NUMERICAL SEMIGROUPS FengRao distances in numerical semigroups - - PowerPoint PPT Presentation

IBERIAN MEETING ON “NUMERICAL SEMIGROUPS” Feng–Rao distances in numerical semigroups and application to AG codes

Porto 2008

Jos´ e Ignacio Farr´ an Mart´ ın

ignfar@eis.uva.es

Departamento de Matem´ atica Aplicada Universidad de Valladolid – Campus de Segovia Escuela Universitaria de Inform´ atica

Feng–Rao distances– p.1/45

Contents

• AG codes
• Numerical semigroups

Feng–Rao distances– p.2/45

Contents

• AG codes
• Numerical semigroups

Feng–Rao distances– p.3/45

AG codes

Feng–Rao distances– p.4/45

Error-correcting codes

Alphabet A = I Fq

Feng–Rao distances– p.5/45

Error-correcting codes

Alphabet A = I Fq Code C ⊆ I Fn

q

Feng–Rao distances– p.5/45

Error-correcting codes

Alphabet A = I Fq Code C ⊆ I Fn

q

“Size” dim C = k ≤ n

Feng–Rao distances– p.5/45

Error-correcting codes

Alphabet A = I Fq Code C ⊆ I Fn

q

“Size” dim C = k ≤ n The difference n − k is called redundancy

Feng–Rao distances– p.5/45

Encoding

Encoding is an injective (linear) map C : I Fk

q ֒

→ I Fn

q

where C is the image of such a map

Feng–Rao distances– p.6/45

Encoding

Encoding is an injective (linear) map C : I Fk

q ֒

→ I Fn

q

where C is the image of such a map It can be described by means of the generator matrix G of C whose rows are a basis of C

Feng–Rao distances– p.6/45

Encoding

Encoding is an injective (linear) map C : I Fk

q ֒

→ I Fn

q

where C is the image of such a map It can be described by means of the generator matrix G 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”

Feng–Rao distances– p.6/45

Errors

encoding decoding

transmitter

− →

CHANNEL

− →

receiver

Feng–Rao distances– p.7/45

Errors

NOISE

↓

encoding decoding

transmitter

− →

CHANNEL

− →

receiver

Feng–Rao distances– p.8/45

Errors

transmitter receiver

↓ ↑

Information Source

NOISE

Decoded Information

↓ ↓ ↑

encoding

error

decoding

↓ ↓ ↑

Encoded Information

− →

CHANNEL

− →

Received Information

Feng–Rao distances– p.9/45

Examples of codes

source I II III IV V 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

Feng–Rao distances– p.10/45

Examples of codes

source I II III IV V 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

Feng–Rao distances– p.11/45

Examples of codes

source I II III IV V 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

Feng–Rao distances– p.12/45

Examples of codes

source I II III IV V 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

Feng–Rao distances– p.13/45

Examples of codes

source I II III IV V 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

Feng–Rao distances– p.14/45

Examples of codes

source I II III IV V 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

Feng–Rao distances– p.15/45

Examples of codes

source I II III IV V 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

Feng–Rao distances– p.16/45

Examples of codes

source I II III IV V 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

Feng–Rao distances– p.17/45

Hamming distance

The Hamming distance in I Fn

q is defined by

d(x, y) . = ♯{i | xi = yi}

Feng–Rao distances– p.18/45

Hamming distance

The Hamming distance in I Fn

q is defined by

d(x, y) . = ♯{i | xi = yi} The minimum distance of C is d . = d(C) . = min {d(c, c′) | c, c′ ∈ C,

c = c′}

Feng–Rao distances– p.18/45

Hamming distance

The Hamming distance in I Fn

q is defined by

d(x, y) . = ♯{i | xi = yi} The minimum distance of C is d . = d(C) . = min {d(c, c′) | c, c′ ∈ C,

c = c′}

The parameters of a code are C ≡ [n, k, d]q length n dimension k minimum distance d

Feng–Rao distances– p.18/45

Error detection and correction

Let d be the minimum distance of the code C

Feng–Rao distances– p.19/45

Error detection and correction

Let d be the minimum distance of the code C C detects up to d − 1 errors

Feng–Rao distances– p.19/45

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 ⌊d − 1 2 ⌋ errors

Feng–Rao distances– p.19/45

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 ⌊d − 1 2 ⌋ errors C corrects up to d − 1 erasures

