decoding algorithms for algebraic geometric codes over rings

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.. The fi

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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 presentthree decoding algorithms for algebraic geometric codes over rings

The first algorithm presented is a modification of the basic algorithm for algebraicgeometric codes over fields, and decodes with respect to the Hamming weight Thesecond algorithm presented is a modification of the Guruswami-Sudan algorithm,

a list decoding algorithm for one-point algebraic geometric codes over fields Thisalgorithm also decodes with respect to the Hamming weight Finally, we show howthe Koetter-Vardy algorithm, a soft-decision decoding algorithm, can be used todecode one-point algebraic geometric codes over rings of the form Z/pr

Z, where p is

a prime, with respect to the squared Euclidean weight

This dissertation would have not been possible without the support and friendship

of my advisor, Judy Walker She has provided tremendous help and encouragementover the last five years I would also like to thank the other members of my

committee, Tom Marley, Lance Perez, Mark Walker and Roger Wiegand for theirhelp and advice throughout my graduate education In particular, I would like tothank Mark Walker for all the time he spent helping me understand algebraicgeometry I would also like to thank Ralf Koetter for his help with writing Chapter5

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 introduced me

to research and the field of coding theory I would also like to thank Judy Greenand Elsa Schaefer They helped prepare me for graduate school and gave me endlessadvice

Several of the graduate students have become my family here at UNL It is partlythrough their support and help that I have succeeded here at UNL In particular, Iwould like to thank Suanne Au, Daniel Buettner, Jennifer Everson, Pari Ford, MattKoetz, Ed Loeb and Melissa Luckas I will miss them greatly

Finally, I would like to thank my family, my parents John and Joan Bartley andsister 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

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 Codes

Chapter 1

Introduction

Every communications channel contains noise which can cause errors to occur Thenoise in a channel can come from several different sources For example, rain or solarflares can cause noise in a satellite link; a scratch on a CD can be thought of as noisefor that channel Error-correcting codes are used when transmitting data across achannel 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 An Theelements of C are called codewords If C is a submodule of An, then C is called alinear 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 linear binary codes, such as the Nordstrom-Robinson code, are nonlinear projections

non-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

1982, Tsfasman, Vlˇadut¸ and Zink [29] used modular curves to prove the existence

of a sequence of codes with asymptotically better parameters then any previouslyknown sequence of codes Algebraic geometric codes over finite fields are constructed

by using a smooth, absolutely irreducible, projective curve X over a finite field Fq.Given a divisor Weil D on X and a set of distinct Fq-rational points P = {P1, , Pn}

on X such that supp D ∩ P = ∅, the algebraic geometric code CL(X, P, D) is defined

by evaluating functions in the Riemann-Roch space of D at points in P, i.e,

CL(X, P, D) = {(f (P1), , f (Pn)) | f ∈ L(D)}

In the late 1990’s, Walker [33] defined algebraic geometric codes over rings, thuscombining two different areas of coding theory Given a local Artinian ring A and asmooth irreducible projective scheme X of relative dimension one over Spec A whoseclosed fiber X is absolutely irreducible, a Cartier divisor on X is the analog of a Weildivisor on X, and the A-module Γ(X, OX(D)) is the analog of the Riemann-Rochspace L(D) Lastly, a set of pairwise disjoint A-points Z = {Z1, , Zn} on X isthe analog of a set of distinct Fq-rational points P = {P1, , Pn} on X Given anA-point Zi ∈ Z, there is a non-canonical isomorphism γi : Γ(Zi, OX(D)|Z i) → A.Any system γ = {γi} of these isomorphisms gives us a map

α : Γ(X, OX(D)) → An

The algebraic geometric code CL(X, Z, OX(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 CL(X, Z, OX(mZ), γ) is called a one-point algebraic geometric code In[33], Walker proved several properties about algebraic geometric codes over rings, butdid not provide any decoding algorithms, although it is believed that many of the

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 tomodify three of the decoding algorithms for algebraic geometric codes over finitefields to make them work for algebraic geometric codes over rings

In Chapter 3, the modification of the basic algorithm for algebraic geometric codesover finite fields is given The basic algorithm is a classical decoding algorithm andworks by finding error locator functions for a received word ~y Although the modi-fied basic algorithm decodes algebraic geometric codes over any finite local ArtinianGorenstein ring A, it does so with respect to the Hamming weight, where the Ham-ming weight of a vector ~x ∈ An is defined to be

wH(~x) = |{i | xi 6= 0}|

But, for codes over Z/4Z, the more pertinent weight measure is the Lee weight, ormore generally, the squared Euclidean weight for codes over rings of the form Z/pr

Z.For A = Z/4Z, the Lee weight wL: Z/4Z → Z is given by

of unity via the map epr : Z/pr

Z → C defined by

epr(x) = e2πixpr

The squared Euclidean weight of x ∈ Z/pr

Z, wE 2(x), is the square of the Euclideandistance between epr(x) and epr(0) in the complex plane Note that for A = Z/4Z,

wL(x) = 12wE2(x) for all x ∈ A

The motivation for decoding codes over rings with respect to the Lee weight orthe squared Euclidean weight is two-fold The first motivation comes from the Graymap The Gray map λ : Z/4Z → F22 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, wL) → (F2n2 , wH) is an isometry.For a code C over Z/4Z, decoding C with respect to the Lee weight is equivalent todecoding λ(C) with respect to the Hamming weight

