AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Algebraic Geometry Codes

Shelly Manber

December 2, 2011

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

References

Main Source: Stichtenoth, Henning. “Algebraic FunctionFields and Codes”. Springer, 2009.

Other Sources:

Høholdt, Lint and Pellikaan. “Algebraic geometry codes.”Handbook of Coding Theory, vol 1, p 871-961,Amsterdam, 2011.

Bartley and Walker. “Algebraic Geometric Codes overRings.” World Scientific Review, June 2008.

Guruswami,Venkatesan. “Notes 2: Gilbert-Varshamovbound.” Jan 2010.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Error Correcting Codes

Natural way of sending information is a stream of bits:011100010100100011...

If one bit is off, the entire message may be 100% different

Error Correcting Codes: a way to send information along anoisy channel so that the original message can berecovered with high probability

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Error Correcting Codes

Natural way of sending information is a stream of bits:011100010100100011...

If one bit is off, the entire message may be 100% different

Error Correcting Codes: a way to send information along anoisy channel so that the original message can berecovered with high probability

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Error Correcting Codes

Natural way of sending information is a stream of bits:011100010100100011...

If one bit is off, the entire message may be 100% different

Error Correcting Codes: a way to send information along anoisy channel so that the original message can berecovered with high probability

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Error Correcting Codes

Natural way of sending information is a stream of bits:011100010100100011...

If one bit is off, the entire message may be 100% different

Error Correcting Codes: a way to send information along anoisy channel so that the original message can berecovered with high probability

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Definitions

Definition: A linear code C is a linear subpace of Fnq.

Definition: The dimension k of a linear code is its dimensionas a vector space

Definition: The Hamming distance between two elements ofFn

q is the number of coefficients on which the two elementsdiffer. The weight of an element, wt(e), is the number ofnonzero coefficients.

Definition: The minimum distance of a code is the minimumHamming distance between any two elements of the linear code(equivalently the minimum weight of a codeword)

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Definitions

Definition: A linear code C is a linear subpace of Fnq.

Definition: The dimension k of a linear code is its dimensionas a vector space

Definition: The Hamming distance between two elements ofFn

q is the number of coefficients on which the two elementsdiffer. The weight of an element, wt(e), is the number ofnonzero coefficients.

Definition: The minimum distance of a code is the minimumHamming distance between any two elements of the linear code(equivalently the minimum weight of a codeword)

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Definitions

Definition: A linear code C is a linear subpace of Fnq.

Definition: The dimension k of a linear code is its dimensionas a vector space

Definition: The Hamming distance between two elements ofFn

q is the number of coefficients on which the two elementsdiffer. The weight of an element, wt(e), is the number ofnonzero coefficients.

Definition: The minimum distance of a code is the minimumHamming distance between any two elements of the linear code(equivalently the minimum weight of a codeword)

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Definitions

Definition: A linear code C is a linear subpace of Fnq.

Definition: The dimension k of a linear code is its dimensionas a vector space

Definition: The Hamming distance between two elements ofFn

q is the number of coefficients on which the two elementsdiffer. The weight of an element, wt(e), is the number ofnonzero coefficients.

Definition: The minimum distance of a code is the minimumHamming distance between any two elements of the linear code(equivalently the minimum weight of a codeword)

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Encoding

An encoding is a map from a message into the code C

A generator matrix is a k × n matrix whose rows are abasis for C

A generator matrix gives an encoding:

(x1 x2 x3

) a1 a2 a3 a4 a5

b1 b2 b3 b4 b5

c1 c2 c3 c4 c5

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Encoding

An encoding is a map from a message into the code C

A generator matrix is a k × n matrix whose rows are abasis for C

A generator matrix gives an encoding:

(x1 x2 x3

) a1 a2 a3 a4 a5

b1 b2 b3 b4 b5

c1 c2 c3 c4 c5

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Encoding

An encoding is a map from a message into the code C

A generator matrix is a k × n matrix whose rows are abasis for C

A generator matrix gives an encoding:

(x1 x2 x3

) a1 a2 a3 a4 a5

b1 b2 b3 b4 b5

c1 c2 c3 c4 c5

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Error Checking

Definition: The dual C⊥ of a code C is its dual as a vectorspace under the canonical inner product.