Feng–Rao distances– p.19/45

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 ⌊d − 1 2 ⌋ errors C corrects up to d − 1 erasures C corrects any configuration of t errors and s erasures, provided 2t + s ≤ d − 1

Feng–Rao distances– p.19/45

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

Feng–Rao distances– p.20/45

Examples

Encode four possible messages {a, b, c, d} Example 2: n = 3 (one control digit) a = 000 b = 011 c = 101 d = 110 (x3 = x1 + x2) d = 2 ⇒ DETECTS one single error

Feng–Rao distances– p.21/45

Examples

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

Feng–Rao distances– p.22/45

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 expect to detect/correct

• In the case of AG codes some numerical semigroup

helps . . .

Feng–Rao distances– p.23/45

One-point AG Codes

χ “curve” over a finite field I F ≡ I Fq

Feng–Rao distances– p.24/45

One-point AG Codes

χ “curve” over a finite field I F ≡ I Fq P and P1, . . . , Pn “rational” points of χ

Feng–Rao distances– p.24/45

One-point AG Codes

χ “curve” over a finite field I F ≡ I Fq P and P1, . . . , Pn “rational” points of χ C∗

m image of the linear map

evD : L(mP) − → I Fn f → (f(P1), . . . , f(Pn))

Feng–Rao distances– p.24/45

One-point AG Codes

χ “curve” over a finite field I F ≡ I Fq P and P1, . . . , Pn “rational” points of χ C∗

m image of the linear map

evD : L(mP) − → I Fn f → (f(P1), . . . , f(Pn)) Cm the orthogonal code of C∗

m

with respecto to the canonical bilinear form a, b . =

n

• i=1

aibi

Feng–Rao distances– p.24/45

Parameters

If we assume that 2g − 2 < m < n, then the encoding evD : L(mP) − → I Fn f → (f(P1), . . . , f(Pn)) is injective and

Feng–Rao distances– p.25/45

Parameters

If we assume that 2g − 2 < m < n, then the encoding evD : L(mP) − → I Fn f → (f(P1), . . . , f(Pn)) is injective and k = n − m + g − 1

Feng–Rao distances– p.25/45

Parameters

If we assume that 2g − 2 < m < n, then the encoding evD : L(mP) − → I Fn f → (f(P1), . . . , f(Pn)) is injective and k = n − m + g − 1 d ≥ m + 2 − 2g (Goppa bound)

Feng–Rao distances– p.25/45

Parameters

If we assume that 2g − 2 < m < n, then the encoding evD : L(mP) − → I Fn f → (f(P1), . . . , f(Pn)) is injective and k = n − m + g − 1 d ≥ m + 2 − 2g (Goppa bound) by using the Riemann-Roch theorem

Feng–Rao distances– p.25/45

Weierstrass semigroup

The Goppa bound can actually be improved by using the Weierstrass semigroup of χ at the point p ΓP . = {m ∈ I

N | ∃f with (f)∞ = mP}

Feng–Rao distances– p.26/45

Weierstrass semigroup

The Goppa bound can actually be improved by using the Weierstrass semigroup of χ at the point p ΓP . = {m ∈ I

N | ∃f with (f)∞ = mP}

k = n − km, where km . = ♯(ΓP ∩ [0, m]) (note that km = m + 1 − g for m >> 0)

Feng–Rao distances– p.26/45

Weierstrass semigroup

The Goppa bound can actually be improved by using the Weierstrass semigroup of χ at the point p ΓP . = {m ∈ I

N | ∃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)

Feng–Rao distances– p.26/45

Weierstrass semigroup

The Goppa bound can actually be improved by using the Weierstrass semigroup of χ at the point p ΓP . = {m ∈ I

N | ∃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

Feng–Rao distances– p.26/45

Contents

• AG codes
• Numerical semigroups

Feng–Rao distances– p.27/45

Numerical semigroups

Feng–Rao distances– p.28/45

Numerical semigroups

S ⊆ I

N such that ♯(I N \ S) < ∞ and 0 ∈ S

The genus is g . = ♯(I

N \ S)