The second motivation for decoding codes over Z/pr

Z with respect to the squaredEuclidean weight comes from communication channels Because of the way that data

is encoded for transmission across a channel, there is a higher probability that asymbol ci of a codeword ~c will get changed to (ci± 1) (mod pr) rather than any ofthe other symbols Decoding algorithms for codes over Z/prZ need to take this intoconsideration We can achieve this goal by decoding with respect to the squaredEuclidean weight

In order to decode algebraic geometric codes over rings with respect to the squaredEuclidean weight, we first look at the Guruswami-Sudan algorithm, a list decodingalgorithm Given an error bound e and a received word ~y, the algorithm returns allcodewords ~c ∈ C such that dH(~y, ~c) ≤ e, where dH denotes the Hamming distance

List decoding algorithms are modern decoding algorithms motivated, in part, by thefact that many classical algorithms may fail to correctly decode all received wordsthat are closest to a unique codeword.

The Guruswami-Sudan algorithm [7] was originally designed for decoding alized Reed-Solomon codes over finite fields, a class of algebraic geometric codes Thealgorithm was then generalized to decode one-point algebraic geometric codes overfinite fields Given a finite field Fq, a set P = {P1, , Pn} of distinct elements of

gener-Fq, a set A = {a1, , an} of (not necessarily distinct) elements of Fq, and a positiveinteger k, 0 < k < n, the generalized Reed-Solomon code GRS(P, k) is defined by

GRS(P, A, k) = {(a1f (P1), , anf (Pn)) | f ∈ Fq[x], deg f ≤ k}

The Guruswami-Sudan algorithm works by solving a polynomial reconstruction lem, which is equivalent to decoding GRS(P, A, k) That is, given a received word ~yand error bound e, the Guruswami-Sudan algorithm finds all f ∈ Fq[x] of degree atmost k such that f (Pi) = yi

prob-a i for at least n−e of the ordered pairs (P1,y1

a 1), , (Pn,yn

a n).The algorithm solves the reconstruction problem by finding a nonzero polynomialQ(x, y) ∈ Fq[x, y] such that the roots of Q(x, y) include all f ∈ A[x] of degree at most

k such that f (Pi) = yi

a i for at least n − e of the ordered pairs (P1,y1

a i), , (Pn,yn

a n).Let A be a local Artinian ring whose maximal ideal is principal In Chapter 4, wemodify the Guruswami-Sudan algorithm to work for generalized Reed-Solomon andone-point algebraic geometric codes over A

By itself, the Guruswami-Sudan algorithm decodes one-point algebraic geometriccodes with respect to the Hamming weight Let A = Z/prZ, where p is a prime InChapter 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

Given the code CL(X, Z, OX(mZ), γ) of length n over A, an error bound e and areceived word ~y, the Koetter-Vardy algorithm first constructs a pr × n multiplicitymatrix M of nonnegative integers As in the Guruswami-Sudan algorithm, a polyno-mial QM(y) ∈ K(X)[y] is then found, where K(X) is the A-module of rational func-tions on X The roots of QM(y) contained in the A-module Γ(X, OX(mZ)) ⊂ K(X)include all functions which correspond to codewords ~c ∈ CL(X, Z, OX(mZ), γ) suchthat wtE2(~y − ~c) ≤ e.

In Chapter 6, we conclude by discussing a two-stage decoder, developed by mand [1], for decoding one-point algebraic geometric codes over rings

Ar-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 areview of the construction of algebraic geometric codes over rings and their properties.Although much of the material in this chapter is well known, a few of the definitions,lemmas and propositions are new

This section recalls the basic definitions regarding codes over rings References forthis section are [8], [30] and [33] Throughout this section, A denotes a local Artinianring

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 of An, then C is called a linearcode Furthermore, if C is a free A-module, then we say that C is free and define thedimension of C to be dim C = rankA(C)

The following definitions define a metric and a symmetric bilinear form on An

Definition 2.1.2 The Hamming distance d : An× An→ Z is given by

d((x1, , xn), (y1, , yn)) = |{i | xi 6= yi}|

The Hamming weight of a vector ~x ∈ An is defined as wt(~x) = d(~x, ~0), where ~0 is the0-vector of length n The minimum Hamming distance d of C is

d = d(C) = min{d(~x, ~y) | ~x 6= ~y ∈ C}

Note that if C is a linear code, then d(C) = min{wt(~x) | ~x 6= ~0 ∈ C}

Definition 2.1.3 Let ~x, ~y ∈ An Define

Definition 2.1.4 Let C be a free code of length n and dimension k over A Then

a generator matrix G for C is a k × n matrix over A whose rows form a basis of C.Thus

C = {~uG | ~u ∈ Ak}

A parity check matrix H for C is an (n − k) × n matrix over A such that

C = {~x ∈ An | H~xT = 0}

Note that any parity check matrix H for a free code C is a generator matrix for

The following theorem from [33] gives the relationship between the minimumHamming distance of C and the minimum Hamming distance of C = π(C)

Theorem 2.1.6 ([33, Theorem 3.4]) Let C be a linear code over a local Artinianring A and let C = π(C) be its coordinatewise projection Let d and d denote theHamming distances of C and C, respectively Then d ≤ d with equality if C is free

This section gives a summary of the construction and properties of algebraic geometriccodes 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) and genus g

Definition 2.2.1 Let P be a closed point of X and let κ(P ) denote the residue field

at P Then κ(P ) = Fq m for some m ∈ Z and we say the degree of P , denoted deg P ,

is m

We denote the set of degree 1 points of X by X(Fq) The elements of X(Fq) arecalled the Fq-rational points of X.

