/k Content 2/15 1. Introduction 2. Hamming weight 3. Rank weight - PowerPoint PPT Presentation

On defining the generalized rank weight Ruud Pellikaan joint work with Relinde Jurrius Autonomous University Barcelona, 6 November 2014 /k

Content 2/15 1. Introduction 2. Hamming weight 3. Rank weight 4. Extended rank weight enumerator /k

Introduction 3/15 1. Error-correction, vectors in F n q , Hamming weight 2. Network coding, matrices in F m × n , rank weight q 3. Wire-tap channel, generalized rank weight /k

� � � � � � � Butterfly network 4/15 Some text S 1 S 2 A 1 A 2 R 2 R 1 /k

Notation 5/15 F q is the finite field with q elements F q m is the finite field extension of F q of degree m An [ n , k ] code over F q is a subspace of F n q of dimension k The inner product on F n q is defined by x · y = x 1 y 1 + · · · + x n y n This inner product is bilinear, symmetric and non-degenerate For an [ n , k ] code C we define the dual or orthogonal code C ⊥ as C ⊥ = { x ∈ F n q | c · x = 0 for all c ∈ C } /k

Support and weight 6/15 The support of x in F n q is defined by supp ( x ) = { j | x j �= 0 } The weight of x is defined by wt ( x ) = | supp ( x ) | that is the number of nonzero entries of x The support of subspace D of F n q is defined by supp ( D ) = { j | x j �= 0 for some x ∈ D } The weight of D is defined by wt ( D ) = | supp ( D ) | /k

Generalized Hamming weight 7/15 Let C be an F q -linear code Then the minimum distance of C is d ( C ) = min { wt ( c ) | 0 �= c ∈ C } The r -th generalized distance of C is d r ( C ) = min { wt ( D ) | D subspace of C , dim ( D ) = r } So d 1 ( C ) = d ( C ) . /k

Network coding 8/15 Gabidulin defined rank weight Applications in network coding Choose a basis α 1 , . . . α m of F q m as a vector space over F q Let C be an F q m -linear code of length n Let c = ( c 1 , . . . , c n ) in C Then M ( c ) is the m × n matrix with entries c ij : m � c j = c ij α i i = 1 /k

Rank support, weight and distance 9/15 Let C be an F q m -linear code of length n and c ∈ C Rsupp ( c ) , the rank support of c is by definition the row space of M ( c ) The rank weight of c is wt R ( c ) = dim ( Rsupp ( c )) The rank distance is defined by d R ( x , y ) = wt R ( x − y ) This defines a metric on F n q m The rank distance of the code is d R ( C ) = min { wt R ( c ) : 0 �= c ∈ C } /k

Dictionary 10/15 The q -analogue of a finite set is a finite dimensional vector space We list the q -analogues of some properties of subsets: I , J subspaces of F n I , J subsets of { 1 , . . . , n } q { 0 } ∅ I ∩ J intersection I ∩ J intersection I ∪ J union I + J sum | I | , size of I dim ( I ) , dimension of I � n � n � � Newton binomial q Gaussian binomial k k Hamming distance on F n Rank distance on F n q m q C an F q -linear code C an F q m -linear code /k

Formalism for extended weight enumerator 11/15 Let C be a linear code over F q For a subset J of { 1 , 2 , . . . , n } define { c ∈ C | c j = 0 for all j ∈ J } C ( J ) = l ( J ) dim C ( J ) = T l ( J ) − 1 B J ( T ) = � B t ( T ) = B J ( T ) | J |= t Note that B J ( q m ) is the number of nonzero codewords in ( C ⊗ F q m )( J ) Then n W C ( X , Y , T ) = X n + � B t ( T )( X − Y ) t Y n − t t = 0 /k

Ambiguous translation of C(J) 12/15 This translation is sometimes ambiguous: I , J subspaces of F n I , J subsets of { 1 , . . . , n } q C an F q -linear code C an F q m -linear code I ∩ J = ∅ I ∩ J = { 0 } J c complement of J J ⊥ orthoplement of J I ⊆ J c I ⊆ J ⊥ C ( J ) = { c ∈ C : c j = 0 , ∀ j ∈ J } ??? C ( J ) = { c ∈ C : supp ( c ) ∩ J = ∅} C ( J ) = { c ∈ C : Rsupp ( c ) ∩ J = { 0 }} C ( J ) = { c ∈ C : supp ( c ) ⊆ J c } C ( J ) = { c ∈ C : Rsupp ( c ) ⊆ J ⊥ } /k

Extended rank weight enumerator - 1 13/15 Let C be an F q m -linear code of length n For an F q -linear subspace J of F n q define { c ∈ C : Rsupp ( c ) ⊆ J ⊥ } C ( J ) = dim C ( J ) l ( J ) = B R T m · l ( J ) J ( T ) = � B R t ( T ) = B J ( T ) dim ( J ) = t Then B R J ( q ) is the number of codewords in C ( J ) /k

Extended rank weight enumerator - 2 14/15 The extended rank weight enumerator is given by n � W R A R w ( T ) X n − w Y w C ( X , Y , T ) = w = 0 Now A R J ( q ) is the number of codewords in C of rank weight w and n � t � ( − 1 ) t − n + w T ( t − n + w 2 ) A R � B R w ( T ) = t ( T ) n − w T t = n − w /k

15/15 THANKS! QUESTIONS? /k