I Feng–Rao distances in numerical semigroups and...

IBERIAN MEETING ON

“NUMERICAL SEMIGROUPS”

Feng–Rao distances in numerical semigroupsand application to AG codes

Porto 2008

Jose Ignacio Farran Martın

[email protected]

Departamento de Matematica Aplicada

Universidad de Valladolid – Campus de Segovia

Escuela Universitaria de InformaticaFeng–Rao distances– p.1/45

http://wmatem.eis.uva.es/~ignfar/

Contents

• AG codes• Numerical semigroups

Feng–Rao distances– p.2/45

Contents



AG codes


Error-correcting codes

Alphabet A = IFq

Code C ⊆ IFnq

“Size” dim C = k ≤ n

The difference n − k is called redundancy



Alphabet A = IFq

Code C ⊆ IFnq





Alphabet A = IFq

Code C ⊆ IFnq





Alphabet A = IFq

Code C ⊆ IFnq




Encoding

Encoding is an injective (linear) map

C : IFkq ↪→ IFn

q

where C is the image of such a map

It can be described by means of the generator matrixG of C whose rows are a basis of C

Thus the encoding has a matrix expression

c = m · G

where m represents to k “information digits”


Encoding


C : IFkq ↪→ IFn

q




c = m · G



Encoding


C : IFkq ↪→ IFn

q




c = m · G



Errors

encoding decoding

transmitter −→ CHANNEL −→ receiver


Errors

NOISE

↓

encoding decoding

transmitter −→ CHANNEL −→ receiver


Errorstransmitter receiver

↓ ↑

Information

SourceNOISE

Decoded

Information

↓ ↓ ↑

encoding error decoding

↓ ↓ ↑

Encoded

Information−→ CHANNEL −→

Received

Information


Examples of codes

source I II III IV V

0 0000 00000000 000000000000 00000 00011

1 0001 00000011 000000000111 00011 11000

2 0010 00001100 000000111000 00101 10100

3 0011 00001111 000000111111 00110 01100

4 0100 00110000 000111000000 01001 10010

5 0101 00110011 000111000111 01010 01010

6 0110 00111100 000111111000 01100 00110

7 0111 00111111 000111111111 01111 10001

8 1000 11000000 111000000000 10001 01001

9 1001 11000011 111000000111 10010 00101


Examples of codes


0 0000 00000000 000000000000 00000 00011

1 0101 00000011 000000000111 00011 11000

2 0010 00001100 000000111000 00101 10100

3 0011 00001111 000000111111 00110 01100

4 0100 00110000 000111000000 01001 10010

5 0101 00110011 000111000111 01010 01010

6 0110 00111100 000111111000 01100 00110

7 0111 00111111 000111111111 01111 10001

8 1000 11000000 111000000000 10001 01001

9 1001 11000011 111000000111 10010 00101


Examples of codes


0 0000 10000000 000000000000 00000 00011

1 0001 00000011 000000000111 00011 11000

2 0010 00001100 000000111000 00101 10100

3 0011 00001111 000000111111 00110 01100

4 0100 00110000 000111000000 01001 10010

5 0101 00110011 000111000111 01010 01010

6 0110 00111100 000111111000 01100 00110

7 0111 00111111 000111111111 01111 10001

8 1000 11000000 111000000000 10001 01001

9 1001 11000011 111000000111 10010 00101


Examples of codes


0 0000 00000000 000000000000 00000 00011

1 0001 00000011 000000000010 00011 11000

2 0010 00001100 000000111000 00101 10100

3 0011 00001111 000000111111 00110 01100

4 0100 00110000 000111000000 01001 10010

5 0101 00110011 000111000111 01010 01010

6 0110 00111100 000111111000 01100 00110

7 0111 00111111 000111111111 01111 10001

8 1000 11000000 111000000000 10001 01001

9 1001 11000011 111000000111 10010 00101


Examples of codes


0 0000 00000000 000000000000 00000 00011

1 0001 00000011 000000000111 00011 11000