Definition 2.2.2 A Weil divisor D on X is an element of the free abelian groupgenerated by the closed points of X That is, D =P

P nPP , where nP is zero for allbut finitely many closed points P of X The support of D, denoted supp D, is theset of points P such that nP 6= 0 The divisor D is said to be effective if nP ≥ 0for all points P of X; in this case we write D ≥ 0 The degree of D is defined asdeg D =P

P nP deg P

Let P be a closed point of X There exists a unique valuation ring OP ⊂ Fq(X)associated to P Since OP is a valuation ring, OP is local with principal maximalideal mP Any element t ∈ Fq(X) such that mP = (t) is called a local parameterfor P Given a nonzero rational function f ∈ Fq(X), we may write f = tmu, where

Then νP is a discrete valuation of Fq(X) with valuation ring OP

Note that νP is well-defined since, if t0 is another local parameter for P , then

t = ut0 for some u ∈ OP× For f , g ∈ Fq(X) and c ∈ Fq \ {0}, νP has the followingproperties:

1 νP(c) = 0

2 νP(f + g) ≥ min {νP(f ), νP(g)}

3 νP(f g) = νP(f ) + νP(g).

Remark 2.2.4 Let f ∈ Fq(X) be nonzero and let P be a closed point of X Let

νP(f ) = m If m > 0, then we say that f has a zero of order m at P If m < 0, then

we say that f has a pole of order −m at P

Definition 2.2.5 Let f ∈ Fq(X) be nonzero The divisor (f ) associated to f is

By [26, Theorem I.4.11], the degree of a principal divisor (f ) is zero

The set Ω(X) of rational differentials on X is a Fq(X)-vector space generated bythe symbols df , where f ∈ Fq(X), with relations given by those of usual differentia-tion

Definition 2.2.6 Let v ∈ Ω(X) be nonzero For each closed point of P on X, let

tP be a local parameter for P and fP ∈ Fq(X) such that v = fPdtP Then the divisor(v) associated to v is

(v) = X

P ∈X

νP(fP)P

The residue of v at P , resP(v), is a−1, where fP =P aiti

P is the Laurent expansion

of fP near P A divisor D is called a canonical divisor if D = (v) for a nonzerodifferential v ∈ Ω(X)

Given v ∈ Ω(X), the value resP(v) and the divisor (v) associated to v are pendent of the choices of local parameters for each closed point P by [26, TheoremIV.3.2, Proposition IV.2.9] The degree of any canonical divisor is 2g−2 [26, CorollaryI.5.16]

inde-Let D be a divisor on X We associate two Fq-vector spaces with D:

L(D) = {f ∈ Fq(X) | (f ) + D ≥ 0} ∪ {0}

and

Ω(D) = {v ∈ Ω(X) | (v) + D ≥ 0} ∪ {0}

We are now ready to define algebraic geometric codes over finite fields

Definition 2.2.7 Let X be a smooth, absolutely irreducible, projective curve over

Fq, P = {P1, , Pn} a set of n distinct Fq-rational points on X, and D a divisor on

X such that supp D ∩ P = ∅ Abusing notation, let P denote the divisor P1+ · · · + Pn.Then there are two algebraic geometric codes associated to X, P and D:

dL≥ n − deg D

2 CΩ is a linear code of length n with dimension kΩ and minimum Hammingdistance dΩ satisfying

kΩ ≥ n − deg D + g − 1and

This section is a review of some properties of curves over rings Throughout thissection, let A be a local Artinian ring with maximal ideal m and finite residue field

Fq Let X ⊂ PrA be a curve over A, by which we mean that X is a smooth irreducibleprojective scheme over Spec A of relative dimension one Let

X = X ×Spec ASpec Fq

be the fibre of X over m As in [33], we assume that X is absolutely irreducible.Additional information on curves over rings can be found in [9] and [10].

The Picard group of X, Pic(X), is the group of isomorphism classes of invertibleline bundles on X In the field case, Pic(X) is isomorphic to the group Div(X) of Weildivisors on X modulo linear equivalence and the construction of algebraic geometriccodes over finite fields implicitly uses this isomorphism Although this isomorphismdoes not hold for curves over rings, we do have an isomorphism between Pic(X) andthe group of Cartier divisors on X modulo linear equivalence It is this isomorphismwhich will be used in the construction of algebraic geometric codes over rings.Definition 2.3.1 ([10]) Let X be a scheme and let K denote the sheaf of totalquotient rings on X Denote by K∗ the sheaf of invertible elements in K and let

O∗ denote the sheaf of invertible elements of OX A Cartier divisor D is a globalsection of the sheaf K∗/O∗ If a Cartier divisor D is in the image of the naturalmap Γ(X, K∗) → Γ(X, K∗/O∗), then D is said to be principal Two Cartier divisors

D and D0 are linearly equivalent if their difference is principal The Cartier divisorclass group, CaCl(X), is the group of Cartier divisors modulo linear equivalence

A Cartier divisor can be represented in the form D = {(Ui, fi)} where {Ui} is anopen cover of X and fi is an element of Γ(Ui, K∗) such that for each i and j,

fi/fj ∈ Γ(Ui∩ Uj, O∗)

Moreover, although the group operation on K∗/O∗ is multiplication, it is standard(see, eg., [10]) to use the language of additive groups when talking about Cartierdivisors in order to preserve the analogy of Cartier divisors with Weil divisors Infact, in the field case, the group of Weil divisors on X is isomorphic to the group

of Cartier divisors [10, Proposition II.6.11] The proof of the next proposition uses

the fact that K is constant [33, Lemma 1] and follows the proof of [10, PropositionII.6.15].