Definition: A parity check matrix M for C is an (n − k)× nmatrix whose rows are a basis for C⊥

Claim: A vector x := (x1, . . . , xn) ∈ Fnq is in C if and only if

Mx = 0.

Proof.

(C⊥)⊥ = C , so

x ∈ C ⇔ 〈x , c〉 = 0 ∀c ∈ C⊥ ⇔ Mx = 0

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Error Checking

Definition: The dual C⊥ of a code C is its dual as a vectorspace under the canonical inner product.

Definition: A parity check matrix M for C is an (n − k)× nmatrix whose rows are a basis for C⊥

Claim: A vector x := (x1, . . . , xn) ∈ Fnq is in C if and only if

Mx = 0.

Proof.

(C⊥)⊥ = C , so

x ∈ C ⇔ 〈x , c〉 = 0 ∀c ∈ C⊥ ⇔ Mx = 0

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Error Checking

Definition: The dual C⊥ of a code C is its dual as a vectorspace under the canonical inner product.

Definition: A parity check matrix M for C is an (n − k)× nmatrix whose rows are a basis for C⊥

Claim: A vector x := (x1, . . . , xn) ∈ Fnq is in C if and only if

Mx = 0.

Proof.

(C⊥)⊥ = C , so

x ∈ C ⇔ 〈x , c〉 = 0 ∀c ∈ C⊥ ⇔ Mx = 0

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Error Checking

Definition: The dual C⊥ of a code C is its dual as a vectorspace under the canonical inner product.

Definition: A parity check matrix M for C is an (n − k)× nmatrix whose rows are a basis for C⊥

Claim: A vector x := (x1, . . . , xn) ∈ Fnq is in C if and only if

Mx = 0.

Proof.

(C⊥)⊥ = C , so

x ∈ C ⇔ 〈x , c〉 = 0 ∀c ∈ C⊥ ⇔ Mx = 0

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Error Checking

Definition: The dual C⊥ of a code C is its dual as a vectorspace under the canonical inner product.

Definition: A parity check matrix M for C is an (n − k)× nmatrix whose rows are a basis for C⊥

Claim: A vector x := (x1, . . . , xn) ∈ Fnq is in C if and only if

Mx = 0.

Proof.

(C⊥)⊥ = C , so

x ∈ C ⇔ 〈x , c〉 = 0 ∀c ∈ C⊥

⇔ Mx = 0

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Error Checking

Definition: The dual C⊥ of a code C is its dual as a vectorspace under the canonical inner product.

Definition: A parity check matrix M for C is an (n − k)× nmatrix whose rows are a basis for C⊥

Claim: A vector x := (x1, . . . , xn) ∈ Fnq is in C if and only if

Mx = 0.

Proof.

(C⊥)⊥ = C , so

x ∈ C ⇔ 〈x , c〉 = 0 ∀c ∈ C⊥ ⇔ Mx = 0

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Decoding

Let C be an [n, k , d ] code. If x ∈ Fnq has Hamming distance

≤ (d − 1)/2 from a codeword c ∈ C then c is the uniquecodeword with minimal distance to x .

Definition: A decoding is an algorithm, given a ∈ Fnq and the

guarantee that a = c + e for some c ∈ C and e ∈ Fnq with

weight ≤ (d − 1)/2, to recover c .

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Decoding

Let C be an [n, k , d ] code. If x ∈ Fnq has Hamming distance

≤ (d − 1)/2 from a codeword c ∈ C then c is the uniquecodeword with minimal distance to x .

Definition: A decoding is an algorithm, given a ∈ Fnq and the

guarantee that a = c + e for some c ∈ C and e ∈ Fnq with

weight ≤ (d − 1)/2, to recover c .

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

To maximize code efficacy:

Should d be higher or lower with respect to n?

Should k be higher or lower with respect to n?

Theorem (Singleton Bound): for all linear codes

k + d ≤ n + 1

Proof.

LetV := (a1, ..., an) ∈ Fn

q | ai = 0 ∀i ≥ d

So dim(V ) = d − 1 and V ∩ C = ∅, so

n = dim(Fnq) ≥ dim(V + C ) = dim(V ) + dim(C )

= (d − 1) + k

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

To maximize code efficacy:

Should d be higher or lower with respect to n?Should k be higher or lower with respect to n?

Theorem (Singleton Bound): for all linear codes

k + d ≤ n + 1

Proof.

