Algebraic Geometric Coding Theory presented by Jake Hustad John Hanson Berit Rollay Nick Bremer Tyler Stelzer Robert Coulson

Algebraic Geometric Coding Theory

presented byJake Hustad

John Hanson

Berit Rollay

Nick Bremer

Tyler Stelzer

Robert Coulson

Contents

I. Review of Codes

II. The Performance Parameters

III. Reed-Solomon Code

IV. Finite Fields/Algebraic Closure

V. The Projective Plane

VI. Bezout’s Theorem

VII. Frobenius Maps

VIII. Nonsingularity and Genus

IX. Goppa’s Construction

X. Conclusion

Review of Codes

Jake Hustad

Introduction

• Basic Overview of Coding TheoryBasic Overview of Coding Theory

- Coding theory is the branch of mathematics concerned with

transmitting data across noisy channels and recovering the

message. Coding theory is about making messages easy

to read and finding efficient ways of encoding data.

What is a Code?Here is the formal definition of a code:

Where is usually a finite field.

A

A code C over an alphabet is simply a subset of

A

An = A x…x A(n copies).

Objectives for efficient codes:

Detection and correction of errors due to noise

Efficient transmission of data

Easy encoding and decoding schemes

The Ideal Code

Ideally, we would like a code that is capable of correcting all errors due to noise. In general, the more errors that a code needs to correct per message digit, the less efficient the transmission and also the more complicated the encoding and decoding schemes.

Code Parameters

The Performance Parameters

“n” is the total number of available symbols for a code word.

“k” is the number of information symbols in a given code word. “k” is the size of the code.

“d” is the distance between two code words. This is a Hamming distance.

The Hamming Distance

The Hamming distance between two words is the number of places where

the digits differ.

Example

u = “111” (Transmitted Code)v = “110” (Received Code) They differ only in the last digit, so the Hamming distance, d(u,v) = 1.

The Hamming Distance

This distance is significant because it gives us an idea of how many errors can be detected. The larger this distance is, the more errors can be detected.

The Minimum Distance

The minimum distance is the smallest Hamming distance between any two

possible code words.


Suppose that the minimum distance for the coding function is 3. Then, given any codeword, at least 3 places in it have to be changed before it gets converted into another codeword. In other words, if up to 2 errors occur, the resulting word will not be a codeword, and we detect the occurrence of errors.


Fact: If the minimum distance between code words is d, then up to d – 1 errors can be detected.

Reed-Solomon Codes

A Case Study

John Hanson

Definitions

Fq ~ field with q elements

Lr := { f Fq[x] | deg(f) r } {0}

r is non-negative note: this is a vector subspace over Fq with dim = r+1

basis = [ 1 x x2 … xr ]

Procedure

Label q-1 nonzero elements of Fq as:1, 2,…, q-1

Pick a k Z such that 1 k q-1

Then we have:

RS(k,q) := { ( f( 1), f( 2),…,f( q-1) )| f Lk-1 }

Notice

RS(k,q) is a subset of := FqxFqx…xFq

q-1 copies

so this is a code over the alphabet Fq

1qqF

Summary

Through our research and presentation last semester, we proved that for Reed-Solomon codes:

n = q – 1k = k (which was chosen)d = n – k + 1

Algebraic Geometry Background

By Nick Bremer and Berit Rollay

Contents

Finite Fields/Algebraic Closure

Projective Planes

Bezout’s Theorem

Algebraically Closed Fields

A field k is algebraically closed if every non-constant polynomial in k[x] has at least one root.

This is not closed under R because i R, but it is closed under C.

ix 1xf(x) 2

Ex)

Definition: Algebraic Closure

Let k be a field. An algebraic closure of k is a field K with k K satisfying:

K is algebraically closed, and

If L is a field such that k L K and L is algebraically closed, then L = K.

Are Algebraic Closures Unique?

It turns out they are. Every field has an unique algebraic closure, up to isomorphism. This theorem allows us to call the algebraic closure of k.k

CR Ex)

Let k be an algebraically closed field and let

be a polynomial of degree n. Then there exists a non-zero u and

(not necessarily distinct) such that:

Theorem

][)( xkxf kn ,...,1

))...(u(xf(x) 1 nx

In particular, counting multiplicity, f has exactly n roots in k.

Diophantine Equations

A Diophantine Equation is a polynomial with integer or rational coefficients, such as

. A useful problem to solve is how many rational solutions does this equation have?

12 22 yx

In order to answer this question, we need to define points at “infinity.”

Projective Plane

Let k be a field. The projective plane is defined as:

where if and only if

there is some non-zero with ,

and .