2 0010 00001100 000000101000 00101 10100

3 0011 00001111 000000111111 00110 01100

4 0100 00110000 000111000000 01001 10010

5 0101 00110011 000111000111 01010 01010

6 0110 00111100 000111111000 01100 00110

7 0111 00111111 000111111111 01111 10001

8 1000 11000000 111000000000 10001 01001

9 1001 11000011 111000000111 10010 00101


Examples of codes


0 0000 00000000 000000000000 00000 00011

1 0001 00000011 000000000111 00011 11000

2 0010 00001100 000000111000 00101 10100

3 0011 00001111 000000111111 00110 01100

4 0100 00110000 000111000000 01001 10010

5 0101 00110011 000111000111 01010 01010

6 0110 00111100 000111111000 01100 00110

7 0111 00111111 000111111111 01111 10001

8 1000 11000000 111000000000 10001 01001

9 1001 11000011 111000000111 10010 00101


Examples of codes


0 0000 00000000 000000000000 01000 00011

1 0001 00000011 000000000111 00011 11000

2 0010 00001100 000000111000 00101 10100

3 0011 00001111 000000111111 00110 01100

4 0100 00110000 000111000000 01001 10010

5 0101 00110011 000111000111 01010 01010

6 0110 00111100 000111111000 01100 00110

7 0111 00111111 000111111111 01111 10001

8 1000 11000000 111000000000 10001 01001

9 1001 11000011 111000000111 10010 00101


Examples of codes


0 0000 00000000 000000000000 00000 00011

1 0001 00000011 000000000111 00011 11000

2 0010 00001100 000000111000 00101 10100

3 0011 00001111 000000111111 00110 01100

4 0100 00110000 000111000000 01001 10010

5 0101 00110011 000111000111 01010 01010

6 0110 00111100 000111111000 01100 00110

7 0111 00111111 000111111111 01111 10001

8 1000 11000000 111000000000 10001 01001

9 1001 11000011 111000000111 10010 00101


Hamming distance

The Hamming distance in IFnq is defined by

d(x, y).= ]{i | xi 6= yi}

The minimum distance of C is

d.= d(C)

.= min {d(c, c′) | c, c′ ∈ C, c 6= c′}

The parameters of a code are C ≡ [n, k, d]q

length n

dimension k

minimum distance d


Hamming distance


d(x, y).= ]{i | xi 6= yi}


d.= d(C)

.= min {d(c, c′) | c, c′ ∈ C, c 6= c′}


length n

dimension k

minimum distance d


Hamming distance


d(x, y).= ]{i | xi 6= yi}


d.= d(C)

.= min {d(c, c′) | c, c′ ∈ C, c 6= c′}


length n

dimension k

minimum distance d


Error detection and correction

Let d be the minimum distance of the code C

C detects up to d − 1 errors

C corrects up to bd − 1

2c errors

C corrects up to d − 1 erasures

C corrects any configuration of t errors and s erasures,provided

2t + s ≤ d − 1






2c errors



2t + s ≤ d − 1






2c errors



2t + s ≤ d − 1






2c errors



2t + s ≤ d − 1






2c errors



2t + s ≤ d − 1


Examples

Encode four possible messages {a, b, c, d}

Example 1: n = k = 2

a = 00

b = 01

c = 10

d = 11

d = 1 ⇒ NO error capability


Examples


Example 2: n = 3 (one control digit)

a = 000

b = 011

c = 101

d = 110

(x3 = x1 + x2)

d = 2 ⇒ DETECTS one single error


Examples


Example 3: n = 5 (three control digits)

a = 00000

b = 01101

c = 10110

d = 11011

x3 = x1 + x2

x4 = x2 + x3

x5 = x3 + x4

d = 3 ⇒ CORRECTS one single error


Conclusion

It is important for decoding to compute either

the exact value of d, or

a lower-bound for d

in order to estimate how many errors (at least) we expectto detect/correct

• In the case of AG codes some numerical semigrouphelps . . .