LetV := (a1, ..., an) ∈ Fn

q | ai = 0 ∀i ≥ d

So dim(V ) = d − 1 and V ∩ C = ∅, so

n = dim(Fnq) ≥ dim(V + C ) = dim(V ) + dim(C )

= (d − 1) + k

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

To maximize code efficacy:

Should d be higher or lower with respect to n?Should k be higher or lower with respect to n?

Theorem (Singleton Bound): for all linear codes

k + d ≤ n + 1

Proof.

LetV := (a1, ..., an) ∈ Fn

q | ai = 0 ∀i ≥ d

So dim(V ) = d − 1 and V ∩ C = ∅, so

n = dim(Fnq) ≥ dim(V + C ) = dim(V ) + dim(C )

= (d − 1) + k

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

To maximize code efficacy:

Should d be higher or lower with respect to n?Should k be higher or lower with respect to n?

Theorem (Singleton Bound): for all linear codes

k + d ≤ n + 1

Proof.

LetV := (a1, ..., an) ∈ Fn

q | ai = 0 ∀i ≥ d

So dim(V ) = d − 1 and V ∩ C = ∅, so

n = dim(Fnq) ≥ dim(V + C ) = dim(V ) + dim(C )

= (d − 1) + k

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

To maximize code efficacy:

Should d be higher or lower with respect to n?Should k be higher or lower with respect to n?

Theorem (Singleton Bound): for all linear codes

k + d ≤ n + 1

Proof.

LetV := (a1, ..., an) ∈ Fn

q | ai = 0 ∀i ≥ d

So dim(V ) = d − 1 and V ∩ C = ∅, so

n = dim(Fnq) ≥ dim(V + C ) = dim(V ) + dim(C )

= (d − 1) + k

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

To maximize code efficacy:

Should d be higher or lower with respect to n?Should k be higher or lower with respect to n?

Theorem (Singleton Bound): for all linear codes

k + d ≤ n + 1

Proof.

LetV := (a1, ..., an) ∈ Fn

q | ai = 0 ∀i ≥ d

So dim(V ) = d − 1 and V ∩ C = ∅, so

n = dim(Fnq) ≥ dim(V + C ) = dim(V ) + dim(C )

= (d − 1) + k

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

To maximize code efficacy:

Should d be higher or lower with respect to n?Should k be higher or lower with respect to n?

Theorem (Singleton Bound): for all linear codes

k + d ≤ n + 1

Proof.

LetV := (a1, ..., an) ∈ Fn

q | ai = 0 ∀i ≥ d

So dim(V ) = d − 1 and V ∩ C = ∅, so

n = dim(Fnq) ≥ dim(V + C ) = dim(V ) + dim(C )

= (d − 1) + k

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Example: Reed-Solomon Codes

Let n = q − 1 and Fq = 0, 1, α, . . . , αn−1. Choose k ≤ n,and define

Lk = f ∈ Fq[X ] | deg f ≤ k − 1

Define

CRS = (f (1), f (α), . . . , f (αn−1) | f ∈ Lk

Claim: CRS is an [n, k , n − k + 1] code.

Proof.

1 Lk → CRS is an injective Fq-linear map

2 A polynomial of degree k − 1 has at most k − 1 zeros, sod ≥ n − k + 1

3 By the Singleton bound, d ≤ n − k + 1

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Example: Reed-Solomon Codes

Let n = q − 1 and Fq = 0, 1, α, . . . , αn−1. Choose k ≤ n,and define

Lk = f ∈ Fq[X ] | deg f ≤ k − 1

Define

CRS = (f (1), f (α), . . . , f (αn−1) | f ∈ Lk

Claim: CRS is an [n, k , n − k + 1] code.

Proof.

1 Lk → CRS is an injective Fq-linear map

2 A polynomial of degree k − 1 has at most k − 1 zeros, sod ≥ n − k + 1

3 By the Singleton bound, d ≤ n − k + 1

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Example: Reed-Solomon Codes

Let n = q − 1 and Fq = 0, 1, α, . . . , αn−1. Choose k ≤ n,and define

Lk = f ∈ Fq[X ] | deg f ≤ k − 1

Define

CRS = (f (1), f (α), . . . , f (αn−1) | f ∈ Lk

Claim: CRS is an [n, k , n − k + 1] code.

Proof.

