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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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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