Algebraic Geometric
Coding Theory
presented by
Jake 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 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:
A code C over an alphabet
is
simply a subset of
An = A x…x A(n copies).
A
A
Where
is usually a finite field.
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”
v = “110”
(Transmitted Code)
(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.
The Minimum Distance
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.
The Minimum Distance
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 Fqq 1:= FqxFqx…xFq
q-1 copies
so this is a code over the alphabet Fq
Summary
Through our research and presentation last
semester, we proved that for Reed-Solomon
codes:
n=q–1
k = 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 nonconstant polynomial in k[x] has at least one root.
Ex) f(x)  x  1
2
x  i
This is not closed under R because i  R, but
it is closed under C.
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 k the algebraic closure of k.
Ex) R  C
Theorem
Let k be an algebraically closed field and let
f ( x)  k[ x] be a polynomial of degree n. Then
there exists a non-zero u and 1 ,..., n  k
(not necessarily distinct) such that:
f(x)  u(x  1 )...( x   n )
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
2
2
x  2 y  1. A useful problem to solve is
how many rational solutions does this
equation have?
In order to answer this question, we need to
define points at “infinity.”
Projective Plane
2
Let k be a field. The projective plane P (k )
is defined as:
P 2 (k ) : (k 3 \ {( 0,0,0)}) / ~
where ( X 0 , Y0 , Z 0 ) ~ ( X 1 , Y1 , Z1 ) if and only if
there is some non-zero  with X 1  X 0 ,
Y1  Y0 and Z1  Z 0 .
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, f ( x, y)  k[ x, y] a
polynomial of degree d, and Cf the curve
associated to f (f(x,y)=0). The projective
closure of the curve Cf is:
Cˆ f : {( X 0 : Y0 : Z 0 )  P 2 | F ( X 0 , Y0 , Z 0 )  0}
Where the homogenization of f is:
X Y
F ( X , Y , Z ) : Z ( , )  k[ X , Y , Z ]
Z Z
d
Example of Homogenization
Consider the curve f ( x, y)  y  x  x  1. If
we take the Homogenization, we get:
2
3

X X 
3 Y 
f ( x, y )  Z          1
 Z   Z   Z  
2
 Y Z  X  XZ  Z
2
3
2
3
3
Now we are ready to solve the Diophantine
Equation.
Remember the Diophantine
Equation?
Now we can find all solutions to a
Diophantine Equation, including solutions
that occur at “infinity.”
Any point in the homogenization that is
of the form ( X 0 : Y0 : Z 0 ) with Z 0  0 is
called a point at infinity.
All other points are called affine points.
Bezout’s Theorem
If f , g  k[ x, y ] are polynomials of degrees
d and e respectively, then C f and C g
intersect in at most de points. Further, Ĉ f
and Ĉ g intersect in exactly de points of
2
P (k ) , 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.
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  Fqn
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)  k x k
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.
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)  k x k
such that f(x0,y0) = 0 and fx(x0,y0) = 0 and fy(x0,y0) = 0.
 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
 f(x,y)
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)  k x k
such that f(x0,y0) = 0 and fx(x0,y0) = 0 and fy(x0,y0) = 0.
 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.
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)  k x k
such that f(x0,y0) = 0 and fx(x0,y0) = 0 and fy(x0,y0) = 0.
 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.
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, and
Divisors 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(Fqn) such that
Pi = σiq,n(P0) for i = 1,…, n-1.
Points, Functions, and
Divisors on Curves

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 D   nqQ 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 D   nq degQ . The support of
D is suppD = {Q | nQ  0}.
Points, Functions, and
Divisors on Curves
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
  g ( X , Y , Z ) g,h  F [X,Y,Z] 

Fq (C ) :  
| are homogeneous   {0}  / ~
of the same degree
h
(
X
,
Y
,
Z
)




q
where g/h ~ g`/ h` if and only if gh` - g`h ‹F› 
Fq[X,Y,Z].
Points, Functions, and
Divisors on Curves
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.


Points, Functions, and
Divisors on Curves
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 ReedSolomon 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.
Definitions for:
C(X,P,D) := {( f (P1) , … ,(Pn) ) | f  L(D) }
Definition of P:
P = {P1,…,Pn}  X(Fq)
i.e. P is the set of n distinct Fq-rational
points on X.
Definitions for:
C(X,P,D) := {( f (P1) , … ,(Pn) ) | f  L(D) }
Definition of D:
D is a divisor on X.
Definitions for:
C(X,P,D) := {( f (P1) , … ,(Pn) ) | f  L(D) }
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 / n
and 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.
Final Thoughts
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.
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/