On the q -Analogue of Cauchy Matrices Alessandro Neri 17-21 June - PowerPoint PPT Presentation

On the q -Analogue of Cauchy Matrices Alessandro Neri 17-21 June 2019 - VUB Alessandro Neri 19 June 2019 1 / 19

On the q -Analogue of Cauchy Matrices Alessandro Neri I am a friend of Finite Geometry! 17-21 June 2019 - VUB Alessandro Neri 19 June 2019 1 / 19

q -Analogues Model Finite set Finite dim vector space over F q � Element 1-dim subspace � ∅ { 0 } � Cardinality Dimension � Intersection Intersection � Union Sum � Alessandro Neri 19 June 2019 1 / 19

Examples 1. Binomials and q -binomials: k − 1 k − 1 1 − q n − i � n � n − i � n � � � = = k − i , 1 − q k − i . k k q i = 0 i = 0 2. (Chu)-Vandermonde and q -Vandermonde identity: � m k k � m + n � � m �� � � m + n � � � n � n � � q j ( m − k + j ) . = = , k − j k − j k j k j q q q j = 0 j = 0 3. Polynomials and q -polynomials: a 0 x + a 1 x q + . . . + a k x q k . a 0 + a 1 x + . . . + a k x k , 4. Gamma and q -Gamma functions: Γ q ( x + 1 ) = 1 − q x Γ( z + 1 ) = z Γ( z ) , 1 − q Γ q ( x ) . Alessandro Neri 19 June 2019 2 / 19

Vandermonde Matrix Let k ≤ n .   1 1 . . . 1 α 1 α 2 . . . α n    α 2 α 2 α 2  . . . V =  1 2 n   . . .  . . .   . . .   α k − 1 α k − 1 α k − 1 . . . n 1 2 Alessandro Neri 19 June 2019 3 / 19

Vandermonde Matrix Let k ≤ n .   1 1 . . . 1 α 1 α 2 . . . α n    α 2 α 2 α 2  . . . V =  1 2 n   . . .  . . .   . . .   α k − 1 α k − 1 α k − 1 . . . n 1 2 For n = k , det ( V ) � = 0 if and only if the α i ’s are all distinct. |{ α 1 , . . . , α n }| = n . In particular, all the k × k minors of V are non-zero. Alessandro Neri 19 June 2019 3 / 19

Moore Matrix Let k ≤ n .  g 1 g 2 . . . g n  g q g q g q . . . 1 2 n   G = . . .    , . . .   . . .  g q k − 1 g q k − 1 g q k − 1 . . . 1 2 n Alessandro Neri 19 June 2019 4 / 19

Moore Matrix Let k ≤ n .  g 1 g 2 . . . g n  g q g q g q . . . 1 2 n   G = . . .   ,  . . .   . . .  g q k − 1 g q k − 1 g q k − 1 . . . 1 2 n For n = k , det ( G ) � = 0 if and only if the g i ’s are all F q -linearly independent. dim F q � g 1 , . . . , g n � F q = n . In particular, for every M ∈ GL n ( F q ) , all the k × k minors of GM are non-zero. Alessandro Neri 19 June 2019 4 / 19

Moore Matrix → x q Let k ≤ n , σ : x �−   g 1 g 2 . . . g n σ ( g 1 ) σ ( g 2 ) . . . σ ( g n )   G = . . .  ,   . . .   . . .  σ k − 1 ( g 1 ) σ k − 1 ( g 2 ) σ k − 1 ( g n ) . . . For n = k , det ( G ) � = 0 if and only if the g i ’s are all F q -linearly independent. dim F q � g 1 , . . . , g n � F q = n . In particular, for every M ∈ GL n ( F q ) , all the k × k minors of GM are non-zero. Alessandro Neri 19 June 2019 5 / 19

Moore matrix 1. Dickson used the Moore matrix for finding the modular invariants of the general linear group over a finite field. 2. It is widely used in the study of normal bases of finite fields. 3. � � det ( G ) = ( c 1 g 1 + · · · + c i − 1 g i − 1 + g i ) . 1 ≤ i ≤ n c 1 ,..., c i − 1 ∈ F q Alessandro Neri 19 June 2019 6 / 19

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 19 June 2019 7 / 19

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 . The matrix X ∈ F r × s defined by q X i , j = c i d j x i − y j is called Generalized Cauchy (GC) Matrix. For j ≤ min { r , s } , all the j × j minors are non-zero. Alessandro Neri 19 June 2019 7 / 19

Question How to define a q -analogue of Cauchy matrices? Alessandro Neri 19 June 2019 8 / 19

Generalized Reed-Solomon Codes [Reed, Solomon ’60] F q [ x ] < k := { f ( x ) ∈ F q [ x ] | deg f < k } . Alessandro Neri 19 June 2019 9 / 19

Generalized Reed-Solomon Codes [Reed, Solomon ’60] 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 19 June 2019 9 / 19