The conductor satisfies c ≤ 2g The last gap is lg = c − 1 (Frobenius number) S is symmetric if c = 2g

Feng–Rao distances– p.29/45

Feng–Rao distance

The Feng–Rao distance in S is defined as the function δFR : S − → I

N

m → δFR(m) . = min{ν(r) | r ≥ m, r ∈ S} where ν is ν : S − → I

N

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

Feng–Rao distances– p.30/45

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}

Feng–Rao distances– p.31/45

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

Feng–Rao distances– p.31/45

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,

Feng–Rao distances– p.31/45

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”

Feng–Rao distances– p.31/45

Two tricks

Find elements m ∈ S with D(m) = 0 WHY?

Feng–Rao distances– p.32/45

Two tricks

Find elements m ∈ S with D(m) = 0 WHY? (a) δFR(m) = ν(m) = m + 1 − 2g for such elements

Feng–Rao distances– p.32/45

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}

Feng–Rao distances– p.32/45

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} [@]

Feng–Rao distances– p.32/45

Symmetric semigroups

Theorem Let 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}

Feng–Rao distances– p.33/45

Symmetric semigroups

Theorem Let 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

Feng–Rao distances– p.33/45

Minimum formula

For a symmetric semigroup S we can find an element m0 so that the “minimum formula” δFR(m) = min{r ∈ S | r ≥ m + 1 − 2g} [@] in the interval (m0, ∞) (Campillo and Farrán)

Feng–Rao distances– p.34/45

Minimum formula

For a symmetric semigroup S we can find an element m0 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

Feng–Rao distances– p.34/45

Minimum formula

For a symmetric semigroup S we can find an element m0 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

Feng–Rao distances– p.34/45

Minimum formula

For a symmetric semigroup S we can find an element m0 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

Feng–Rao distances– p.34/45

Minimum formula

For a symmetric semigroup S we can find an element m0 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

Feng–Rao distances– p.34/45

Minimum formula

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

Feng–Rao distances– p.35/45

Minimum formula

For telescopic (free) semigroups, there was an estimate for this m0 in terms of the generators (Kirfel and Pellikaan) S = 8, 10, 12, 13

Feng–Rao distances– p.35/45

Minimum formula

For telescopic (free) semigroups, there was an estimate for this m0 in terms of the generators (Kirfel and Pellikaan) S = 8, 10, 12, 13 c = 28 and [@] holds for m ≥ 31 Instead of m ≥ 42 !

Feng–Rao distances– p.35/45

Minimum formula

For telescopic (free) semigroups, there was an estimate for this m0 in terms of the generators (Kirfel and Pellikaan) S = 8, 10, 12, 13 c = 28 and [@] holds for m ≥ 31 Instead of m ≥ 42 ! S = 6, 10, 15

Feng–Rao distances– p.35/45

Minimum formula

For telescopic (free) semigroups, there was an estimate for this m0 in terms of the generators (Kirfel and Pellikaan) S = 8, 10, 12, 13 c = 28 and [@] holds for m ≥ 31 Instead of m ≥ 42 ! S = 6, 10, 15 c = 30 and [@] holds for m ≥ 30 Instead of m ≥ 44 !

Feng–Rao distances– p.35/45

Arf semigroups

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

Feng–Rao distances– p.36/45

Arf semigroups

S is called an Arf semigroup if for every m, n, k ∈ S with m ≥ n ≥ k, we have m + n − k ∈ S Equivalently: for every couple m, n ∈ S, with m ≥ n, we have 2m − n ∈ S

Feng–Rao distances– p.36/45

Arf semigroups

We can represent the semigroup as S = {ρ1 = 0 < ρ2 < ρ3 < · · · }

Feng–Rao distances– p.37/45

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 = 0 Then 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

Feng–Rao distances– p.37/45

Inductive semigroups

A particular case of Arf semigroups are the so-called inductive semigroups defined by sequences of semigroups of the form S1 = I

N

and for m > 1 Sm = amSm−1 ∪ {n ∈ I

N | n ≥ ambm−1}

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

Feng–Rao distances– p.38/45

Inductive semigroups

A particular case of Arf semigroups are the so-called inductive semigroups defined by sequences of semigroups of the form S1 = I