)(2 kP

~/)})0,0,0{(\(:)( 32 kkP ),,(~),,( 111000 ZYXZYX

01 XX

01 ZZ 01 YY

Projective Plane, con’t

To further understand the projective plane, consider the following illustration:

The projective plane can be described as all of the lines in R3 that pass through the origin. Further, we can say that lines that intersect the plane P shown above represent “real” points, and lines that do not intersect the plane represent points at “infinity.” These terms will be defined in more depth momentarily.

Homogenization

Let k be a field, a polynomial of degree d, and Cf the curve associated to f (f(x,y)=0). The projective closure of the curve Cf is:

],[),( yxkyxf

}0),,(|)::{(:ˆ000

2000 ZYXFPZYXC f

],,[),(:),,( ZYXkZ

Y

Z

XZZYXF d

Where the homogenization of f is:

Example of Homogenization

Consider the curve . If we take the Homogenization, we get:

1),(

323

Z

X

Z

X

Z

YZyxf

3232 ZXZXZY

1),( 32 xxyyxf

Now we are ready to solve the Diophantine Equation.

Now we can find all solutions to a Diophantine Equation, including solutions that occur at “infinity.”

Remember the Diophantine Equation?

Any point in the homogenization that is of the form with is called a point at infinity.

)::( 000 ZYX 00 Z

All other points are called affine points.

Bezout’s Theorem

If are polynomials of degrees d and e respectively, then and intersect in at most de points. Further,

and intersect in exactly de points of

, when points are counted with multiplicity. This is used in a classical proof of algebraic geometry, but we will not go into that at this time.

fC gC],[, yxkgf

fC

gC

)(2 kP

Points, Divisors and Rational Functions

By Tyler Stelzer and

Bob Coulson

Frobenius Maps

Suppose Fq is a finite field (recall this means that q must be a prime power) and that n >= 1. The Frobenius Automorphism is the map

σq,n : Fqn Fq

n

defined by

σq,n(α) = αq, for any α Fqn

Relative and Absolute Frobenius

If q = pr where p is prime and r >= 2, then

• the map σq,n is often called the relative Frobenius

• the function σp,n if often called the absolute Frobenius

Composing Frobenius with Itself

The symbol σjq,n represents the map obtained by

composing σq,n with itself j times.

For example:

σ2q,n (α) = σq,n (σq,n(α)).

Nonsingularity

When constructing a code, one of the elements needed is a nonsingular projective plane curve. A projective plane curve, Cf,

is nonsingular when no singular points exist on it.

Singular Points

A singular point of Cf is a point (x0,y0) such

that f (x0,y0) = 0 and fx(x0,y0) = 0 and fy(x0,y0) = 0.Or if F(X,Y,Z) is the homogenization of f(x,y), then (X0:Y0:Z0) is a singular point of Cf if the point is on the curve and:

F (X0:Y0:Z0) = Fx (X0:Y0:Z0)

= Fy (X0:Y0:Z0)

= Fz (X0:Y0:Z0)

= 0.

k x k

What are f(x,y), fx(x,y) f y(x,y)?

What is f(x,y)? A singular point of Cf is a point (x0,y0)

such that f(x0,y0) = 0 and fx(x0,y0) = 0 and fy(x0,y0) = 0.

f(x,y) k[x,y] where k[x,y] is the set of polynomials having coefficients from k and two variables, x and y.

Example: Let k = Z5 = {0,1,2,3,4}. Then one possible polynomial is:

f(x,y) = 2x2y + xy3 + x2 + 2y

k x k

What are f(x,y), fx(x,y) f y(x,y)?

What is fx(x,y)? A singular point of Cf is a point (x0,y0)


fx(x,y) is the partial derivative of f(x,y) with respect to x.

Example: Let f(x,y) = 2x2y + xy3 + x2 + 2y

Then fx(x,y) = 4xy + y3 + 2x.

k x k

What are K, f(x,y), fx(x,y) f y(x,y)?

What is f y(x,y)? A singular point of Cf is a point (x0,y0)


fy(x,y) is the partial derivative of f(x,y) with respect to y.

Example: Let f(x,y) = 2x2y + xy3 + x2 + 2y

Then fy(x,y) = 2x2 + 3xy2 + 2.

k x k

Genus and the Plϋcker Formula

A nonsingular curve can be realized as a torus-like object with one or more holes in R3. This torus has a certain number of holes which is called the topological genus (g). The genus is given by the formula g = (d-1)(d-2)/2 where d is the degree of the polynomial which makes the curve nonsingular. This formula is called the Plϋcker Formula.

