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A NACHRICHTENTECHNIK July 10, 2019 Carmen Sippel, Cornelia Ott, - PowerPoint PPT Presentation

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Ulm University Institute of Communications Engineering ReedSolomon Codes over Fields of Characteristic Zero Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris A NACHRICHTENTECHNIK July 10, 2019 Carmen Sippel,

  1. Ulm University Institute of Communications Engineering Reed–Solomon Codes over Fields of Characteristic Zero Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris A NACHRICHTENTECHNIK July 10, 2019 Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 1

  2. Ulm University Institute of Communications Engineering A Motivation NACHRICHTENTECHNIK We know Reed–Solomon Codes over F q C Elements are represented with Floating point operations are a fixed number of bits used Operations cost a constant Problem: Rounding errors number of bit operations Aim Reed–Solomon Codes over arbitrary fields with exact calculations during Encoding and Decoding. Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 2

  3. Ulm University Institute of Communications Engineering A Motivation NACHRICHTENTECHNIK We know Reed–Solomon Codes over F q C Elements are represented with Floating point operations are a fixed number of bits used Operations cost a constant Problem: Rounding errors number of bit operations Aim Reed–Solomon Codes over arbitrary fields with exact calculations during Encoding and Decoding. Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 2

  4. Ulm University Institute of Communications Engineering A GRS Codes over arbitrary Fields NACHRICHTENTECHNIK Definition: Generalization of Definition 5.1.1 in [Rot06] Let K be a field and k , n ∈ N such that k ≤ n . Choose α 1 , . . . , α n ∈ K \{ 0 } to be distinct and v 1 , . . . , v n ∈ K \ { 0 } . We define the generalized Reed–Solomon Code C GRS ⊆ K n with parity check matrix     1 1 . . . 1 v 1 α 1 α 2 . . . α n v 2         H Vandermonde =  . . . . ...  . . .    . . . . . .    α n − k − 1 α n − k − 1 . . . α n − k − 1 v n 1 2 n Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 3

  5. Ulm University Institute of Communications Engineering A GRS Codes over arbitrary Fields NACHRICHTENTECHNIK Generator Matrix A generator matrix is of the form     v ′ 1 1 . . . 1 1 v ′ α 1 α 2 . . . α n     2     G Vandermonde =  , . . . ...  . . .    . . . . . .    α k − 1 α k − 1 α k − 1 v ′ . . . n 1 2 n where the v ′ i ∈ K \ { 0 } , given by the following linear system of equations: n � i v i v ′ α r i = 0 ∀ r = 0 , . . . , n − 2 . i =1 Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 4

  6. Ulm University Institute of Communications Engineering A Coefficient Growth NACHRICHTENTECHNIK over Fields of Characteristic Zero If the underlying field is of characteristic zero the coefficients during Encoding and Decoding will grow. Example: Euclidean Algorithm f 0 , f 1 ∈ F 1789 [ x ] g 0 , g 1 ∈ Q [ t ] f 0 ( x ) = x 10 − 3 g 0 ( t ) = t 10 − 3 f 1 ( x ) = 3 x 9 − 2 g 1 ( t ) = 3 t 9 − 2 Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 5

  7. Ulm University Institute of Communications Engineering A Coefficient Growth NACHRICHTENTECHNIK over Fields of Characteristic Zero Example: Euclidean Algorithm - Step 1 ( x 10 − 3) / (3 x 9 − 2) ( t 10 − 3) / (3 t 9 − 2) = 1 = 1193 x 3 t Remainder: 597 x + 1786 Remainder: 2 3 t − 3 Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 6

  8. Ulm University Institute of Communications Engineering A Coefficient Growth NACHRICHTENTECHNIK over Fields of Characteristic Zero Example: Euclidean Algorithm - Step 2 (3 t 9 − 2) / (2 (3 x 9 − 2) / (597 x + 1786) 3 t − 3) = 899 x 8 + 1362 x 7 + 762 x 6 = 9 2 t 8 + 81 4 t 7 + 729 8 t 6 + 6561 16 t 5 + 1640 x 5 + 224 x 4 + 1008 x 3 + 59049 t 4 + 531441 + 958 x 2 + 733 x + 615 t 3 32 64 Remainder: 54 + 4782969 t 2 + 43046721 t 128 256 + 387420489 512 Remainder: 1162260443 512 Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 7

  9. Ulm University Institute of Communications Engineering A Coefficient Growth NACHRICHTENTECHNIK over Fields of Characteristic Zero Example: Euclidean Algorithm - Step 3 (2 3 t − 3) / 1162260443 (597 x + 1786) / 54 512 = 508 x + 1292 1024 1536 = 3486781329 t − Remainder: 0 1162260443 Remainder: 0 Question: Is it possible to derive bounds for the growth of the coefficients during Encoding and Decoding? → Solution with the help of already known results from computer algebra. Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 8

  10. Ulm University Institute of Communications Engineering A Coefficient Growth NACHRICHTENTECHNIK over Fields of Characteristic Zero Example: Euclidean Algorithm - Step 3 (2 3 t − 3) / 1162260443 (597 x + 1786) / 54 512 = 508 x + 1292 1024 1536 = 3486781329 t − Remainder: 0 1162260443 Remainder: 0 Question: Is it possible to derive bounds for the growth of the coefficients during Encoding and Decoding? → Solution with the help of already known results from computer algebra. Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 8