Proposition 2.3.2 Let X be a curve over A Then

CaCl(X) ' Pic(X)

The following proposition from [10] will be used to throughout

Proposition 2.3.3 ([10, Proposition II.6.13]) Let X be a scheme Then

1 For any Cartier divisor D, OX(D) is an invertible sheaf on X The map

D 7→ OX(D) gives a 1-1 correspondence between Cartier divisors and invertiblesubsheaves of K

2 OX(D1− D2) ' OX(D1) ⊗ 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 points Note that, since X is a curve over A, there exists a structure morphism

¿From the above definition, it follows that Γ(Z, OX|Z) ' A for any A-point Z of

X It is noted in [33] that every closed point P ∈ X which is an Fq-rational point

of X is contained in an A-point of X Furthermore, since A/m = Fq, the uniqueclosed point of any A-point Z is an Fq-rational point of X If Z1 and Z2 are A-points

containing Fq-rational points P1 and P2 respectively, then Z1 and Z2 are disjoint if

P1 6= P2

Given an A-point Z on X, there is a unique, well-defined Cartier divisor (which

we will also denote by Z) associated to Z The next proposition is needed in order togive an explicit expression for the Cartier divisor associated to Z The proposition isoriginally found in [9]; a complete proof is given in [31]

Proposition 2.3.5 ([9]) Let Z be an A-point on X and let P be the unique closedpoint contained in Z Denote by OX,P the stalk of OX at P and by mP the maximalideal of this ring Then there is an element t ∈ OX,P with the following properties:

1 OX,P/(t) ' A

2 t is a non-zero-divisor in OX,P

3 t ∈ mP \ m2

P

4 OX,P/(tn) is a free A-module with basis 1, t, , tn−1

5 The total quotient ring of OX,P is generated as an OX,P-module by 1, t−1, t−2, Definition 2.3.6 Let Z be an A-point on X and let P be the unique closed pointcontained in Z Any element t ∈ OX,P that satisfies the properties of Proposition2.3.5 is called a local parameter for Z (or for P with respect to Z.)

Let Z be an A-point of X and let P be the closed point contained in Z Let

U = Spec B be an affine open neighborhood of P on which the ideal for Z is principaland let t be a local parameter for Z on U Then B/(t) ' A and t is a unit on theset U \ {P } Let V =X \ {P } Then t is a unit on U ∩ V = U \ {P } and the Cartierdivisor for Z can be expressed as {(U, t), (V, 1)}

Definition 2.3.7 Let D = {(Ui, fi)} be a Cartier divisor on X The subsheaf OX(D)

of K is given by

Γ(Ui, OX(D)) = fi−1Γ(Ui, OX) = 1

fiOX(Ui)

Definition 2.3.8 ([33, Definition 5.3]) Let D be a Cartier divisor on X, and let

P ∈ X be a closed point which is a rational point of X = X ×Spec ASpec Fq We saythat P is not in the support of D if we can write D = {(Ui, fi)}, where fi ∈ OX(Ui)×for some i such that P ∈ Ui

Remark 2.3.9 If Z is an A-point containing the closed point P and D is a Cartierdivisor on X such that P is not in the support of D, then it is shown in [33, Section5] that we may view the composite map

Z = (z0 : : zr) Since A is local and z0, , zr generate the unit ideal of A, some zi

is a unit Without loss of generality, we may assume that z0 = 1 and U is contained inthe standard affine open subset of Pr defined by z0 = 1 Then J is the ideal generated

by z1− x1, , zn− xn, and the evaluation map (2.1) is given by

s 7→ h(1, z1, , zn)

f (1, z1, , zn) ∈ A

From this point on, we will write s(Z) to represent the image of s ∈ Γ(X, OX(D))under the composite map (2.1), i.e s(Z) = γ(s|Z).

Definition 2.3.10 Let D be a Cartier divisor on X and let D0 be a Weil divisor on

X such that φ∗(OX(D)) = OX(D0), where φ : X → X is the natural map Then D issaid to be effective if D0 is an effective divisor

The following information on invertible line bundles will be used throughout thefollowing chapters

Definition 2.3.11 Let L be a line bundle on X and let D0 be a Weil divisor on Xsuch that φ∗(L) = OX(D0) Then the degree of L is

deg L = deg D0

Let L1 and L2 be line bundles on X Then φ∗(L1 ⊗ L2) = φ∗(L1) ⊗ φ∗(L2) anddeg (L1⊗ L2) = deg L1+ deg L2 Furthermore, if ω is the canonical line bundle on X(see [10, Section II.8] for a definition), then since X is smooth, φ∗(ω) is the canonicalline bundle on X Hence, in this situation, deg ω = deg φ∗(ω) = 2g − 2

The proof of the next lemma is found in [33]

