A note on the weight distribution of the Schubert code C ( 2 , m ) - PowerPoint PPT Presentation

Basic Notions A note on the weight distribution of the Schubert code C α ( 2 , m ) F . Piñero (joint work with Prashant Singh) Department of Mathematics University of Puerto Rico at Ponce Fifth Irsee Conference on Finite Geometries, September 2017 Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Definition (Grassmannian) The Grassmannian of all ℓ –subspaces of F m q is given by G ℓ, m := { W ⊆ F m q : W is a subspace and dim W = ℓ } . For each W ∈ G , we pick an ℓ × m matrix, M W whose rowspace is W . Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Definition I ( ℓ, m ) := { ( α 1 < α 2 < · · · < α ℓ ) | 1 ≤ α i ≤ m } Definition For a generic ℓ × m matrix X , and I ∈ I ( ℓ, m ) denote by det I ( X ) the ℓ × ℓ minor given by the columns indexed by I . Definition We denote by � ∆( ℓ, m ) := { f I det I ( X ) | f I ∈ F q } . I ∈ I ( ℓ, m ) Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Definition (Evaluation Map) ev G ℓ, m : ∆( ℓ, m ) → F q f �→ ( f ( M W )) W ∈ G ℓ, m Definition (Grassmann code) C α ( ℓ, m ) := { ev G ℓ, m ( f ) | W ∈ G ℓ, m , f ∈ ∆( ℓ, m ) } ⊆ F G ℓ, m q C α ( ℓ, m ) is a linear code obtained from the Plücker embedding of G ℓ, m onto a linear space. Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Definition (Bruhat Order) For α = ( α 1 , α 2 , . . . , α ℓ ) and β = ( β 1 , β 2 , . . . , β ℓ ) we say that α ≤ β if and only if α i ≤ β i for all i . Definition (Schubert variety) Let α ∈ I ( ℓ, m ) . The Schubert variety may be defined as: Ω α ( ℓ, m ) := { W ∈ G ℓ, m | det I ( M W ) = 0 , ∀ I �≤ α } Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Definition (Evaluation Map (for Schubert varieties) ev Ω α ( ℓ, m ) : ∆( ℓ, m ) → F q f �→ ( f ( M W )) W ∈ Ω α ( ℓ, m ) Definition (Schubert codes) C α ( ℓ, m ) := { ev Ω α ( ℓ, m ) ( f ) | f ∈ ∆( ℓ, m ) } ⊆ F Ω α ( ℓ, m ) q Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Ryan introduced Grassmann codes for q = 2 in 1988. Nogin generalized Grassmann codes to arbitrary q in 1996. �� m , q ℓ ( m − ℓ ) � � m � � Grassmann codes are q , codes. ℓ ℓ Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Nogin determined the weight distribution for ℓ = 2. � . . . � x i . . . x j . . . He observed that for M W = , the . . . y i . . . y j . . . function det { i , j } ( M w ) = xDy T where most of the entries of D are zero except for D i , j = − D j , i = 1. Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Definition Let i m � � f = f i , j det { i , j } ( X ) ∈ ∆( 2 , m ) . i = 1 j = i + 1 Define F to be the skew–symmetric matrix corresponding to f , that is F a 1 , a 2 = − F a 2 , a 1 = f { ( a 1 , a 2 ) } . This means that if the rows of M W are x and y then f ( M W ) = xFy T Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Theorem (Nogin) Let f ∈ ∆( ℓ, m ) . Let F be the matrix corresponding to f. Suppose the rank of F is r. Then wt ( ev G ℓ, m ( f )) = q 2 ( m − r + 1 ) q 2 r − 1 q 2 − 1 . Counting the number of skew–symmetric matrices of a particular rank will give the weight distribution for the Grassmann code in the case ℓ = 2. Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Let A 1 = { e 1 , e 2 , . . . , e a 1 } . For the case ℓ = 2 we give an alternative definition of Ω α ( ℓ, m ) , where α = ( a 1 , m ) . Definition (Schubert variety) Ω α ( ℓ, m ) := { W ∈ G ℓ, m | dim( W ∩ A i ) ≥ 1 } Definition (Schubert variety) Ω α ( ℓ, m ) := { W ∈ G ℓ, m | det I ( M W ) = 0 , ∀ I = ( b 1 , b 2 ) , b 1 > a 1 } Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions We know the parameters of C α ( 2 , m ) where α = ( a 1 , m ) The length is ( q m − 1 )( q m − 1 − 1 ) m − a 1 + 1 j q 2 m − j − 2 − i . − � � ( q 2 − 1 )( q − 1 ) j = 1 i = 1 The dimension is m ( m − 1 ) − ( m − a 1 )( m − a 1 − 1 ) . 