One-point AG Codes

χ “curve” over a finite field IF ≡ IFq

P and P1, . . . , Pn “rational” points of χ

C∗

m image of the linear map

evD : L(mP ) −→ IFn

f 7→ (f(P1), . . . , f(Pn))

Cm the orthogonal code of C∗

m

with respecto to the canonical bilinear form

〈a,b〉.=

n∑

i=1

aibi


One-point AG Codes



C∗



f 7→ (f(P1), . . . , f(Pn))


m


〈a,b〉.=

n∑

i=1

aibi


One-point AG Codes



C∗



f 7→ (f(P1), . . . , f(Pn))


m


〈a,b〉.=

n∑

i=1

aibi


One-point AG Codes



C∗



f 7→ (f(P1), . . . , f(Pn))


m


〈a,b〉.=

n∑

i=1

aibi


Parameters

If we assume that 2g − 2 < m < n, then the encoding


f 7→ (f(P1), . . . , f(Pn))

is injective and

k = n − m + g − 1

d ≥ m + 2 − 2g (Goppa bound)

by using the Riemann-Roch theorem


Parameters



f 7→ (f(P1), . . . , f(Pn))

is injective and

k = n − m + g − 1




Parameters



f 7→ (f(P1), . . . , f(Pn))

is injective and

k = n − m + g − 1




Parameters



f 7→ (f(P1), . . . , f(Pn))

is injective and

k = n − m + g − 1




Weierstrass semigroup

The Goppa bound can actually be improved by using theWeierstrass semigroup of χ at the point p

ΓP.= {m ∈ IN | ∃f with (f)∞ = mP}

k = n − km, where km.= ](ΓP ∩ [0, m])

(note that km = m + 1 − g for m >> 0)

d ≥ δ(m + 1) (the so-called Feng–Rao distance)

We have an improvement, since

δ(m + 1) ≥ m + 2 − 2g

and they coincide for m >> 0




ΓP.= {m ∈ IN | ∃f with (f)∞ = mP}

k = n − km, where km.= ](ΓP ∩ [0, m])




δ(m + 1) ≥ m + 2 − 2g





ΓP.= {m ∈ IN | ∃f with (f)∞ = mP}

k = n − km, where km.= ](ΓP ∩ [0, m])




δ(m + 1) ≥ m + 2 − 2g





ΓP.= {m ∈ IN | ∃f with (f)∞ = mP}

k = n − km, where km.= ](ΓP ∩ [0, m])




δ(m + 1) ≥ m + 2 − 2g



Contents



Numerical semigroups


Numerical semigroups

S ⊆ IN such that ](IN \ S) < ∞ and 0 ∈ S

The genus is g.= ](IN \ S)

The conductor satisfies c ≤ 2g

The last gap is lg = c − 1 (Frobenius number)

S is symmetric if c = 2g


Feng–Rao distance

The Feng–Rao distance in S is defined as the function

δFR : S −→ IN

m 7→ δFR(m).= min{ν(r) | r ≥ m, r ∈ S}

where ν isν : S −→ IN

r 7→ ν(r).= ]{(a, b) ∈ S2 | a + b = r}


Basic results

(i) ν(m) = m + 1 − 2g + D(m) for m ≥ c, where

D(m).= ]{(x, y) | x, y /∈ S and x + y = m}

(ii) ν(m) = m + 1 − 2g for m ≥ 2c − 1

(iii) δFR(m) ≥ m + 1 − 2g.= d∗(m − 1) ∀m ∈ S,

“and equality holds for m ≥ 2c − 1”


