Generalized Gabidulin Codes and their Generator Matrices Alessandro - PowerPoint PPT Presentation

Generalized Gabidulin Codes and their Generator Matrices Alessandro Neri 23 July 2018 - LAWCI Alessandro Neri 11 June 2018 1 / 6

Generalized Reed-Solomon Codes Reed, Solomon (1960) F q [ x ] < k := { f ( x ) ∈ F q [ x ] | deg f < k } . Alessandro Neri 11 June 2018 1 / 6

Generalized Reed-Solomon Codes Reed, Solomon (1960) F q [ x ] < k := { f ( x ) ∈ F q [ x ] | deg f < k } . α 1 , . . . , α n ∈ F q distinct elements b 1 , . . . , b n ∈ F ∗ q C = { ( b 1 f ( α 1 ) , b 2 f ( α 2 ) , . . . , b n f ( α n )) | f ∈ F q [ x ] < k } Alessandro Neri 11 June 2018 1 / 6

Generalized Reed-Solomon Codes Reed, Solomon (1960) F q [ x ] < k := { f ( x ) ∈ F q [ x ] | deg f < k } . α 1 , . . . , α n ∈ F q distinct elements b 1 , . . . , b n ∈ F ∗ q C = { ( b 1 f ( α 1 ) , b 2 f ( α 2 ) , . . . , b n f ( α n )) | f ∈ F q [ x ] < k } THEN C is the Generalized Reed-Solomon code GRS n , k ( α , b ) Alessandro Neri 11 June 2018 1 / 6

Generator matrix for GRS codes  1 1 1  . . . α 1 α 2 . . . α n    α 2 α 2 α 2  . . . V k ( α ) =  1 2  n ,  . . .  . . .   . . .   α k − 1 α k − 1 α k − 1 . . . 1 2 n Alessandro Neri 11 June 2018 2 / 6

Generator matrix for GRS codes  1 1 1  . . . α 1 α 2 . . . α n    α 2 α 2 α 2  . . . V k ( α ) =  1 2  n ,  . . .  . . .   . . .   α k − 1 α k − 1 α k − 1 . . . 1 2 n Weighted Vandermonde (WV) matrix   b 1 b 2 b n . . . b 1 α 1 b 2 α 2 b n α n . . .    b 1 α 2 b 2 α 2 b n α 2  . . . GRS n , k ( α , b ) = rowsp ( V k ( α ) diag ( b )) =  1 2 n   . . .  . . .   . . .   b 1 α k − 1 b 2 α k − 1 b n α k − 1 . . . 1 2 n Alessandro Neri 11 June 2018 2 / 6

Generalized Cauchy matrix (1) x 1 , . . . , x r ∈ F q pairwise distinct, (2) y 1 , . . . , y s ∈ F q pairwise distinct, (3) y 1 , . . . , y s ∈ F q \ { x 1 , . . . , x r } , (4) c 1 , . . . , c r , d 1 , . . . , d s ∈ F ∗ q . Alessandro Neri 11 June 2018 3 / 6

Generalized Cauchy matrix (1) x 1 , . . . , x r ∈ F q pairwise distinct, (2) y 1 , . . . , y s ∈ F q pairwise distinct, (3) y 1 , . . . , y s ∈ F q \ { x 1 , . . . , x r } , (4) c 1 , . . . , c r , d 1 , . . . , d s ∈ F ∗ q . defined by X i , j = c i d j The matrix X ∈ F r × s x i − y j is called Generalized Cauchy q (GC) Matrix. Theorem [Roth, Seroussi (1985)] There is a 1-1 correspondence between GC matrices and GRS codes codes given by X ← → rowsp ( I k | X ) . Alessandro Neri 11 June 2018 3 / 6