Lemma 2.3.12 ([33, Lemma 4.6]) For any open affine U ⊂ X and line bundle L on

Proposition 2.3.13 Let X and A be as above and let L be a line bundle on X Set

L0 = φ∗(L) Then

Γ(X, L0) = Γ(X, L) ⊗AFq.Proof Let U and V be affine open sets of X such that X = U ∪ V Such affine sets

U and V exist by [31, Lemma 4.11] We then have the following exact sequences:

A-The next lemma is used throughout

Lemma 2.3.15 Let L be a line bundle on X If deg L < 0, then Γ(X, L) = {0}.Proof Note that deg L = deg φ∗(L) Thus deg φ∗(L) < 0, so Γ(X, φ∗(L)) = {0} ByProposition 2.3.13, Γ(X, L)⊗AFq = Γ(X, φ∗(L)) = {0} By Remark 2.3.14, Γ(X, L) isfinitely generated So, by Nakayama’s Lemma [4, Proposition 2.6], Γ(X, L) = {0}.Let Z1, , Zn be disjoint A-points of X We will often be interested in divisors

of the form m1Z1+ + mnZn, where m1, , mn∈ Z

Lemma 2.3.16 Let Z1, , Zn be disjoint A-points of X and let P1, , Pn be theclosed points contained in Z1, , Zn Then

φ∗(OX(m1Z1+ · · · + mnZn)) = OX(m1P1 + · · · + mnPn)

In particular, deg(OX(m1Z1+ · · · + mnZn)) = m1 + · · · + mn

Proof By Proposition 2.3.3, it is enough to show that if Z is an A-point of X and

P is the closed point contained in Z, then φ∗(OX(Z)) = OX(P ) Let U = Spec B

be an affine open set containing P on which the ideal for Z is principal and let t

be a local parameter for Z on U Let {Vi = Spec Bi} be an open affine cover of

V = X \ {P } Then we may write Z = {(U, t), (Vi, 1)} Note Γ(U, OX(Z)) = 1tB andfor all i, Γ(Vi, OX(Z)) = Bi Set U0 = U ×Spec ASpec Fq and Vi0 = Vi ×Spec A Spec Fq

By Lemma 2.3.12, U0 is an affine open set of X and Vi0 is an affine open set of X for all

i By Lemma 2.3.12, Γ(U0, φ∗(OX(Z))) = 1tB/mB and Γ(Vi0, φ∗(OX(Z))) = Bi/mBifor all i, where t is the image of t in B/mB Since t is a local parameter for P , wemay write P = {(U0, t), (Vi0, 1)} Hence,

We define the following “valuation” functions on MZ.

Definition 2.4.1 Let Z be an A-point on X and f ∈ MZ Define

If f 6= 0 and νZ(f ) = −m, we say that f has a pole of order m at Z For any A-point

W on X disjoint from Z, define νZ,W : MZ → N ∪ {∞, 0} by

where νZ(f ) = −m If f 6= 0 and νZ(f ) = l, we say that f has a zero of order l at W

If A is a field, then on the set MZ, the functions defined in Definition 2.4.1 areequivalent to the discrete valuation defined in Definition 2.2.3

Remark 2.4.2 Let Z be an A-point of X, let f ∈ MZ and let W be an A-point on

X disjoint from Z Then f ∈ Γ(X, OX(jZ − iW )) if and only if 0 ≤ i ≤ νZ,W(f ) and

j ≥ −νZ(f )

The functions νZ and νZ,W have properties similar to those of valuations

Lemma 2.4.3 Let Z be an A-point of X and let W be an A-point of X disjoint from

Z Let r, s ∈ MZ and let a ∈ A \ {0} Then

1 νZ(a) = 0 and νZ,W(a) = 0

as before, that νZ,W(a) = 0

(2) Note that r ∈ Γ(X, OX(mZ)) and s ∈ Γ(X, OX(nZ)) Let l = max {m, n}

By Remark 2.4.2, we have that r, s ∈ Γ(X, OX(lZ)), and so r + s ∈ Γ(X, OX(lZ)).Since −l = min {−m, −n}, it follows that νZ(r + s) ≥ min {νZ(r), νZ(s)}

(3) Let P be the closed point contained in Z and let U = Spec B be an openaffine neighborhood of P on which the ideal for Z is principal Let t be a localparameter for Z on U and let {Vi = Spec Bi} be an affine open cover of X \ {P }.Then we may write Z = {(U, t), (Vi, 1)} Since νZ(r) = −m and νZ(s) = −n we have

r ∈ t1mB and s ∈ t1nB Thus rs ∈ tm+n1 B Since r, s ∈ Bi for all i, rs ∈ Bi Hence

rs ∈ Γ(X, OX((m + n)Z)) and νZ(rs) ≥ −m − n

(4) Note that r ∈ Γ(X, OX(mZ − jW )) and s ∈ Γ(X, OX(nZ − kW )) Let

l = max {m, n} and h = min {j, k} Note that r, s ∈ Γ(X, OX(lZ − hW )) by Remark2.4.2 Hence r+s ∈ Γ(X, OX(lZ −hW )), and so νZ,W(r+s) ≥ min {νZ,W(r), νZ,W(s)}

