Decoding F q -linear codes over erasure channels Sara D. Cardell - PowerPoint PPT Presentation

Decoding F q -linear codes over erasure channels Sara D. Cardell Universidad de Alicante SPCoding School S.D. Cardell Decoding F q -linear codes over erasure channels

Erasure channel We consider as model of errors the erasure channel introduced by Elias. P. E LIAS . Coding for noisy channels. IRE International Convention Record, pt. 4 , 37–46 (1955). 1 − p e 1 1 p e e ? p e 0 0 1 − p e S.D. Cardell Decoding F q -linear codes over erasure channels

Codes over extension alphabets Consider the code C F 2 whose (systematic) generator matrix is:   1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0   G =   0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 0 Here [ N , K , D ] = [ 8 , 4 , 3 ] and C F 2 is not an MDS code. We can consider the block matrix:   1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0   G =   0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 0 Over F 2 2 , the code C F 2 2 has parameters [ n , k , d ] = [ 4 , 2 , 3 ] . Therefore, it is an MDS F 2 -linear code. S.D. Cardell Decoding F q -linear codes over erasure channels

Codes over extension alphabets Consider the code C F 2 whose (systematic) generator matrix is:   1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0   G =   0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 0 Here [ N , K , D ] = [ 8 , 4 , 3 ] and C F 2 is not an MDS code. We can consider the block matrix:   1 0 0 0 1 1 1 1 0 1 0 0 1 0 1 0   G =   0 0 1 0 1 1 0 1 0 0 0 1 0 1 1 0 Over F 2 2 , the code C F 2 2 has parameters [ n , k , d ] = [ 4 , 2 , 3 ] . Therefore, it is an MDS F 2 -linear code. S.D. Cardell Decoding F q -linear codes over erasure channels

Construction C is the companion matrix of a primitive polynomial of degree b over F q [ x ] . Using the isomorphism ψ : F q b − → F q [ C ] α − → C , we construct a family of MDS F q -linear codes over F b q . S.D. Cardell Decoding F q -linear codes over erasure channels

Decoding information � �� � The codeword: v = u G = [ v 1 v 2 . . . v k | v k + 1 . . . v n ] � �� � redundancy If the number of erased symbols is t ≤ n − k , then we propose a decoding algorithm based on solving linear reduced systems. S.D. Cardell Decoding F q -linear codes over erasure channels

Conclusions So far ◮ Construction of MDS F q − linear codes, based on the isomorphism ψ : F q b − → F q [ C ] . ◮ Decoding algorithm for these codes over the erasure channel, based on solving a linear system over F q . In the future ◮ Analyze the complexity. ◮ Try to adapt this algorithm for other codes. ◮ More applications. S.D. Cardell Decoding F q -linear codes over erasure channels

Conclusions So far ◮ Construction of MDS F q − linear codes, based on the isomorphism ψ : F q b − → F q [ C ] . ◮ Decoding algorithm for these codes over the erasure channel, based on solving a linear system over F q . In the future ◮ Analyze the complexity. ◮ Try to adapt this algorithm for other codes. ◮ More applications. S.D. Cardell Decoding F q -linear codes over erasure channels

Decoding F q -linear codes over erasure channels Sara D. Cardell Universidad de Alicante SPCoding School S.D. Cardell Decoding F q -linear codes over erasure channels