Basic results


D(m).= ]{(x, y) | x, y /∈ S and x + y = m}

(ii) ν(m) = m + 1 − 2g for m ≥ 2c − 1

(iii) δFR(m) ≥ m + 1 − 2g.= d∗(m − 1) ∀m ∈ S,



Basic results


D(m).= ]{(x, y) | x, y /∈ S and x + y = m}

(ii) ν(m) = m + 1 − 2g for m ≥ 2c − 1

(iii) δFR(m) ≥ m + 1 − 2g.= d∗(m − 1) ∀m ∈ S,



Basic results


D(m).= ]{(x, y) | x, y /∈ S and x + y = m}

(ii) ν(m) = m + 1 − 2g for m ≥ 2c − 1

(iii) δFR(m) ≥ m + 1 − 2g.= d∗(m − 1) ∀m ∈ S,



Two tricks

Find elements m ∈ S with D(m) = 0

WHY?

(a) δFR(m) = ν(m) = m + 1 − 2g for such elements(b) For all m ∈ S one has

δFR(m) = min{ν(m), ν(m + 1), . . . , ν(m′)}

where m′.= min{r ∈ S | r ≥ m and D(r) = 0}

Find elements m ∈ S satisfying the formula

δFR(m) = min{r ∈ S | r ≥ m + 1 − 2g} [@]


Two tricks


WHY?

(a) δFR(m) = ν(m) = m + 1 − 2g for such elements

(b) For all m ∈ S one has

δFR(m) = min{ν(m), ν(m + 1), . . . , ν(m′)}



δFR(m) = min{r ∈ S | r ≥ m + 1 − 2g} [@]


Two tricks


WHY?


δFR(m) = min{ν(m), ν(m + 1), . . . , ν(m′)}



δFR(m) = min{r ∈ S | r ≥ m + 1 − 2g} [@]


Two tricks


WHY?


δFR(m) = min{ν(m), ν(m + 1), . . . , ν(m′)}



δFR(m) = min{r ∈ S | r ≥ m + 1 − 2g} [@]


Symmetric semigroups

TheoremLet S be a symmetric semigroup; then

δFR(m) = ν(m) = m − lg = m + 1 − 2g = e

for all m = 2g − 1 + e with e ∈ S \ {0}

Proof: D(m) = 0 for such elements


Symmetric semigroups

TheoremLet S be a symmetric semigroup; then

δFR(m) = ν(m) = m − lg = m + 1 − 2g = e

for all m = 2g − 1 + e with e ∈ S \ {0}

Proof: D(m) = 0 for such elements


Minimum formula

For a symmetric semigroup S we can find an elementm0 so that the “minimum formula”

δFR(m) = min{r ∈ S | r ≥ m + 1 − 2g} [@]

in the interval (m0,∞) (Campillo and Farrán)

S = 〈9, 12, 15, 17, 20, 23, 25, 28〉c = 32 and [@] holds for m ≥ 38

S = 〈6, 8, 10, 17, 19〉c = 22 and [@] holds for m ≥ 24


Minimum formula


δFR(m) = min{r ∈ S | r ≥ m + 1 − 2g} [@]


S = 〈9, 12, 15, 17, 20, 23, 25, 28〉

c = 32 and [@] holds for m ≥ 38

S = 〈6, 8, 10, 17, 19〉c = 22 and [@] holds for m ≥ 24


Minimum formula


δFR(m) = min{r ∈ S | r ≥ m + 1 − 2g} [@]


S = 〈9, 12, 15, 17, 20, 23, 25, 28〉c = 32 and [@] holds for m ≥ 38

S = 〈6, 8, 10, 17, 19〉c = 22 and [@] holds for m ≥ 24


Minimum formula


δFR(m) = min{r ∈ S | r ≥ m + 1 − 2g} [@]


S = 〈9, 12, 15, 17, 20, 23, 25, 28〉c = 32 and [@] holds for m ≥ 38

S = 〈6, 8, 10, 17, 19〉

c = 22 and [@] holds for m ≥ 24


Minimum formula


δFR(m) = min{r ∈ S | r ≥ m + 1 − 2g} [@]


S = 〈9, 12, 15, 17, 20, 23, 25, 28〉c = 32 and [@] holds for m ≥ 38

S = 〈6, 8, 10, 17, 19〉c = 22 and [@] holds for m ≥ 24


Minimum formula

For telescopic (free) semigroups, there was anestimate for this m0 in terms of the generators(Kirfel and Pellikaan)

S = 〈8, 10, 12, 13〉

c = 28 and [@] holds for m ≥ 31Instead of m ≥ 42 !