1 Lk → CRS is an injective Fq-linear map

2 A polynomial of degree k − 1 has at most k − 1 zeros, sod ≥ n − k + 1

3 By the Singleton bound, d ≤ n − k + 1

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Example: Reed-Solomon Codes

Let n = q − 1 and Fq = 0, 1, α, . . . , αn−1. Choose k ≤ n,and define

Lk = f ∈ Fq[X ] | deg f ≤ k − 1

Define

CRS = (f (1), f (α), . . . , f (αn−1) | f ∈ Lk

Claim: CRS is an [n, k , n − k + 1] code.

Proof.

1 Lk → CRS is an injective Fq-linear map

2 A polynomial of degree k − 1 has at most k − 1 zeros, sod ≥ n − k + 1

3 By the Singleton bound, d ≤ n − k + 1

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Example: Reed-Solomon Codes

Let n = q − 1 and Fq = 0, 1, α, . . . , αn−1. Choose k ≤ n,and define

Lk = f ∈ Fq[X ] | deg f ≤ k − 1

Define

CRS = (f (1), f (α), . . . , f (αn−1) | f ∈ Lk

Claim: CRS is an [n, k , n − k + 1] code.

Proof.

1 Lk → CRS is an injective Fq-linear map

2 A polynomial of degree k − 1 has at most k − 1 zeros, sod ≥ n − k + 1

3 By the Singleton bound, d ≤ n − k + 1

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Definition

Given:

X a curve of genus g over Fq with function field F

P1, . . . ,Pn distinct places of F of degree one

D := P1 + · · ·+ Pn a divisor of X

G a divisor of X such that Supp G ∩ Supp D = ∅Define

CL (D,G ) := (x(P1), . . . , x(Pn)) | x ∈ L (G ) ⊂ Fnq

Claim: k = `(G )− `(G − D) and d ≥ n − deg G .

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Definition

Given:

X a curve of genus g over Fq with function field F

P1, . . . ,Pn distinct places of F of degree one

D := P1 + · · ·+ Pn a divisor of X

G a divisor of X such that Supp G ∩ Supp D = ∅Define

CL (D,G ) := (x(P1), . . . , x(Pn)) | x ∈ L (G ) ⊂ Fnq

Claim: k = `(G )− `(G − D) and d ≥ n − deg G .

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Definition

Given:

X a curve of genus g over Fq with function field F

P1, . . . ,Pn distinct places of F of degree one

D := P1 + · · ·+ Pn a divisor of X

G a divisor of X such that Supp G ∩ Supp D = ∅Define

CL (D,G ) := (x(P1), . . . , x(Pn)) | x ∈ L (G ) ⊂ Fnq

Claim: k = `(G )− `(G − D) and d ≥ n − deg G .

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Definition

Given:

X a curve of genus g over Fq with function field F

P1, . . . ,Pn distinct places of F of degree one

D := P1 + · · ·+ Pn a divisor of X

G a divisor of X such that Supp G ∩ Supp D = ∅Define

CL (D,G ) := (x(P1), . . . , x(Pn)) | x ∈ L (G ) ⊂ Fnq

Claim: k = `(G )− `(G − D) and d ≥ n − deg G .

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Definition

Given:

X a curve of genus g over Fq with function field F

P1, . . . ,Pn distinct places of F of degree one

D := P1 + · · ·+ Pn a divisor of X

G a divisor of X such that Supp G ∩ Supp D = ∅Define

CL (D,G ) := (x(P1), . . . , x(Pn)) | x ∈ L (G ) ⊂ Fnq

Claim: k = `(G )− `(G − D) and d ≥ n − deg G .

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Some Nice Properties

If deg G < n then

1 k = `(G ) ≥ deg(G ) + 1− g (Riemann-Roch)

2 L (G )→ CL (D,G ) is injective

3 For a basis x1, . . . , xk of L (G ), the matrix:x1(P1) x1(P2) . . . x1(Pn)x2(P1) x2(P2) . . . x2(Pn). . . . . . . . . . . .

xn(P1) xn(P2) . . . xn(Pn)

is a generator matrix for CL .