Generalized Reed-Solomon Codes [Reed, Solomon ’60] 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 19 June 2019 9 / 19

Generalized Reed-Solomon Codes [Reed, Solomon ’60] F q [ x ] < k := { f ( x ) ∈ F q [ x ] | deg f < k } . α 1 , . . . , α n ∈ F q s.t. |{ α 1 , . . . , α n }| = n 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 19 June 2019 9 / 19

Canonical Generator Matrix for GRS codes Consider the canonical monomial basis { 1 , x , . . . , x k − 1 } and evaluate it. We get the following generator matrix Weighted Vandermonde (WV) matrix   1 1 . . . 1 α 1 α 2 . . . α n    α 2 α 2 α 2  . . . diag ( b )  1 2 n   . . .  . . .   . . .   α k − 1 α k − 1 α k − 1 . . . n 1 2 Alessandro Neri 19 June 2019 10 / 19

GRS Codes and Cauchy Matrices 1. Every [ n , k ] q code has many generator matrices G ∈ F k × n . q 2. Every [ n , k ] q code has a unique generator matrix in Reduced Row Echelon Form (RREF). Alessandro Neri 19 June 2019 11 / 19

GRS Codes and Cauchy Matrices 1. Every [ n , k ] q code has many generator matrices G ∈ F k × n . q 2. Every [ n , k ] q code has a unique generator matrix in Reduced Row Echelon Form (RREF). 3. If C is MDS the generator matrix in RREF is of the form ( I k | X ) . 4. GRS codes are MDS. Alessandro Neri 19 June 2019 11 / 19

GRS Codes and Cauchy Matrices 1. Every [ n , k ] q code has many generator matrices G ∈ F k × n . q 2. Every [ n , k ] q code has a unique generator matrix in Reduced Row Echelon Form (RREF). 3. If C is MDS the generator matrix in RREF is of the form ( I k | X ) . 4. GRS codes are MDS. Alessandro Neri 19 June 2019 11 / 19

GRS Codes and Cauchy Matrices 1. Every [ n , k ] q code has many generator matrices G ∈ F k × n . q 2. Every [ n , k ] q code has a unique generator matrix in Reduced Row Echelon Form (RREF). 3. If C is MDS the generator matrix in RREF is of the form ( I k | X ) . 4. GRS codes are MDS. Theorem [Roth, Seroussi ’85] There is a 1-1 correspondence between GC matrices and GRS codes codes given by X ← → rowsp ( I k | X ) . Alessandro Neri 19 June 2019 11 / 19

General Setting � F q m extension field of degree m of a finite field F q . � F q m ∼ = F m q as vector spaces over F q . q m ∼ � F n = F m × n as vector spaces over F q . q Alessandro Neri 19 June 2019 12 / 19

General Setting � F q m extension field of degree m of a finite field F q . � F q m ∼ = F m q as vector spaces over F q . q m ∼ � F n = F m × n as vector spaces over F q . q Rank Distance The rank distance d R on F m × n is defined by q X , Y ∈ F m × n d R ( X , Y ) := rk ( X − Y ) , . q The rank distance d R on F n q m is defined by d R ( u , v ) := dim � u 1 − v 1 , u 2 − v 2 , . . . , u n − v n � F q . A rank metric code C is a subset of F n q m equipped with the rank distance. Alessandro Neri 19 June 2019 12 / 19

Linearized Polynomials and Gabidulin Codes [Delsarte ’78], [Gabidulin ’85], [Roth ’91], [Kshevetskiy, Gabidulin ’05] m − 1 f i x q i a linearized polynomial over F q m , � i = 0 f 0 x + f 1 x q + . . . + f k − 1 x q k − 1 | f i ∈ F q m � � G k := . Alessandro Neri 19 June 2019 13 / 19

Linearized Polynomials and Gabidulin Codes [Delsarte ’78], [Gabidulin ’85], [Roth ’91], [Kshevetskiy, Gabidulin ’05] m − 1 f i x q i a linearized polynomial over F q m , � i = 0 f 0 x + f 1 x q + . . . + f k − 1 x q k − 1 | f i ∈ F q m � � G k := . g 1 , . . . , g n ∈ F q m linearly independent over F q C = { ( f ( g 1 ) , f ( g 2 ) , . . . , f ( g n )) | f ∈ G k } Alessandro Neri 19 June 2019 13 / 19

Linearized Polynomials and Gabidulin Codes [Delsarte ’78], [Gabidulin ’85], [Roth ’91], [Kshevetskiy, Gabidulin ’05] m − 1 f i x q i a linearized polynomial over F q m , � i = 0 f 0 x + f 1 x q + . . . + f k − 1 x q k − 1 | f i ∈ F q m � � G k := . g 1 , . . . , g n ∈ F q m linearly independent over F q C = { ( f ( g 1 ) , f ( g 2 ) , . . . , f ( g n )) | f ∈ G k } THEN C is the Gabidulin code G k ( g 1 , . . . , g n ) Alessandro Neri 19 June 2019 13 / 19