Weight enumeration of codes from finite spaces Relinde Jurrius - PowerPoint PPT Presentation

Weight enumeration of codes from finite spaces Relinde Jurrius Eindhoven University of Technology, The Netherlands Finite Geometries, Third Irsee conference, June 19–25, 2011 1/15

Outline Projective systems and weight enumeration Generalized weight enumerator Codes form finite spaces Finite projective space: simplex code Finite affine space: 1-st order Reed-Muller code Extended weight enumerator Further questions and applications 2/15

Projective systems projective system n -tuple P = ( P 1 , . . . , P n ) points P j ∈ PG( r, q ) in general position Matrix G P with coordinates of points of P as columns generates linear [ n, r + 1] code.     equiv. classes of equiv. classes of     linear [ n, k ] codes ← → proj. systems of length n over GF( q ) over PG( k − 1 , q )     3/15

Weights in linear codes ✞ ☎ 0 ✝ ✆ 1 × k k × n 1 × n message m generator matrix G codeword c 4/15

Weights in linear codes ✞ ☎ 0 ✝ ✆ 1 × k k × n 1 × n message m generator matrix G codeword c Theorem c j = 0 ⇐ ⇒ P j is in nullspace of m We can determine weights by counting points P j on hyperplanes. 4/15

Generalized weight enumerator For a subcode D ⊆ C we define support supp( D ) union of the support of all words in D zero set zero( D ) complement of support, i.e., all coordinates that are always zero weight wt( D ) size of the support Generalized weight enumerators Polynomials counting for every dimension the number of subcodes of a given weight: W ( r ) � X | zero( D ) | Y | supp( D ) | C ( X, Y ) = D ∈ C dim( D )= r 5/15

Weight enumeration of subcodes ✞ ☎ ✞ ☎ ✝ ✆ ✝ ✆ r × k k × n r × n nullspace = Π generator matrix G generates D Π ⊆ PG( k − 1 , q ) D ⊆ C ← → codim(Π) = r dim( D ) = r 6/15

Weight enumeration of subcodes ✞ ☎ ✞ ☎ ✝ ✆ ✝ ✆ r × k k × n r × n nullspace = Π generator matrix G generates D Π ⊆ PG( k − 1 , q ) D ⊆ C ← → codim(Π) = r dim( D ) = r Theorem j ∈ zero( D ) ⇐ ⇒ P j ∈ Π 6/15

Codes from finite spaces Let P contain all points in PG( s − 1 , q ) . The corresponding code is the simplex code S q ( s ) . It has length q s − 1 q − 1 and dimension s . Example S 2 (3) has generator matrix   1 0 0 1 1 1 0 G P = 0 1 0 1 1 0 1   0 0 1 1 0 1 1 7/15

Codes from finite spaces Π ⊆ PG( k − 1 , q ) D ⊆ C ← → codim(Π) = r dim( D ) = r • j ∈ zero( D ) ⇐ ⇒ P j ∈ Π • P contains all points in PG( s − 1 , q ) • | zero( D ) | = | Π | for all D Theorem � s � W ( r ) q X ( q s − r − 1) / ( q − 1) Y ( q s − q s − r ) / ( q − 1) S q ( s ) ( X, Y ) = r 8/15

Codes from finite spaces Let P contain all points in AG( s − 1 , q ) , viewed as all points in PG( s − 1 , q ) not on a hyperplane H . The corresponding code is the 1-st order Reed Muller code RM q (1 , s − 1) . It has length q s − 1 and dimension s . Example RM 2 (1 , 3) has generator matrix  1 1 1 1 1 1 1 1  0 1 0 0 1 1 1 0   G P =   0 0 1 0 1 1 0 1   0 0 0 1 1 0 1 1 9/15