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Some Nice Properties

If deg G < n then

1 k = `(G ) ≥ deg(G ) + 1− g (Riemann-Roch)

2 L (G )→ CL (D,G ) is injective

3 For a basis x1, . . . , xk of L (G ), the matrix:x1(P1) x1(P2) . . . x1(Pn)x2(P1) x2(P2) . . . x2(Pn). . . . . . . . . . . .

xn(P1) xn(P2) . . . xn(Pn)

is a generator matrix for CL .

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Some Nice Properties

If deg G < n then

1 k = `(G ) ≥ deg(G ) + 1− g (Riemann-Roch)

2 L (G )→ CL (D,G ) is injective

3 For a basis x1, . . . , xk of L (G ), the matrix:x1(P1) x1(P2) . . . x1(Pn)x2(P1) x2(P2) . . . x2(Pn). . . . . . . . . . . .

xn(P1) xn(P2) . . . xn(Pn)

is a generator matrix for CL .

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

If deg G < n then k ≥ deg(G ) + 1− g and d ≥ n − deg G so

k + d ≥ n + 1− g

But recall thatk + d ≤ n + 1

So for genus 0 curves,

k + d = n + 1

Unfortunately, for genus zero curves, n ≤ q + 1, so over a fixedalphabet, we can’t send very large messages.

Goal of coding theory: To construct asymptotically goodcurves

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

If deg G < n then k ≥ deg(G ) + 1− g and d ≥ n − deg G so

k + d ≥ n + 1− g

But recall thatk + d ≤ n + 1

So for genus 0 curves,

k + d = n + 1

Unfortunately, for genus zero curves, n ≤ q + 1, so over a fixedalphabet, we can’t send very large messages.

Goal of coding theory: To construct asymptotically goodcurves

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

If deg G < n then k ≥ deg(G ) + 1− g and d ≥ n − deg G so

k + d ≥ n + 1− g

But recall thatk + d ≤ n + 1

So for genus 0 curves,

k + d = n + 1

Unfortunately, for genus zero curves, n ≤ q + 1, so over a fixedalphabet, we can’t send very large messages.

Goal of coding theory: To construct asymptotically goodcurves

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

If deg G < n then k ≥ deg(G ) + 1− g and d ≥ n − deg G so

k + d ≥ n + 1− g

But recall thatk + d ≤ n + 1

So for genus 0 curves,

k + d = n + 1

Unfortunately, for genus zero curves, n ≤ q + 1, so over a fixedalphabet, we can’t send very large messages.

Goal of coding theory: To construct asymptotically goodcurves

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

If deg G < n then k ≥ deg(G ) + 1− g and d ≥ n − deg G so

k + d ≥ n + 1− g

But recall thatk + d ≤ n + 1

So for genus 0 curves,

k + d = n + 1

Unfortunately, for genus zero curves, n ≤ q + 1, so over a fixedalphabet, we can’t send very large messages.

Goal of coding theory: To construct asymptotically goodcurves

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

If deg G < n then k ≥ deg(G ) + 1− g and d ≥ n − deg G so

k + d ≥ n + 1− g

But recall thatk + d ≤ n + 1

So for genus 0 curves,

k + d = n + 1

Unfortunately, for genus zero curves, n ≤ q + 1, so over a fixedalphabet, we can’t send very large messages.

Goal of coding theory: To construct asymptotically goodcurves

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Asymptotically Good Curves

Definition: For an [n, k , d ] code:

R = k/n is called the information rate, and

δ = d/n is called the relative distance

Theorem (Gilbert-Varshamov bound): For any fixed q andδ ≤ 1− 1/q, and an arbitrarily small ε > 0 there exists aninfinite family of codes with

R ≥ 1− hq(δ)− ε

where hq(x) is the entropy function:

hq(x) := xlogq(q − 1)− xlogq(x)− (1− x)logq(1− x)

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Asymptotically Good Curves

Definition: For an [n, k , d ] code:

R = k/n is called the information rate, and

δ = d/n is called the relative distance

Theorem (Gilbert-Varshamov bound): For any fixed q andδ ≤ 1− 1/q, and an arbitrarily small ε > 0 there exists aninfinite family of codes with

R ≥ 1− hq(δ)− ε

where hq(x) is the entropy function:

hq(x) := xlogq(q − 1)− xlogq(x)− (1− x)logq(1− x)

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Residues of Differentials

Let P be a place of F with local parameter t.

Claim: Any x ∈ F can be written uniquely as∑∞

i=m ai ti with

ai ∈ Fq for some integer m.