Points, Functions, andDivisors on Curves

Let k be a field, and let C be the projective plane curve defined by F = 0, where F = F(X,Y,Z) k[X,Y,Z] is a homogenous polynomial. For any field K containing k, we define a K-rational point on C to be a point (X0 : Y0 : Z0) P2(K) such that F(X0,Y0,Z0) = 0. The set of all K-rational points on C is denoted C(K). Elements of C(k) are called points of degree one or simply rational points.

Points, Functions, and Divisors on Curves

Let C be a nonsingular projective plane curve. A point of degree n on C over Fq is a set P = {P0,…,Pn-1} of n distinct points in C(Fq

n) such that Pi =

σiq,n(P0) for i = 1,…, n-1.


A divisor D on X, a nonsingular projective plane curve, over Fq is an element of the free abelian group on the set of points on X over Fq. Every divisor is of the form where the nQ are integers and each Q is a point on X. If nQ 0 for all Q, D is effective and D 0. The degree of . The support of D is suppD = {Q | nQ 0}.

QnD q

QnD q deg


Let F(X,Y,Z) be the polynomial which defines the nonsingular projective plane cure C over the field Fq. The field of rational functions on C is

where g/h ~ g`/ h` if and only if gh` - g`h ‹F› Fq[X,Y,Z].

~ / {0} | ),,(

),,(:)(

ZYXh

ZYXgCFq

g,h Fq[X,Y,Z]

are homogeneous of the same degree


Let C be a curve defined over Fq and let f := g/h Fq(C). The divisor of f is defined to be div(f) := Σ P – Σ Q, where Σ P is the intersection divisor C Cg and Σ Q is the intersection divisor C Ch.


Let D be a divisor on the nonsingular projective plane curve C defined over the field Fq. Then the space of rational functions associated to D is

L(D) := {f Fq(C) | div(f) + D >= 0} {0}.

Riemann-Roch Theorem

Let C be a nonsingular projective plane curve of genus g defined over the field Fq and let D be a divisor on X. Then

dim L(D) >= deg D + 1 – g.

Further, if deg D > 2g – 2, then

dim L(D) = deg D + 1 – g.

Algebraic Geometric Reed-Solomon Codes

Goppa’s Construction

Goppa’s construction is extended from the Reed-Solomon Code formula:

RS(k,q) := { ( f( 1), f( 2),…,f( q-1) )| f Lk-1 }

Goppa’s idea behind this construction was to generalize the formula. This new formula is:

C(X,P,D) := {( f (P1) , … ,(Pn) ) | f L(D) }

Definitions for: C(X,P,D) := {( f (P1) , … ,(Pn) ) | f L(D) }

Definition of X:

X is a projective nonsingular plane curve over Fq, a finite field with q number of elements.


Definition of P:

P = {P1,…,Pn} X(Fq)

i.e. P is the set of n distinct Fq-rational points on X.


Definition of D:

D is a divisor on X.


Definition of L(D):

L(D) is the space of rational functions associated with the defined divisor D over X.

Goppa’s Construction - History

In 1981, Goppa derived a class of linear codes from algebraic curves over finite fields which

are quite general as codes

have parameters circumscribed by the Riemann-Roch theorem

History (cont…)

The discovery of these codes also gave renewed stimulus to investigations on the number of rational points on an algebraic curve for a particular genus as well as to asymptotic values of the ratio of the number of points to the genus.

Goppa – The Problem

The three most important parameters of a linear code over the finite field are

the length n which gives the speed of transmission

the dimension k which gives the number of words in the code

and the minimum distance d which gives the number of errors that can be corrected.

“Good” Codes

“Good” codes have the following properties:a large information rate R = k / nand a large relative distance δ = d / n

The relation between R and δ as n gets large is given by the Gilbert-Varshamov bound. Good codes can be constructed from an algebraic curve of genus g, and particular examples show that the G-V bound is not best possible. This brings to the fore the problem of determining the limit β of n / g for a sequence of curves.

In the past two years, the goal of finding explicit codes which reach the limits predicted by early coding mathematicians has been achieved. The constructions require techniques from a surprisingly wide range of pure mathematics: linear algebra, the theory of fields and algebraic geometry all play a vital role. Not only has coding theory helped to solve problems of vital importance in the world outside mathematics, it has enriched other branches of mathematics, with new problems as well as new solutions.

Final Thoughts

References

Codes and Curves by Judy L. Walker

- American Mathematical Society, 2000

Coding theory: the first 50 years - http://pass.maths.org.uk/issue3/codes/

Documents

Algebraic Geometric Coding Theory presented by Jake Hustad John Hanson Berit Rollay Nick Bremer Tyler Stelzer Robert Coulson