S = 〈6, 10, 15〉



Minimum formula


S = 〈8, 10, 12, 13〉


S = 〈6, 10, 15〉



Minimum formula


S = 〈8, 10, 12, 13〉


S = 〈6, 10, 15〉



Minimum formula


S = 〈8, 10, 12, 13〉


S = 〈6, 10, 15〉



Minimum formula


S = 〈8, 10, 12, 13〉


S = 〈6, 10, 15〉



Arf semigroups

S is called an Arf semigroup if for every m, n, k ∈ S withm ≥ n ≥ k, we have

m + n − k ∈ S

Equivalently: for every couple m, n ∈ S, with m ≥ n, wehave

2m − n ∈ S


Arf semigroups

S is called an Arf semigroup if for every m, n, k ∈ S withm ≥ n ≥ k, we have

m + n − k ∈ S

Equivalently: for every couple m, n ∈ S, with m ≥ n, wehave

2m − n ∈ S


Arf semigroups

We can represent the semigroup as

S = {ρ1 = 0 < ρ2 < ρ3 < · · · }

Theorem: Assume S is Arf and c = ρr

Let li = r + ρi+1 − 2 for i = 1, · · · , r − 1, and l0 = 0Then for a positive integer l, we have:(a) if li−1 < l ≤ li ≤ lr−1, then

δFR(ρl) = 2(i − 1)

(b) if c + r − 2 = lr−1 ≤ l, then

δFR(ρl) = l − g


Arf semigroups

We can represent the semigroup as

S = {ρ1 = 0 < ρ2 < ρ3 < · · · }

Theorem: Assume S is Arf and c = ρr

Let li = r + ρi+1 − 2 for i = 1, · · · , r − 1, and l0 = 0Then for a positive integer l, we have:(a) if li−1 < l ≤ li ≤ lr−1, then

δFR(ρl) = 2(i − 1)

(b) if c + r − 2 = lr−1 ≤ l, then

δFR(ρl) = l − gFeng–Rao distances– p.37/45

Inductive semigroups

A particular case of Arf semigroups are the so-calledinductive semigroups defined by sequences ofsemigroups of the form

S1 = IN

and for m > 1

Sm = amSm−1 ∪ {n ∈ IN | n ≥ ambm−1}

for some sequence of positive integers (am) and (bm)

This kind of semigroups appears in asymptoticallygood sequences of codes (García and Stichtenoth)


Inductive semigroups

A particular case of Arf semigroups are the so-calledinductive semigroups defined by sequences ofsemigroups of the form

S1 = IN

and for m > 1

Sm = amSm−1 ∪ {n ∈ IN | n ≥ ambm−1}

for some sequence of positive integers (am) and (bm)

This kind of semigroups appears in asymptoticallygood sequences of codes (García and Stichtenoth)


Apéry sets

For e ∈ S \ {0} define the Apéry set of S related to e by

{a0, a1, . . . , ae−1}

where ai.= min{m ∈ S | m ≡ i (mod e)} for 0 ≤ i ≤ e − 1

The index i is identified to an element in ZZ/(e)

In fact, one has a disjoint union

S =e−1⋃

i=0

(ai + eIN)

and thus {a1, . . . , ae−1, e} is a generator system for the

semigroup S, called the Apéry system of S related to e


Apéry sets


{a0, a1, . . . , ae−1}




S =e−1⋃

i=0

(ai + eIN)


semigroup S, called the Apéry system of S related to e


Apéry sets


{a0, a1, . . . , ae−1}




S =

e−1⋃

i=0

(ai + eIN)


semigroup S, called the Apéry system of S related to eFeng–Rao distances– p.39/45