(5) Let P and Q be the closed points of Z and W respectively Let Ui = Spec Bi

be an open affine cover of X Note that we may choose U1 and U2 so that P ∈ U1,

Q ∈ U2, the ideal for Z is principal on U1 and the ideal for W is principal on U2.Furthermore, we may pick U1 and U2 small enough so that Q 6∈ U1 and P 6∈ U2.Let tP be a local parameter for Z on U1 and tQ be a local parameters for W on U2.Then we may write Z = {(Ui, fi)} and W = {(Ui, gi)}, where f1 = tP, g2 = tQ,

fj = 1 for j 6= 1 and gj = 1 for j 6= 2 Since r ∈ Γ(X, OX(mZ − jW )), we have

f m 2

B2 = tjQB2, and r ∈ g

j i

f m i

Bi = Bi for all i > 2 Similarly,

s ∈ gk1

t nB1 = t1nB1, s ∈ t

k Q

f n

2 B2 = tk

QB2 and s ∈ gki

f n i

Bi = Bi for all i > 2 So rs ∈ 1

tm+nP B1,

rs ∈ tj+kQ B2 and rs ∈ Bi for all i > 2 Thus rs ∈ Γ(X, OX((m + n)Z − (j + k)W ))and so νZ,W(rs) ≥ νZ,W(r) + νZ,W(s)

Remark 2.4.4 For an A-point Z containing the closed point P with local parameter

tP, OX,P/(tP) ' A Since A is Artinian, A is a field if and only if A is a domain.Thus if A is not a field, then tP is not a prime element of OX,P Equality is thereforenot guaranteed in parts (3) and (5) of Lemma 2.4.3 For an example of this, notethat if r, s ∈ MZ such that r|Z ∈ m and s|Z ∈ m, then it is possible for (rs)|Z = 0even if r|Z 6= 0 and s|Z 6= 0

The proof of the following proposition follows the spirit of the proof of [33, orem 5.4]

The-Proposition 2.4.5 Let Z be an A-point of X and let {Z1, Zn} be a set of pairwisedisjoint A-points of X, all disjoint from Z Let h ∈ MZ If νZ(h) ≥ −m and

νZ,Zi(h) ≥ ri for 1 ≤ i ≤ n, for some nonnegative integers m, r1, , rn, then

h ∈ Γ(X, OX(mZ − r1Z1− · · · − rnZn))

Proof Let h ∈ MZ such that νZ(h) ≥ −m and νZ,Z(h) ≥ ri for 1 ≤ i ≤ n, for

some nonnegative integers m, r1, ., rn Write Z = {(Uj, fj)} and Zi = {(Uj, g(i)j )}where {Ui = Spec Bi} is an affine open cover of X and refinements have been taken

if necessary To show that h ∈ Γ(X, OX(mZ − r1Z1− · · · − rnZn)), we will show thatfor each j,

h ∈ (g

(i)

j )r 1· · · (g(n)j )r n

fm j

OX(Uj)

Since νZ(h) ≥ −m and νZ,Zi(h) ≥ ri, by Remark 2.4.2, h ∈ Γ(X, OX(mZ − riZi)) for

1 ≤ i ≤ n Thus, for each i and j, h ∈ (g

(i)

j ) ri

f m

j OX(Uj) Hence fjmh ∈ (g(i)j )riOX(Uj).Since the Zi are disjoint, for each j,

OX(Uj)

In this section we recall the definition of algebraic geometric codes over local Artinianrings 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 andlet L be a line bundle on X For each i, let γibe an isomorphism, γi : Γ(Zi, L|Zi) → A,and let γ = {γ1, , γn} be the system of these isomorphisms

Definition 2.5.1 ([31, Definition 5.1]) Let A, X, Z, L, and γ be as above Let

CL(X, Z, L, γ) be the image of the composition α

is called an algebraic geometric code over A.

Suppose ϕ = {ϕi : Γ(Zi, L|Zi) → A | 1 ≤ i ≤ n} is another system of phisms Then, for 1 ≤ i ≤ n, there exists ai ∈ A× such that

isomor-γi(s|Z i) = aiϕi(s|Z i) 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 metric codes over A We restate them without proof

geo-Theorem 2.5.2 ([33, geo-Theorem 5.4-Corollary 5.7]) Let X, L, Z and γ be as above.Let g denote the genus of X, and suppose 2g − 2 < deg L < n Then CL(X, Z, L, γ) is

a free code of length n, dimension k = deg L + 1 − g and minimum Hamming distance

d ≥ n − deg L

In [33], it is shown that when A is a Gorenstein ring, the class of algebraic ric codes over A is closed under duals The rings that we are primarily interested inare Galois rings, in particular Galois rings of the form Z/prZ, where p is a prime AsGalois rings are local Artinian Gorenstein rings, we have that the class of algebraicgeometric codes over a given Galois ring is closed under duals In recalling this result

geomet-we will look at residues The following discussion comes from [33, Section 4] Let P

be an Fq-rational point of X and let Z be an A-point containing P Let t be a localparameter for P with respect to Z Let ω = ωX denote the canonical line bundle on

X and let η be the generic point of X Let ν ∈ ωη, where ωη is the stalk of ω at η,and let ν be the image of ν in ωη/ωP Then we can expand ν in a neighborhood of

Z to get

ν =X

j<0

ajtjdt,

where aj ∈ A (see [9, Chapter 7]) Define the residue of ν at Z to be

Γ(X, E )

resZi

##G G G G

Theorem 2.5.3 ([33, Theorem 5.12]) Let X, L, Z, γ, and ξ be described as above.Then

CL(X, Z, L, γ)⊥ = CL(X, Z, ω ⊗ OX(Z) ⊗ L−1, ξ)

It should be noted that, if γi is thought of as the evaluation map, then ξi may be

thought of as the residue map.