2 2 (H. Chen (2000) and Guerra–Vincenti (2002)) The minimum distance is q m + α 1 − 3 . (Guerra – Vincenti 2004) determined the weight spectrum of C α ( 2 , m ) . Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions For the Schubert code C α ( 2 , m ) we may consider the codewords corresponding to block matrices of the form: � � A B F = − B T 0 m − a 1 × m − a 1 where A is a skew–symmetric a 1 × a 1 matrix with zeros on the diagonal and B is a a 1 × m − a 1 matrix. Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions We investigate the weight of xFy T where x ∈ A 1 . We may represent xFy T as � � y T � � � A B = x 1 A y T 1 + x 1 B y T � 1 x 1 0 − B T y T 2 0 2 where y 2 � = 0 . Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Suppose that A has corank r 1 , B has corank r 2 and their left kernels intersect in a space of dimension r . For how many vectors x ∈ A 1 and y ∈ F m q \ A 1 is x 1 A y T 1 + x 1 B y T 2 � = 0? # x 1 # y 1 × # y 2 q r x 1 A = 0, x 1 B = 0 0 q r 2 − q r ( q a 1 − q a 1 − 1 )( q m − a 1 − 1 ) x 1 A � = 0, x 1 B = 0 q m − q m − 1 q r 1 − q r x 1 A = 0, x 1 B � = 0 q m − q r 1 − q r 2 + q r ( q a 1 − q a 1 − 1 )( q m − a 1 − 1 ) x 1 A � = 0, x 1 B � = 0 In total we have ( q m − q r 1 )( q a 1 − q a 1 − 1 )( q m − a 1 − 1 ) + ( q r 1 − q r )( q m − q m − 1 ) pairs of vectors. Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Suppose that A has corank r 1 , B has corank r 2 and their left kernels intersect in a space of dimension r . For how many vectors x ∈ A 1 and y ∈ F m q \ A 1 is x 1 A y T 1 + x 1 B y T 2 � = 0? # x 1 # y 1 × # y 2 q r x 1 A = 0, x 1 B = 0 0 q r 2 − q r ( q a 1 − q a 1 − 1 )( q m − a 1 − 1 ) x 1 A � = 0, x 1 B = 0 q m − q m − 1 q r 1 − q r x 1 A = 0, x 1 B � = 0 q m − q r 1 − q r 2 + q r ( q a 1 − q a 1 − 1 )( q m − a 1 − 1 ) x 1 A � = 0, x 1 B � = 0 In total we have ( q m − q r 1 )( q a 1 − q a 1 − 1 )( q m − a 1 − 1 ) + ( q r 1 − q r )( q m − q m − 1 ) pairs of vectors. Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Suppose that A has corank r 1 , B has corank r 2 and their left kernels intersect in a space of dimension r . For how many vectors x ∈ A 1 and y ∈ F m q \ A 1 is x 1 A y T 1 + x 1 B y T 2 � = 0? # x 1 # y 1 × # y 2 q r x 1 A = 0, x 1 B = 0 0 q r 2 − q r ( q a 1 − q a 1 − 1 )( q m − a 1 − 1 ) x 1 A � = 0, x 1 B = 0 q m − q m − 1 q r 1 − q r x 1 A = 0, x 1 B � = 0 q m − q r 1 − q r 2 + q r ( q a 1 − q a 1 − 1 )( q m − a 1 − 1 ) x 1 A � = 0, x 1 B � = 0 In total we have ( q m − q r 1 )( q a 1 − q a 1 − 1 )( q m − a 1 − 1 ) + ( q r 1 − q r )( q m − q m − 1 ) pairs of vectors. Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Theorem Let f ∈ ∆( 2 , m ) such that its associated matrix, F is of the form � � A B F = − B T 0 m − a 1 × m − a 1 where F is as before, then the weight of the codeword of C α ( 2 , m ) corresponding to f is wt ( c f ) = ( q a 1 − q r 1 )( q a 1 − q a 1 − 1 )( q m − a 1 − 1 ) (1) ( q − 1 )( q 2 − q ) +( q r 1 − q r )( q m − q m − 1 ) (2) ( q − 1 )( q ( q − 1 )) + q 2 ( r 1 − 1 ) q a 1 − r 1 − 1 (3) q 2 − 1 Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Definition Let f ∈ ∆( 2 , m ) such that its associated matrix, F is of the form � � A B F = − B T 0 m − a 1 × m − a 1 where A is a skew–symmetric a 1 × a 1 matrix B is a a 1 × m − a 1 matrix, A has corank r 1 , B has corank r 2 and their left kernels intersect in a space of dimension r . We denote by d ( r 1 , r ) := wt ( ev ( f )) . w ( c f ) depends only on r 1 and r . and ( s , s 1 ) � = ( r , r 1 ) implies d ( s , s 1 ) � = d ( r , r 1 ) . Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )

Basic Definitions Basic Notions Main Result For α = ( a 1 , m ) , the weight distribution of C α ( 2 , m ) is given by � α 1 a 1 r 1 � � r 1 � � � � X d ( r 1 , r ) N 1 ( α 1 , α 1 − r 1 ) Λ( α 1 , α 1 − r 2 )∆( α 1 , r 1 , r 2 , r ) r q r 1 = 0 r = 0 r 2 = r Piñero, Fernando weight distribution of Schubert codes of C α ( 2 , m )