Definition: For any differential ω of F , write

ω = f dt

f =∞∑

i=m

ai ti

Define resP(ω) := a−1

Claim: Residue is well-defined.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Residues of Differentials

Let P be a place of F with local parameter t.

Claim: Any x ∈ F can be written uniquely as∑∞

i=m ai ti with

ai ∈ Fq for some integer m.

Definition: For any differential ω of F , write

ω = f dt

f =∞∑

i=m

ai ti

Define resP(ω) := a−1

Claim: Residue is well-defined.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Residues of Differentials

Let P be a place of F with local parameter t.

Claim: Any x ∈ F can be written uniquely as∑∞

i=m ai ti with

ai ∈ Fq for some integer m.

Definition: For any differential ω of F , write

ω = f dt

f =∞∑

i=m

ai ti

Define resP(ω) := a−1

Claim: Residue is well-defined.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Residues of Differentials

Let P be a place of F with local parameter t.

Claim: Any x ∈ F can be written uniquely as∑∞

i=m ai ti with

ai ∈ Fq for some integer m.

Definition: For any differential ω of F , write

ω = f dt

f =∞∑

i=m

ai ti

Define resP(ω) := a−1

Claim: Residue is well-defined.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Another Algebraic Geometry Code

Given:

P1, . . . ,Pn distinct places of F of degree one

D = P1 + · · ·+ Pn a divisor of X

G a divisor of X such that Supp G ∩ Supp D = ∅as before, define

CΩ(D,G ) := (resP1(ω), . . . , resPn(ω)) | ω ∈ ΩF (G − D)

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Duals

Proposition: CΩ(D,G ) = CL (D,G )⊥

Proposition: There exists a Weil Differential η which can beexplicitly computed such that

CΩ(D,G )⊥ = CL (D,G ) = CΩ(D,D − G + (η))

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Duals

Proposition: CΩ(D,G ) = CL (D,G )⊥

Proposition: There exists a Weil Differential η which can beexplicitly computed such that

CΩ(D,G )⊥ = CL (D,G ) = CΩ(D,D − G + (η))

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Hermitian Codes

Let

F := Fq2(x , y) with yq + y = xq+1

Pα,β be the unique place such that x(Pα,β) = α andy(Pα,β) = β and P∞ be the common pole of x and y .

D =∑

βq+β=αq+1 Pα,β

For each 0 < r < q3 + q2 − q − 2 define

Cr := CL (D, rP∞)

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Hermitian Codes

Let

F := Fq2(x , y) with yq + y = xq+1

Pα,β be the unique place such that x(Pα,β) = α andy(Pα,β) = β and P∞ be the common pole of x and y .

D =∑

βq+β=αq+1 Pα,β

For each 0 < r < q3 + q2 − q − 2 define

Cr := CL (D, rP∞)

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Hermitian Codes

Let

F := Fq2(x , y) with yq + y = xq+1

Pα,β be the unique place such that x(Pα,β) = α andy(Pα,β) = β and P∞ be the common pole of x and y .

D =∑

βq+β=αq+1 Pα,β

For each 0 < r < q3 + q2 − q − 2 define

Cr := CL (D, rP∞)

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Generating Hermitian Codes

Proposition: For each r ≥ 0, the elements of the form x iy j

with

0 ≤ i

0 ≤ j ≤ q − 1

iq + j(q + 1) ≤ r

form a basis for L (rP∞)

Corollary: The generating matrix for Cr is the matrix whoserows are (αiβj)βq+β=αq+1 where i and j satisfy the aboveconditions.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Generating Hermitian Codes

Proposition: For each r ≥ 0, the elements of the form x iy j

with

0 ≤ i

0 ≤ j ≤ q − 1

iq + j(q + 1) ≤ r

form a basis for L (rP∞)

Corollary: The generating matrix for Cr is the matrix whoserows are (αiβj)βq+β=αq+1 where i and j satisfy the aboveconditions.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Bounds

Let N(r) be the number of pairs i , j satisfying the propertiespreviously mentioned.