N

and for m > 1 Sm = amSm−1 ∪ {n ∈ I

N | n ≥ ambm−1}

for some sequence of positive integers (am) and (bm) This kind of semigroups appears in asymptotically good sequences of codes (García and Stichtenoth)

Feng–Rao distances– p.38/45

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

Feng–Rao distances– p.39/45

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 Z

Z/(e)

Feng–Rao distances– p.39/45

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 Z

Z/(e)

In fact, one has a disjoint union S =

e−1

• i=0

(ai + eI

N)

and thus {a1, . . . , ae−1, e} is a generator system for the semigroup S, called the Apéry system of S related to e

Feng–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 ∈ Z

Ze and l ≥ 0

Feng–Rao distances– p.40/45

Apéry coordinates and relations

Every m ∈ S can be written in a unique way as m = ai + le with i ∈ Z

Ze and l ≥ 0

Thus, we can associate to m two Apéry coordinates (i, l) ∈ Z

Ze × I N

Feng–Rao distances– p.40/45

Apéry coordinates and relations

Every m ∈ S can be written in a unique way as m = ai + le with i ∈ Z

Ze and l ≥ 0

Thus, we can associate to m two Apéry coordinates (i, l) ∈ Z

Ze × I N

Let i, j ∈ Z

Z/(e) ≡ Z Ze and consider i + j ∈ Z Ze

ai + aj = ai+j + αi,je with αi,j ≥ 0, by definition of the Apéry set

Feng–Rao distances– p.40/45

Apéry coordinates and relations

Every m ∈ S can be written in a unique way as m = ai + le with i ∈ Z

Ze and l ≥ 0

Thus, we can associate to m two Apéry coordinates (i, l) ∈ Z

Ze × I N

Let i, j ∈ Z

Z/(e) ≡ Z Ze and consider i + j ∈ Z Ze

ai + aj = ai+j + αi,je with αi,j ≥ 0, by definition of the Apéry set The numbers αi,j are called Apéry relations

Feng–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)

Feng–Rao distances– p.41/45

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) e then l1 + l2 = l − αi1,i2

Feng–Rao distances– p.41/45

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) e then l1 + l2 = l − αi1,i2 Write i1 = k and i2 = i − k If l < αk,i−k the equality m = a + b is not possible So we are interested in the case αk,i−k ≤ l

Feng–Rao distances– p.41/45

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) e then l1 + l2 = l − αi1,i2 Write i1 = k and i2 = i − k If l < αk,i−k the equality m = a + b is not possible So 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 ∈ Z

Ze}

Feng–Rao distances– p.41/45

Feng–Rao distances with Apéry sets

Theorem: ν(m) = B(0)

i

+ B(1)

i

+ . . . + B(l)

i

Feng–Rao distances– p.42/45

Feng–Rao distances with Apéry sets

Theorem: ν(m) = B(0)

i

+ B(1)

i

+ . . . + B(l)

i

Proof: If αk,i−k = h ≤ l then it has been considered 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 ✷

Feng–Rao distances– p.42/45

Feng–Rao distances with Apéry sets

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

Feng–Rao distances– p.43/45

Feng–Rao distances with Apéry sets

In order to compute the Feng–Rao distance, note that ν(m) is increasing in l, because of the previous formula Thus it suffices to calculate a minimum in the coordinate i, what gives only a finite number of possibilities More precisely, one obtains the following result . . .

Feng–Rao distances– p.43/45

Feng–Rao distances with Apéry sets

Theorem: Set m = ai + le For each j ∈ Z

Ze , take mj = aj + tje, where tj is the

minimum integer such that tj ≥ max ai − aj e + l, 0

• Then one has

δFR(m) = min{ν(mj) | j ∈ Z

Ze}

Feng–Rao distances– p.44/45

Feng–Rao distances with Apéry sets

Theorem: Set m = ai + le For each j ∈ Z

Ze , take mj = aj + tje, where tj is the

minimum integer such that tj ≥ max ai − aj e + l, 0

• Then one has

δFR(m) = min{ν(mj) | j ∈ Z

Ze}

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

Feng–Rao distances– p.44/45

Feng–Rao distances– p.45/45