  11. Ulm University Institute of Communications Engineering A Coefficient Growth NACHRICHTENTECHNIK over Fields of Characteristic Zero Example: Euclidean Algorithm - Step 3 (2 3 t − 3) / 1162260443 (597 x + 1786) / 54 512 = 508 x + 1292 1024 1536 = 3486781329 t − Remainder: 0 1162260443 Remainder: 0 Question: Is it possible to derive bounds for the growth of the coefficients during Encoding and Decoding? → Solution with the help of already known results from computer algebra. Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 8

  12. Ulm University Institute of Communications Engineering The bit width - a Measure of Coefficient Growth A NACHRICHTENTECHNIK We define the bit width λ ( a ) : (Generalization of [vzGG13] p. 142) a ∈ Z : � ⌊ log 2 ( | a | ) ⌋ + 1 , if a � = 0 λ ( a ) := 0 , if a = 0 a = b c ∈ Q with b, c ∈ Z , c � = 0 , and gcd( b, c ) = 1 : λ ( a ) := max { λ ( b ) , λ ( c ) } . a ( x ) = � r b · x i ∈ Q [ x ] with a i ∈ Z and b ∈ N \ { 0 } such a i i =0 that gcd( a 0 , . . . , a r , b ) = 1 : λ ( a ( x )) := max { λ ( a 0 ) , . . . , λ ( a r ) , λ ( b ) } . NEW: A = ( a ij ) ∈ Q k × r : λ ( A ) = max { λ ( a ij ) : i = 1 , . . . , k and j = 1 , . . . , r } . Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 9

  13. Ulm University Institute of Communications Engineering The bit width - a Measure of Coefficient Growth A NACHRICHTENTECHNIK We define the bit width λ ( a ) : (Generalization of [vzGG13] p. 142) a ∈ Z : � ⌊ log 2 ( | a | ) ⌋ + 1 , if a � = 0 λ ( a ) := 0 , if a = 0 a = b c ∈ Q with b, c ∈ Z , c � = 0 , and gcd( b, c ) = 1 : λ ( a ) := max { λ ( b ) , λ ( c ) } . a ( x ) = � r b · x i ∈ Q [ x ] with a i ∈ Z and b ∈ N \ { 0 } such a i i =0 that gcd( a 0 , . . . , a r , b ) = 1 : λ ( a ( x )) := max { λ ( a 0 ) , . . . , λ ( a r ) , λ ( b ) } . NEW: A = ( a ij ) ∈ Q k × r : λ ( A ) = max { λ ( a ij ) : i = 1 , . . . , k and j = 1 , . . . , r } . Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 9

  14. Ulm University Institute of Communications Engineering The bit width - a Measure of Coefficient Growth A NACHRICHTENTECHNIK We define the bit width λ ( a ) : (Generalization of [vzGG13] p. 142) a ∈ Z : � ⌊ log 2 ( | a | ) ⌋ + 1 , if a � = 0 λ ( a ) := 0 , if a = 0 a = b c ∈ Q with b, c ∈ Z , c � = 0 , and gcd( b, c ) = 1 : λ ( a ) := max { λ ( b ) , λ ( c ) } . a ( x ) = � r b · x i ∈ Q [ x ] with a i ∈ Z and b ∈ N \ { 0 } such a i i =0 that gcd( a 0 , . . . , a r , b ) = 1 : λ ( a ( x )) := max { λ ( a 0 ) , . . . , λ ( a r ) , λ ( b ) } . NEW: A = ( a ij ) ∈ Q k × r : λ ( A ) = max { λ ( a ij ) : i = 1 , . . . , k and j = 1 , . . . , r } . Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 9

  15. Ulm University Institute of Communications Engineering The bit width - a Measure of Coefficient Growth A NACHRICHTENTECHNIK We define the bit width λ ( a ) : (Generalization of [vzGG13] p. 142) a ∈ Z : � ⌊ log 2 ( | a | ) ⌋ + 1 , if a � = 0 λ ( a ) := 0 , if a = 0 a = b c ∈ Q with b, c ∈ Z , c � = 0 , and gcd( b, c ) = 1 : λ ( a ) := max { λ ( b ) , λ ( c ) } . a ( x ) = � r b · x i ∈ Q [ x ] with a i ∈ Z and b ∈ N \ { 0 } such a i i =0 that gcd( a 0 , . . . , a r , b ) = 1 : λ ( a ( x )) := max { λ ( a 0 ) , . . . , λ ( a r ) , λ ( b ) } . NEW: A = ( a ij ) ∈ Q k × r : λ ( A ) = max { λ ( a ij ) : i = 1 , . . . , k and j = 1 , . . . , r } . Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 9

  16. Ulm University Institute of Communications Engineering The bit width - a Measure of Coefficient Growth A NACHRICHTENTECHNIK Examples (i) λ (127) = ⌊ log 2 ( | 127 | ) ⌋ + 1 = 7 (ii) λ ( 3 64 ) = max { λ (3) , λ (64) } = max { 1 , 7 } = 7 ���� � �� � ⌊ log 2 ( | 3 | ) ⌋ +1 ⌊ log 2 ( | 64 | ) ⌋ +1 (iii) λ (2 x 3 + 2 5 x 2 + 1 8 ) = λ ( 80 x 3 +16 x 2 +5 ) 40 = max { λ (80) , λ (16) , λ (5) , λ (40) } = λ (80) = ⌊ log 2 ( | 80 | ) ⌋ + 1 = 7 Carmen Sippel, Cornelia Ott, Sven Puchinger, Martin Bossert ISIT 2019, Paris 10

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