Numerical Semigroups and Codes Cortona 2014 Jos e Ignacio Farr an - PowerPoint PPT Presentation

I NTERNATIONAL M EETING ON N UMERICAL S EMIGROUPS Numerical Semigroups and Codes Cortona 2014 Jos´ e Ignacio Farr´ an Mart´ ın jifarran@eii.uva.es Departamento de Matem´ atica Aplicada Universidad de Valladolid – Campus de Segovia Escuela Universitaria de Inform´ atica Joint work with: M. Delgado, P. A. Garc´ ıa-S´ anchez, and D. Llena Numerical Semigroups and Codes– p. 1/48

Contents • AG codes • Numerical semigroups Numerical Semigroups and Codes– p. 2/48

Contents • AG codes • Numerical semigroups Numerical Semigroups and Codes– p. 3/48

AG codes Numerical Semigroups and Codes– p. 4/48

Error-correcting codes Alphabet A = F q Code C ⊆ F n q Dimension dim C = k ≤ n The difference n − k is called redundancy Numerical Semigroups and Codes– p. 5/48

Encoding Encoding is an injective (linear) map C : F k → F n q ֒ 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” Numerical Semigroups and Codes– p. 6/48

Errors transmitter receiver ↓ ↑ Information Decoded NOISE Source Information ↓ ↓ ↑ error encoding decoding ↓ ↓ ↑ Encoded Received − → − → CHANNEL Information Information Numerical Semigroups and Codes– p. 7/48

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 Numerical Semigroups and Codes– p. 8/48

Examples of codes source I II III IV V 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 Numerical Semigroups and Codes– p. 9/48

Examples of codes source I II III IV V 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 Numerical Semigroups and Codes– p. 10/48

Examples of codes source I II III IV V 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 Numerical Semigroups and Codes– p. 11/48

Examples of codes source I II III IV V 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 Numerical Semigroups and Codes– p. 12/48

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 Numerical Semigroups and Codes– p. 13/48

Examples of codes source I II III IV V 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 Numerical Semigroups and Codes– p. 14/48

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 Numerical Semigroups and Codes– p. 15/48

Hamming distance The Hamming distance in F n q is defined by d ( x , y ) . = ♯ { i | x i � = y i } 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 Numerical Semigroups and Codes– p. 16/48

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 ⌋ errors 2 C corrects up to d − 1 erasures C corrects any configuration of t errors and s erasures, provided 2 t + s ≤ d − 1 Numerical Semigroups and Codes– p. 17/48

Examples Encode four possible messages { a, b, c, d } n = k = 2 Example 1: a = 00 b = 01 c = 10 d = 11 d = 1 ⇒ NO error capability Numerical Semigroups and Codes– p. 18/48

Examples Encode four possible messages { a, b, c, d } n = 3 (one control digit) Example 2: a = 000 b = 011 ( x 3 = x 1 + x 2 ) c = 101 d = 110 d = 2 ⇒ DETECTS one single error Numerical Semigroups and Codes– p. 19/48

Examples Encode four possible messages { a, b, c, d } n = 5 (three control digits) Example 3: a = 00000   x 3 = x 1 + x 2 b = 01101 x 4 = x 2 + x 3   c = 10110 x 5 = x 3 + x 4 d = 11011 d = 3 ⇒ CORRECTS one single error Numerical Semigroups and Codes– p. 20/48

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 . . . Numerical Semigroups and Codes– p. 21/48

One-point AG Codes χ “curve” over a finite field F ≡ F q P and P 1 , . . . , P n “rational” points of χ C ∗ m image of the linear map F n ev D : L ( mP ) − → f �→ ( f ( P 1 ) , . . . , f ( P n )) C m the orthogonal code of C ∗ m with respect to the canonical bilinear form n � a , b � . � = a i b i i =1 Numerical Semigroups and Codes– p. 22/48

Parameters If we assume that 2 g − 2 < m < n , then the encoding F n ev D : L ( mP ) − → f �→ ( f ( P 1 ) , . . . , f ( P n )) is injective and k = n − m + g − 1 d ≥ m + 2 − 2 g (Goppa bound) by using the Riemann-Roch theorem Numerical Semigroups and Codes– p. 23/48

Weierstrass semigroup The Goppa bound can actually be improved by using the Weierstrass semigroup of χ at the point p Γ P . = { m ∈ N | ∃ f with ( f ) ∞ = mP } Note that Γ P = N \ { ℓ 1 , . . . , ℓ g } where g is the genus of χ and the numbers ℓ i are called the Weierstrass gaps of χ at P k = n − k m , where k m . = ♯ (Γ P ∩ [0 , m ]) (note that k m = 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 − 2 g , and they coincide for m >> 0 Numerical Semigroups and Codes– p. 24/48

Generalized Hamming weights Define the support of a linear code C as supp( C ) := { i | c i � = 0 for some c ∈ C } The r -th generalized weight of C is defined by d r ( C ) := min { ♯ supp( C ′ ) | C ′ ≤ C with dim( C ′ ) = r } The above definition only makes sense if r ≤ k , where k = dim ( C ) The set of numbers GHW( C ) := { d 1 , . . . , d k } is called the weight hierarchy of the code C Numerical Semigroups and Codes– p. 25/48

Generalized Feng-Rao distances It is possible to generalize the generalized Feng-Rao distance for higher order r It is also known that for a one-point AG code C m one has d r ( C m ) ≥ δ r FR ( m + 1) The details on Feng-Rao distances are given later Numerical Semigroups and Codes– p. 26/48