Proposition:

n = q3

k = dim Cr =

N(r) 0 ≤ r < q3

n − N(r) q3 ≤ r ≤ q3 + q2 − q − 2

d ≥ n − r

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Numerical Example: q = 2

Let F4 = 0, 1, a, a + 1, C defined by y2z + yz2 + x3, i.e.F = Frac(Fq[x , y , z ]/(y2z + yz2 + x3))

Rational points:[(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (1 : a : 1), (1 : a + 1 : 1), (a :a : 1), (a : a + 1 : 1), (a + 1 : a : 1), (a + 1 : a + 1 : 1)]

Let r = 5. Then the basis for L (r(0 : 1 : 0)) is

1, x , y , x2, xy

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Numerical Example: q = 2

Let F4 = 0, 1, a, a + 1, C defined by y2z + yz2 + x3, i.e.F = Frac(Fq[x , y , z ]/(y2z + yz2 + x3))

Rational points:[(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (1 : a : 1), (1 : a + 1 : 1), (a :a : 1), (a : a + 1 : 1), (a + 1 : a : 1), (a + 1 : a + 1 : 1)]

Let r = 5. Then the basis for L (r(0 : 1 : 0)) is

1, x , y , x2, xy

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Numerical Example: q = 2

Let F4 = 0, 1, a, a + 1, C defined by y2z + yz2 + x3, i.e.F = Frac(Fq[x , y , z ]/(y2z + yz2 + x3))

Rational points:[(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (1 : a : 1), (1 : a + 1 : 1), (a :a : 1), (a : a + 1 : 1), (a + 1 : a : 1), (a + 1 : a + 1 : 1)]

Let r = 5. Then the basis for L (r(0 : 1 : 0)) is

1, x , y , x2, xy

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Numerical Example: generating matrix

Generating matrix for C :1 1 1 1 1 1 1 11 1 a a + 1 a a + 1 a 00 0 1 1 a a a + 1 00 0 a a + 1 a + 1 1 1 00 0 1 1 a + 1 a + 1 a 0

Sum of third and fifth rows is(0 0 0 0 1 1 1 0

)So d = 3.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Numerical Example: generating matrix

Generating matrix for C :1 1 1 1 1 1 1 11 1 a a + 1 a a + 1 a 00 0 1 1 a a a + 1 00 0 a a + 1 a + 1 1 1 00 0 1 1 a + 1 a + 1 a 0

Sum of third and fifth rows is(

0 0 0 0 1 1 1 0)

So d = 3.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Decoding Overview

Fix t ≥ 0 and let a = c + e with c ∈ CΩ and wt(e) ≤ t

We want an algorithm to compute c given a

Write e = (e1, . . . , en)

Algorithm Steps:

1 Construct f ∈ F such that f (Pi ) = 0 when ei 6= 0.

2 Compute ei for all i such that f (Pi ) = 0.

3 Since ei = 0 for all i such that f (Pi ) 6= 0, we havecomputed e. Then c = a− e.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Decoding Overview

Fix t ≥ 0 and let a = c + e with c ∈ CΩ and wt(e) ≤ t

We want an algorithm to compute c given a

Write e = (e1, . . . , en)

Algorithm Steps:

1 Construct f ∈ F such that f (Pi ) = 0 when ei 6= 0.

2 Compute ei for all i such that f (Pi ) = 0.

3 Since ei = 0 for all i such that f (Pi ) 6= 0, we havecomputed e. Then c = a− e.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Decoding Overview

Fix t ≥ 0 and let a = c + e with c ∈ CΩ and wt(e) ≤ t

We want an algorithm to compute c given a

Write e = (e1, . . . , en)

Algorithm Steps:

1 Construct f ∈ F such that f (Pi ) = 0 when ei 6= 0.

2 Compute ei for all i such that f (Pi ) = 0.

3 Since ei = 0 for all i such that f (Pi ) 6= 0, we havecomputed e. Then c = a− e.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Decoding Overview

Fix t ≥ 0 and let a = c + e with c ∈ CΩ and wt(e) ≤ t

We want an algorithm to compute c given a

Write e = (e1, . . . , en)

Algorithm Steps:

1 Construct f ∈ F such that f (Pi ) = 0 when ei 6= 0.

2 Compute ei for all i such that f (Pi ) = 0.

3 Since ei = 0 for all i such that f (Pi ) 6= 0, we havecomputed e. Then c = a− e.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Decoding Overview

Fix t ≥ 0 and let a = c + e with c ∈ CΩ and wt(e) ≤ t

We want an algorithm to compute c given a

Write e = (e1, . . . , en)

Algorithm Steps:

