squared Euclidean weight. In order to decode algebraic geometric codes over rings with respect to the squared Euclidean weight, we first look at the Guruswami-Sudan algorithm, a list decoding algorithm. Given an error bound e and a received word y, the algorithm returns all codewords c ∈ C such that d H (y,c) ≤ e, where d H denotes the Hamming distance. 5 List decoding algorithms are modern decoding algorithms motivated, in part, by the fact that many classical algorithms may fail to correctly decode all received words that are closest to a unique co deword. The Guruswami-Sudan algorithm [7] was originally designed for decoding gener- alized Reed-Solomon codes over finite fields, a class of algebraic geometric codes. The algorithm was then generalized to decode one-point algebraic geometric codes over finite fields . Given a finite field F q , a set P = {P 1 , . . . , P n } of distinct elements of F q , a set A = {a 1 , . . . , a n } of (not necessarily distinct) elements of F q , and a positive integer k, 0 < k < n, the generalized Reed-Solomon code GRS(P, k) is defined by GRS(P, A, k) = {(a 1 f(P 1 ), . . . , a n f(P n )) | f ∈ F q [x], deg f ≤ k}. The Guruswami-Sudan algorithm works by solving a polynomial reconstruction prob- lem, which is equivalent to decoding GRS(P, A, k). That is, given a received word y and error bound e, the Guruswami-Sudan algorithm finds all f ∈ F q [x] of degree at most k such that f (P i ) = y i a i for at least n−e of the ordered pairs (P 1 , y 1 a 1 ), . . . , (P n , y n a n ). The algorithm solves the reconstruction problem by finding a nonzero polynomial Q(x, y) ∈ F q [x, y] such that the roots of Q(x, y) include all f ∈ A[x] of degree at most k such that f (P i ) = y i a i for at least n − e of the ordered pairs (P 1 , y 1 a i ), . . . , (P n , y n a n ). Let A be a local Artinian ring whose maximal ideal is principal. In Chapter 4, we modify the Guruswami-Sudan algorithm to work for generalized Reed-Solomon and one-point algebraic geometric codes over A. By itself, the Guruswami-Sudan algorithm decodes one-point algebraic geometric codes with respect to the Hamming weight. Let A = Z/p r Z, where p is a prime. In Chapter 5, we look at how the Koetter-Vardy algorithm [17],[20] is used to decode one- point algebraic geometric codes over A with respect to the squared Euclidean weight. [...]... two-stage decoder, developed by Armand [1], for decoding one-point algebraic geometric codes over rings 7 Chapter 2 Background Information The purpose of this chapter is to give an overview of codes over rings and to provide a summary of the basic properties of curves over rings This chapter also includes a review of the construction of algebraic geometric codes over rings and their properties Although much... code, so the algorithm decodes all algebraic geometric codes over A The algorithm is a generalization of the basic algorithm for decoding algebraic geometric codes over finite fields Presentations of the basic algorithm, which itself is a generalization of the Arimoto-Peterson algorithm for decoding Reed-Solomon codes, can be found in [11], [13] and [24] 3.1 The Generalized Basic Decoding Algorithm We begin... (X, Z, L, γ) be a residue code over A The designed Hamming distance δΩ of CΩ is δΩ = deg L − 2g + 2 29 Chapter 3 The Basic Decoding Algorithm for Algebraic Geometric Codes over Rings This section describes a decoding algorithm for a residue code over a finite local Artinian Gorenstein ring A with respect to the Hamming distance By Remark 2.5.5, any algebraic geometric code over A is equivalent to a residue... is shown that when A is a Gorenstein ring, the class of algebraic geometric codes over A is closed under duals The rings that we are primarily interested in are Galois rings, in particular Galois rings of the form Z/pr Z, where p is a prime As Galois rings are local Artinian Gorenstein rings, we have that the class of algebraic geometric codes over a given Galois ring is closed under duals In recalling... 2.5 (n) (gj )r1 ···(gj )rn OX (Uj ) m fj Algebraic Geometric Codes over Rings In this section we recall the definition of algebraic geometric codes over local Artinian rings These codes were first introduced in the late 1990’s by Walker [33] Let A and X be as in Section 2.3 Let Z = {Z1 , , Zn } be a set of disjoint A-points on X and let L be a line bundle on X For each i, let γi be an isomorphism, γi... equivalence and the construction of algebraic geometric codes over finite fields implicitly uses this isomorphism Although this isomorphism does not hold for curves over rings, we do have an isomorphism between Pic(X) and the group of Cartier divisors on X modulo linear equivalence It is this isomorphism which will be used in the construction of algebraic geometric codes over rings Definition 2.3.1 ([10]) Let... is called an algebraic geometric code over A Suppose ϕ = {ϕi : Γ(Zi , L|Zi ) → A | 1 ≤ i ≤ n} is another system of isomorphisms Then, for 1 ≤ i ≤ n, there exists ai ∈ A× such that γi (s|Zi ) = ai ϕi (s|Zi ) for all s ∈ Γ(X, L) Therefore CL (X, Z, L, γ) and CL (X, Z, L, ϕ) are equivalent codes In [33], Walker proved the following facts about the parameters of algebraic geometric codes over A We restate... C, respectively Then d ≤ d with equality if C is free 2.2 Algebraic Geometric Codes over Finite Fields This section gives a summary of the construction and properties of algebraic geometric codes over finite fields A more extensive discussion of these codes can be found in [26] and [28] Let X be a smooth, absolutely irreducible, projective curve over the finite field Fq with rational function field Fq (X)... ) ⊗ OX (D2 )−1 3 D1 ∼ D2 if and only if OX (D1 ) OX (D2 ) as sheaves The analog of Fq -rational points for algebraic geometric codes over rings are Apoints Note that, since X is a curve over A, there exists a structure morphism f : X → Spec A Definition 2.3.4 ([33, Definition 4.3]) Let X be a curve over A and let Z be a zerodimensional closed subscheme of X Let i : Z → X be inclusion and f : X → Spec... this chapter is well known, a few of the definitions, lemmas and propositions are new 2.1 Codes over Rings This section recalls the basic definitions regarding codes over rings References for this section are [8], [30] and [33] Throughout this section, A denotes a local Artinian ring Definition 2.1.1 A code C of length n over A is a subset of An The elements of C are called codewords If C is a submodule . decoding algorithms for algebraic geometric codes over finite fields to make them work for algebraic geometric codes over rings. In Chapter 3, the modification of the basic algorithm for algebraic geometric. the 3 algorithms for decoding algebraic geometric codes over finite fields can be modified to decode algebraic geometric codes over rings. The goal of this dissertation is to modify three of the decoding. . 21 2.5 Algebraic Geometric Codes over Rings . . . . . . . . . . . . . . . . . 24 3 The Basic Decoding Algorithm for Algebraic Geometric Codes over Rings 29 3.1 The Generalized Basic Decoding