NUMERICAL SEMIGROUPS FengRao distances in numerical semigroups - PowerPoint PPT Presentation

I BERIAN 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 F q Feng–Rao distances– p.5/45

Error-correcting codes Alphabet A = I F q F n Code C ⊆ I q Feng–Rao distances– p.5/45

Error-correcting codes Alphabet A = I F q F n Code C ⊆ I q “Size” dim C = k ≤ n Feng–Rao distances– p.5/45

Error-correcting codes Alphabet A = I F q F n Code C ⊆ I 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 F k F n C : I q ֒ → I q where C is the image of such a map Feng–Rao distances– p.6/45

Encoding Encoding is an injective (linear) map F k F n C : I q ֒ → I 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 F k F n C : I q ֒ → I 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 Decoded NOISE Source Information ↓ ↓ ↑ error encoding decoding ↓ ↓ ↑ Encoded Received − → − → CHANNEL Information Information Feng–Rao distances– p.9/45

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 Feng–Rao distances– p.10/45

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 Feng–Rao distances– p.11/45

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 Feng–Rao distances– p.12/45

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 Feng–Rao distances– p.13/45

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 Feng–Rao distances– p.14/45

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 Feng–Rao distances– p.15/45

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 Feng–Rao distances– p.16/45

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 Feng–Rao distances– p.17/45

Hamming distance F n The Hamming distance in I q is defined by d ( x , y ) . = ♯ { i | x i � = y i } Feng–Rao distances– p.18/45

Hamming distance F n The Hamming distance in I 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 ′ } Feng–Rao distances– p.18/45

Hamming distance F n The Hamming distance in I 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 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 ⌋ errors 2 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 ⌋ errors 2 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 ⌋ 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 Feng–Rao distances– p.19/45

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 Feng–Rao distances– p.20/45

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 Feng–Rao distances– p.21/45

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 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 F q Feng–Rao distances– p.24/45