1 Construct f ∈ F such that f (Pi ) = 0 when ei 6= 0.

2 Compute ei for all i such that f (Pi ) = 0.

3 Since ei = 0 for all i such that f (Pi ) 6= 0, we havecomputed e. Then c = a− e.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Decoding Overview

Fix t ≥ 0 and let a = c + e with c ∈ CΩ and wt(e) ≤ t

We want an algorithm to compute c given a

Write e = (e1, . . . , en)

Algorithm Steps:

1 Construct f ∈ F such that f (Pi ) = 0 when ei 6= 0.

2 Compute ei for all i such that f (Pi ) = 0.

3 Since ei = 0 for all i such that f (Pi ) 6= 0, we havecomputed e. Then c = a− e.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Algorithm Conditions

Let G1 be a divisor such that

Supp G1 ∩ Supp D = ∅deg G1 < deg G − (2g − 2)− t

`(G1) > t

and let

[a, f ] :=n∑

i=1

ai · f (Pi )

Fix bases

f1, . . . , f` of L (G1)

g1, . . . , gk of L (G − G1)

h1, . . . , hm of L (G )

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Algorithm Conditions

Let G1 be a divisor such that

Supp G1 ∩ Supp D = ∅deg G1 < deg G − (2g − 2)− t

`(G1) > t

and let

[a, f ] :=n∑

i=1

ai · f (Pi )

Fix bases

f1, . . . , f` of L (G1)

g1, . . . , gk of L (G − G1)

h1, . . . , hm of L (G )

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Algorithm Conditions

Let G1 be a divisor such that

Supp G1 ∩ Supp D = ∅deg G1 < deg G − (2g − 2)− t

`(G1) > t

and let

[a, f ] :=n∑

i=1

ai · f (Pi )

Fix bases

f1, . . . , f` of L (G1)

g1, . . . , gk of L (G − G1)

h1, . . . , hm of L (G )

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Algorithm Specifics

Proposition: For each 1 ≤ ρ ≤ k, the linear system

∑λ=1

[a, fλgρ] · xλ = 0

has a nontrivial solution (α1, . . . , α`).

Let

f :=∑λ=1

αλfλ

Then f (Pi ) = 0 whenever ei 6= 0

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Algorithm Specifics

Proposition: For each 1 ≤ ρ ≤ k, the linear system

∑λ=1

[a, fλgρ] · xλ = 0

has a nontrivial solution (α1, . . . , α`).

Let

f :=∑λ=1

αλfλ

Then f (Pi ) = 0 whenever ei 6= 0

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Algorithm Specifics

Proposition: For each 1 ≤ µ ≤ m, the linear system∑i | f (Pi )=0

hµ(Pi ) · zi = [a, hµ]

has a unique solution agreeing with e on all i such thatf (Pi ) = 0.

Conclusion: Solving the two linear systems completelycomputes e and hence we can recover c = a− e.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Algorithm Specifics

Proposition: For each 1 ≤ µ ≤ m, the linear system∑i | f (Pi )=0

hµ(Pi ) · zi = [a, hµ]

has a unique solution agreeing with e on all i such thatf (Pi ) = 0.

Conclusion: Solving the two linear systems completelycomputes e and hence we can recover c = a− e.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Decoding Hermitian Codes

Proposition: The dual of Cr is Cq3+q2−q−2−r

Corollary: CL (D, rP∞) = CΩ(D, (q3 + q2 − q − 2− r)P∞)

We can decode CΩ(D, (q3 + q2 − q − 2− r)P∞) with thealgorithm given.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Decoding Hermitian Codes

Proposition: The dual of Cr is Cq3+q2−q−2−r

Corollary: CL (D, rP∞) = CΩ(D, (q3 + q2 − q − 2− r)P∞)

We can decode CΩ(D, (q3 + q2 − q − 2− r)P∞) with thealgorithm given.

AlgebraicGeometry

Codes

Shelly Manber

Linear Codes

AlgebraicGeometryCodes

Example:HermitianCodes

Decoding

Decoding Hermitian Codes

Proposition: The dual of Cr is Cq3+q2−q−2−r

Corollary: CL (D, rP∞) = CΩ(D, (q3 + q2 − q − 2− r)P∞)

We can decode CΩ(D, (q3 + q2 − q − 2− r)P∞) with thealgorithm given.