Definition 2.5.4 Given the parameters X, Z, L and γ, we define the residue code

is any system of isomorphisms

The following corollary summarizes the proprieties of CΩ(X, Z, L, γ)

Corollary 2.5.6 Let X, Z, L and γ be as before If 2g − 2 < deg L < n then

CΩ(X, Z, L, γ) is a free code of dimension kΩ = n + g − 1 − deg L and minimumHamming distance dΩ ≥ deg L − 2g + 2

Proof Recall that deg ω = 2g − 2 Since

deg (ω ⊗ OX(Z) ⊗ L−1) = 2g − 2 + n − deg L

and 2g − 2 < deg L < n, we have

2g − 2 < deg (ω ⊗ OX(Z) ⊗ L−1) < n

Hence, by Theorem 2.5.2, CΩ(X, Z, L, γ) is a free code of dimension

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 tinian 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 code, so the algorithmdecodes all algebraic geometric codes over A The algorithm is a generalization of thebasic algorithm for decoding algebraic geometric codes over finite fields Presenta-tions of the basic algorithm, which itself is a generalization of the Arimoto-Petersonalgorithm for decoding Reed-Solomon codes, can be found in [11], [13] and [24]

We begin this section with a proposition that provides motivation for the basic rithm The proposition is the analog of [11, Proposition 2.4] for the ring case

algo-Proposition 3.1.1 Let C ⊂ Anbe a free code with parity check matrix H, let ~y ∈ An,and let J ⊂ {1, , n} be a set of size |J | < d(C) If there exist ~c ∈ C and ~e ∈ An

such that ~y = ~c + ~e and {j | ej 6= 0} ⊂ J, then ~x = ~e is the unique solution of thesystem of linear equations given by:

For the rest of this section let A be a finite local Artinian Gorenstein ring withmaximal ideal m and finite residue field Fq, and let X be a curve over A of genus g Let

D be a Cartier divisor on X such that 2g − 2 < deg OX(D) < n, let Z = {Z1, , Zn}

be a set of pairwise disjoint A-points on X, and let γ = {γi : Γ(Zi, OX(D)|Zi) → A}

be a system of isomorphisms For 1 ≤ i ≤ n, let Pi be the closed point contained

in Zi, and let P = {P1, , Pn} We will assume that, for each Pi ∈ P, Pi is not insupport of D

Let CΩ = CΩ(X, Z, OX(D), γ) Since 2g − 2 < deg OX(D) < n, by Corollary2.5.6, CΩ is a free code with minimum distance at least δΩ = deg OX(D) − 2g + 2.Hence, by Proposition 3.1.1, a received word ~y = ~c + ~e, where ~c ∈ CΩ and ~e ∈ An,can be correctly decoded if we can find a set J ⊂ {1, , n} such that j ∈ J if ej 6= 0

and |J | < δΩ We shall find this set under that condition that wt(~e) ≤ j(δΩ −1)

2

kbyfinding an error locator function for ~y

Definition 3.1.2 Let ~y = ~c + ~e, where ~c ∈ CΩ and wt(~e) ≤j(δΩ −1)

2

k Set

I = {i | ei 6= 0, 1 ≤ i ≤ n}

Let F be a Cartier divisor on X and let δ = {δi : Γ(Zi, OX(F )|Zi) → A} be a system

of isomorphisms A function

s ∈ Γ(X, OX(F )) \ mΓ(X, OX(F ))

is an error locator function for ~y if δi(s|Zi) ∈ m for all i ∈ I

If A is a field, then an error locator function for ~y is simply a rational function that

is zero at all error positions For the rest of this section, unless otherwise stated, let ~y

be a received word ~y = ~c + ~e such that ~c ∈ CΩ and wt(~e) ≤j(δΩ −1)

2

k, let F be Cartierdivisor on X with support disjoint from P, and let δ = {δi : Γ(Zi, OX(F )|Zi) → A}

be a system of isomorphisms The following lemma is needed to describe the set oferror locator functions for ~y

Lemma 3.1.3 Let X, Z, F , γ = {γi} and δ = {δi} be as before Then, for each

Zi ∈ Z, there exists an isomorphism

τi : Γ(Zi, OX(D − F )|Zi) → Asuch that, for s ∈ Γ(X, OX(F )) and v ∈ Γ(X, OX(D − F )),

δi(s|Z)τi(v|Z) = γi(sv|Z)

Proof Let Zi ∈ Z Let s ∈ Γ(X, OX(F )) and let v ∈ Γ(X, OX(D − F )) Write

F = {(Uj, fj)} and D = {(Uj, gj)}, where {Uj} is an open affine cover of X andrefinements have been taken if necessary For each j, we have s ∈ f1

jOX(Uj) and

v ∈ fj

g jOX(Uj), and so sv ∈ g1

jOX(Uj) for all j Choose Uj such that Pi ∈ Uj Then,

as in Remark 2.3.9, Uj = Spec B and Zi = Spec B/J for some ideal J of B such thatB/J ' A Hence,

Γ(Zi, OX(D)|Z i) = 1

gjB/J,

Γ(Zi, OX(F )|Zi) = 1

fjB/Jand

Γ(Zi, OX(D − F )|Zi) = fj

gjB/Jwhere gj and fj are the images in B/J of gj and fj respectively Since Zi is neither inthe support of F nor in the support of D, we may assume both gj and fj are units of