Linearized Polynomials and Gabidulin Codes Delsarte (1978), Gabidulin (1985), Kshevetskiy and Gabidulin (2005) m − 1 f i x [ i ] a linearized polynomial over F q m , [ i ] := q i . � i = 0 f 0 x + f 1 x [ s ] + . . . + f k − 1 x [ s ( k − 1 )] | f i ∈ F q m � � G k , s := . Alessandro Neri 11 June 2018 4 / 6

Linearized Polynomials and Gabidulin Codes Delsarte (1978), Gabidulin (1985), Kshevetskiy and Gabidulin (2005) m − 1 f i x [ i ] a linearized polynomial over F q m , [ i ] := q i . � i = 0 f 0 x + f 1 x [ s ] + . . . + f k − 1 x [ s ( k − 1 )] | f i ∈ F q m � � G k , s := . g 1 , . . . , g n ∈ F q m linearly independent over F q s is integer coprime to m C = { ( f ( g 1 ) , f ( g 2 ) , . . . , f ( g n )) | f ∈ G k , s } Alessandro Neri 11 June 2018 4 / 6

Linearized Polynomials and Gabidulin Codes Delsarte (1978), Gabidulin (1985), Kshevetskiy and Gabidulin (2005) m − 1 f i x [ i ] a linearized polynomial over F q m , [ i ] := q i . � i = 0 f 0 x + f 1 x [ s ] + . . . + f k − 1 x [ s ( k − 1 )] | f i ∈ F q m � � G k , s := . g 1 , . . . , g n ∈ F q m linearly independent over F q s is integer coprime to m C = { ( f ( g 1 ) , f ( g 2 ) , . . . , f ( g n )) | f ∈ G k , s } THEN C is the Generalized Gabidulin code G k , s ( g 1 , . . . , g n ) of parameter s Alessandro Neri 11 June 2018 4 / 6

Generator matrix for G k , s ( g 1 , . . . , g n ) Let σ : F q m − → F q m x q �− → x be the q -Frobenius automorphism. Alessandro Neri 11 June 2018 5 / 6

Generator matrix for G k , s ( g 1 , . . . , g n ) Let σ : F q m − → F q m x q �− → x be the q -Frobenius automorphism. s -Moore Matrix  g 1 g 2 g n  . . . σ s ( g 1 ) σ s ( g 2 ) σ s ( g n ) . . .   G k , s ( g 1 , . . . , g n ) = rowsp  . . .   , . . .   . . .  σ s ( k − 1 ) ( g 1 ) σ s ( k − 1 ) ( g 2 ) σ s ( k − 1 ) ( g n ) . . . Alessandro Neri 11 June 2018 5 / 6

Hamming vs Rank distance: Recap d H d R Alessandro Neri 11 June 2018 6 / 6

Hamming vs Rank distance: Recap d H GRS codes d R Alessandro Neri 11 June 2018 6 / 6

Hamming vs Rank distance: Recap d H GRS codes �  � d R Gabidulin codes Alessandro Neri 11 June 2018 6 / 6

Hamming vs Rank distance: Recap d H GRS codes ← → WV matrices �  � d R Gabidulin codes Alessandro Neri 11 June 2018 6 / 6

Hamming vs Rank distance: Recap d H GRS codes ← → WV matrices � �   � � ← → d R Gabidulin codes Moore matrices Alessandro Neri 11 June 2018 6 / 6

Hamming vs Rank distance: Recap d H GRS codes ← → WV matrices ← → GC matrices � �   � � ← → d R Gabidulin codes Moore matrices Alessandro Neri 11 June 2018 6 / 6

Hamming vs Rank distance: Recap d H GRS codes ← → WV matrices ← → GC matrices � � �    � � � ← → ← → d R Gabidulin codes Moore matrices ??? Alessandro Neri 11 June 2018 6 / 6

Hamming vs Rank distance: Recap d H GRS codes ← → WV matrices ← → GC matrices � � �    � � � ← → ← → d R Gabidulin codes Moore matrices ??? The answer? It’s in the poster! Alessandro Neri 11 June 2018 6 / 6