Apéry coordinates and relations

Every m ∈ S can be written in a unique way as

m = ai + le

with i ∈ ZZe and l ≥ 0

Thus, we can associate to m two Apéry coordinates(i, l) ∈ ZZe × IN

Let i, j ∈ ZZ/(e) ≡ ZZe and consider i + j ∈ ZZe

ai + aj = ai+j + αi,je

with αi,j ≥ 0, by definition of the Apéry set

The numbers αi,j are called Apéry relations




m = ai + le










m = ai + le










m = ai + le






The numbers αi,j are called Apéry relationsFeng–Rao distances– p.40/45

Feng–Rao distances with Apéry sets

In order to compute

ν(m) = ]{(a, b) ∈ S × S | a + b = m}

set m ≡ (i, l), a ≡ (i1, l1) and b ≡ (i2, l2)

Since m = a + b = ai1+i2 + (l1 + l2 + αi1,i2) ethen l1 + l2 = l − αi1,i2

Write i1 = k and i2 = i − kIf l < αk,i−k the equality m = a + b is not possibleSo we are interested in the case αk,i−k ≤ l

Thus, for 0 ≤ i ≤ e − 1 and h ≥ 0 define

B(h)i

.= ]{αk,i−k ≤ h | k ∈ ZZe}



In order to compute

ν(m) = ]{(a, b) ∈ S × S | a + b = m}





B(h)i

.= ]{αk,i−k ≤ h | k ∈ ZZe}



In order to compute

ν(m) = ]{(a, b) ∈ S × S | a + b = m}





B(h)i

.= ]{αk,i−k ≤ h | k ∈ ZZe}



In order to compute

ν(m) = ]{(a, b) ∈ S × S | a + b = m}





B(h)i

.= ]{αk,i−k ≤ h | k ∈ ZZe}



Theorem: ν(m) = B(0)i + B

(1)i + . . . + B

(l)i

Proof: If αk,i−k = h ≤ l then it has beenconsidered at the right-hand sum in the sets defining

B(h)i , B

(h+1)i , . . . , B

(l)i

that is l − h + 1 times

On the other hand, the equality l1 + l2 = l − αk,i−k

holds for l − h + 1 possible pairs l1, l2

2



Theorem: ν(m) = B(0)i + B

(1)i + . . . + B

(l)i

Proof: If αk,i−k = h ≤ l then it has beenconsidered at the right-hand sum in the sets defining

B(h)i , B

(h+1)i , . . . , B

(l)i

that is l − h + 1 times

On the other hand, the equality l1 + l2 = l − αk,i−k

holds for l − h + 1 possible pairs l1, l2

2



In order to compute the Feng–Rao distance, note thatν(m) is increasing in l, because of the previous formula

Thus it suffices to calculate a minimum in thecoordinate i, what gives only a finite number ofpossibilities

More precisely, one obtains the following result . . .



In order to compute the Feng–Rao distance, note thatν(m) is increasing in l, because of the previous formula

Thus it suffices to calculate a minimum in thecoordinate i, what gives only a finite number ofpossibilities

More precisely, one obtains the following result . . .



Theorem: Set m = ai + leFor each j ∈ ZZe , take mj = aj + tje, where tj is theminimum integer such that

tj ≥ max

{

ai − aj

e+ l, 0

}

Then one has

δFR(m) = min{ν(mj) | j ∈ ZZe}

Proof: mj is the minimum element of S with first Apérycoordinate equal to j such that mj ≥ m

2



Theorem: Set m = ai + leFor each j ∈ ZZe , take mj = aj + tje, where tj is theminimum integer such that

tj ≥ max

{

ai − aj

e+ l, 0

}

Then one has

δFR(m) = min{ν(mj) | j ∈ ZZe}

Proof: mj is the minimum element of S with first Apérycoordinate equal to j such that mj ≥ m

2Feng–Rao distances– p.44/45


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