Definition 3.1.4 Let X, Z, P, CΩ(X, Z, OX(D), γ), F and δ = {δi} be as before.Let ~y ∈ An be any received word The set K(~y, F, δ) is defined to be

K(~y, F, δ) = s ∈ Γ(X, OX(F )) \ mΓ(X, OX(F ))

Pn i=1yiδi(s|Zi)τi(v|Zi) = 0for all v ∈ Γ(X, OX(D − F ))

where {τi} is the system of isomorphisms given in Lemma 3.1.3

We shall show that, under certain conditions, the elements of K(~y, F, δ) are ror locator functions for ~y (see Theorem 3.1.6) Note that since A is finite andboth Γ(X, OX(F )) and Γ(X, OX(D − F )) are finitely generated by Remark 2.3.14,the A-modules Γ(X, OX(F )) and Γ(X, OX(D − F )) contain finitely many elements.Therefore we may calculate K(~y, F, δ) by exhaustive search If A is a field, then it

er-is shown in [25] that the elements of K(~y, F, δ) can be found by solving a system oflinear equations At this time however, it has not been investigated whether or notthis method of finding elements of K(~y, F, δ) holds when A is not a field

Lemma 3.1.5 Let X, Z, P, CΩ(X, Z, OX(D), γ) and ~y = ~c + ~e be as before Let

t = wt(~e), and let I = {i | ei 6= 0} Let Q = {Zi | i ∈ I}, and let Q be the Cartierdivisor obtained by adding up the points of Q Let F be a Cartier divisor on X Then

1 If deg OX(D − F ) > t + 2g − 2, then CΩ(X, Q, OX(D − F ), τ ) = {~0}, where

τ = {τi : Γ(Zi, OX(D − F )|Z i) → A} is any system of isomorphisms

2 If deg OX(D) > t + g, then Γ(X, OX(F − Q)) \ mΓ(X, OX(F − Q)) 6= ∅

If F has support disjoint from P, then

3 For any s ∈ Γ(X, OX(F − Q)) and i ∈ I, we have δi(s|Zi) = 0 for any system

of isomorphisms {δi : Γ(Zi, OX(F − Q)|Zi) → A}

4 If s ∈ Γ(X, OX(F − Q)) \ mΓ(X, OX(F − Q)), then s ∈ K(~y, F, δ), where

δ = {δi : Γ(Zi, OX(F )|Zi) → A} is any system of isomorphisms

Proof (1) Assume deg OX(D − F ) > t + 2g − 2 Since X is smooth, deg ω = 2g − 2,where ω is the canonical line bundle on X Since Q is the sum of t A-points, byLemma 2.3.16, deg OX(Q) = t Therefore

deg (ω ⊗ OX(Q) ⊗ (OX(D − F ))−1) = 2g − 2 + t − deg OX(D − F ) < 0

Hence, by Lemma 2.3.15,

Γ(X, ω ⊗ OX(Q) ⊗ (OX(D − F ))−1) = {0},

and so CΩ(X, Q, OX(D − F ), τ ) = {~0}

(2) As usual, let X = X×Spec ASpec Fqbe the fibre of X over m, and let φ : X → X

be the natural map By the Riemann-Roch Theorem [10, Theorem IV.1.3],

dim H0(X, φ∗(OX(F − Q)))−dim H1(X, φ∗(OX(F − Q))) = deg φ∗(OX(F − Q))+1−g

But deg φ∗(OX(F )) = deg OX(F ) ≥ g + t Since deg φ∗(OX(Q)) = deg OX(Q) = t, itfollows that deg φ∗(OX(F − Q)) ≥ g Thus,

(3) Write F = {(Uj, fj)} and Zi = {(Uj, g(i)j )} for all i ∈ I, where {Uj = Spec Bj}

is an affine open cover of X and refinements have been taken if necessary Then

Given Zi, where i ∈ I, let Uj be an open set containing the unique closed point of Zi.Without loss of generality, we may assume that Uj does not contain any other closed

points of P and that g(i)j is a local parameter for Zi on Uj Then Zi = Spec Bj/(g(i)j ).Thus, if fjs ∈ gj(i)OX(Uj) = g(i)j Bj, then fjs|Zi = 0 By our choice of F , we mayassume fj is a unit of Bj, and so s|Zi = 0 Hence δi(s|Zi) = 0 Therefore, it issufficient to show that, if s ∈ Γ(X, OX(F − Q)), then fjs ∈ g(i)j OX(Uj) for all j andfor all i ∈ I Let s ∈ Γ(OX(F − Q)) Then

fj OX(Uj)for all j and for all i ∈ I Hence fjs ∈ gj(i)OX(Uj) for all j and for all i ∈ I

(4) Recall that ~y = ~c + ~e, where ~c ∈ CΩ(X, Z, OX(D), γ) Let

fj OX(Uj )for all j and for all i ∈ I Hence fjs ∈ gj(i)OX(Uj) for all j and for all i ∈ I

(4) Recall that... δi(s|Zi) = Therefore, it issufficient to show that, if s ∈ Γ(X, OX(F − Q)), then fjs ∈ g(i)j OX(Uj) for all j andfor all i ∈ I... F has support disjoint from P, then

3 For any s ∈ Γ(X, OX(F − Q)) and i ∈ I, we have δi(s|Zi) = for any system

of isomorphisms {δi