Codes from finite spaces Π ⊆ PG( k − 1 , q ) D ⊆ C ← → codim(Π) = r dim( D ) = r • j ∈ zero( D ) ⇐ ⇒ P j ∈ Π • if Π ⊆ H , no P j is in Π , so wt( D ) = n • if Π �⊆ H , all of the P j in Π form a subspace of AG( s − 1 , q ) of codimension r Theorem W ( r ) RM q (1 ,s − 1) ( X, Y ) = � s − 1 � � s − 1 � Y n + q r X q s − 1 − r Y q s − 1 − q s − 1 − r r − 1 r q q 10/15

Extended weight enumerator For every linear [ n, k ] code C with generator matrix G we have: Extension code [ n, k ] code C ⊗ GF( q m ) over some extension field GF( q m ) generated by the words of C . Generator matrix All the extension codes of C have the same generator matrix G . 11/15

Extended weight enumerator For every linear [ n, k ] code C with generator matrix G we have: Extension code [ n, k ] code C ⊗ GF( q m ) over some extension field GF( q m ) generated by the words of C . Generator matrix All the extension codes of C have the same generator matrix G . Extended weight enumerator Polynomial counting “for all extension codes” the number of words of a given weight: n � A w ( T ) X n − w Y w . W C ( X, Y, T ) = w =0 So for T = q m we have W C ( X, Y, q m ) = W C ⊗ GF( q m ) ( X, Y ) . 11/15

Extended weight enumerator The etended weight enumerator is completely determined by the set of generalized weight enumerators (and vice versa): Theorem   k r − 1  W ( r ) � � ( T − q j ) W C ( X, Y, T ) = C ( X, Y ) .  r =0 j =0 Moreover, their sets of supports are the same. 12/15

Supports Theorem (Simplex code) Let c ∈ S q ( s ) ⊗ GF( q m ) with wt( c ) = ( q s − q s − r ) / ( q − 1) , r < m . Then the points in P indexed by zero( c ) are all the points in a subspace of PG( s − 1 , q ) of codimension r . 13/15

Supports Theorem (Simplex code) Let c ∈ S q ( s ) ⊗ GF( q m ) with wt( c ) = ( q s − q s − r ) / ( q − 1) , r < m . Then the points in P indexed by zero( c ) are all the points in a subspace of PG( s − 1 , q ) of codimension r . Theorem (1-st order Reed-Muller code) Let c ∈ RM q (1 , s ) ⊗ GF( q m ) with wt( c ) = q s − 1 − q s − 1 − r , r < m . Then the points in P indexed by zero( c ) are all the points in a subspace of AG( s − 1 , q ) of codimension r . 13/15

Supports Theorem (Simplex code) Let c ∈ S q ( s ) ⊗ GF( q m ) with wt( c ) = ( q s − q s − r ) / ( q − 1) , r < m . Then the points in P indexed by zero( c ) are all the points in a subspace of PG( s − 1 , q ) of codimension r . Theorem (1-st order Reed-Muller code) Let c ∈ RM q (1 , s ) ⊗ GF( q m ) with wt( c ) = q s − 1 − q s − 1 − r , r < m . Then the points in P indexed by zero( c ) are all the points in a subspace of AG( s − 1 , q ) of codimension r . So: the sets zero( c ) for all codewords contains the incidence design of a finite geometry. 13/15

Further questions and applications • Weight enumeration of higher order Reed-Muller codes 14/15

Further questions and applications • Weight enumeration of higher order Reed-Muller codes • Generalized Hamming weights of Reed-Muller codes 14/15

Further questions and applications • Weight enumeration of higher order Reed-Muller codes • Generalized Hamming weights of Reed-Muller codes • Link with perfect matroid designs and associated polynomials 14/15

Further questions and applications • Weight enumeration of higher order Reed-Muller codes • Generalized Hamming weights of Reed-Muller codes • Link with perfect matroid designs and associated polynomials • Two weight codes: quadratic extension code of simplex code 14/15

Further questions and applications • Weight enumeration of higher order Reed-Muller codes • Generalized Hamming weights of Reed-Muller codes • Link with perfect matroid designs and associated polynomials • Two weight codes: quadratic extension code of simplex code • Dimension of a design (generalization of p -rank) 14/15

Thank you for your attention. 15/15