Maximum rank distance codes: constructions, classifications, and - PowerPoint PPT Presentation

Maximum rank distance codes: constructions, classifications, and applications John Sheekey UCD, Dublin, Ireland Dagstuhl, August 2016

Rank metric codes A rank metric code is a set C ⊂ M m × n ( F q ) of m × n matrices ( m ≤ n ) over F q with the distance function d ( X , Y ) := rank ( X − Y ) .

Rank metric codes A rank metric code is a set C ⊂ M m × n ( F q ) of m × n matrices ( m ≤ n ) over F q with the distance function d ( X , Y ) := rank ( X − Y ) . A code is F -linear if it is a subspace over F ≤ F q .

MRD codes Singleton-like bound: A subspace of matrices where every nonzero element has rank at least d has dimension at most n ( m − d + 1 ) .

MRD codes Singleton-like bound: A subspace of matrices where every nonzero element has rank at least d has dimension at most n ( m − d + 1 ) . A code meeting this bound is said to be a Maximum Rank Distance (MRD) code.

MRD codes Singleton-like bound: A subspace of matrices where every nonzero element has rank at least d has dimension at most n ( m − d + 1 ) . A code meeting this bound is said to be a Maximum Rank Distance (MRD) code. If C is an MRD-code which is linear over F q with minimum rank-distance d , we say it has parameters [ m × n , n ( m − d + 1 ) , d ] q .

MRD codes Singleton-like bound: A subspace of matrices where every nonzero element has rank at least d has dimension at most n ( m − d + 1 ) . A code meeting this bound is said to be a Maximum Rank Distance (MRD) code. If C is an MRD-code which is linear over F q with minimum rank-distance d , we say it has parameters [ m × n , n ( m − d + 1 ) , d ] q . Delsarte (1978), and later Gabidulin (1985), constructed examples for all parameters using linearized polynomials.

Applications Rank-metric codes have seen renewed interest in recent years due to new potential applications.

Applications Rank-metric codes have seen renewed interest in recent years due to new potential applications. ◮ Code-based cryptography; ◮ Subspace codes (random network coding); ◮ Index coding; ◮ ...?

Applications Rank-metric codes have seen renewed interest in recent years due to new potential applications. ◮ Code-based cryptography; ◮ Subspace codes (random network coding); ◮ Index coding; ◮ ...? For this reason we would like to find more examples of MRD-codes.

Equivalence of rank metric codes Two codes C 1 and C 2 are said to be equivalent if there exist invertible matrices A and B , a matrix D , and an automorphism ρ of F q , such that C 2 = { AX ρ B + D : X ∈ C 1 }

Equivalence of rank metric codes Two codes C 1 and C 2 are said to be equivalent if there exist invertible matrices A and B , a matrix D , and an automorphism ρ of F q , such that C 2 = { AX ρ B + D : X ∈ C 1 } Clearly operations of this form preserve rank distance.

Equivalence of rank metric codes Two codes C 1 and C 2 are said to be equivalent if there exist invertible matrices A and B , a matrix D , and an automorphism ρ of F q , such that C 2 = { AX ρ B + D : X ∈ C 1 } Clearly operations of this form preserve rank distance. Can also include the transpose operation when n = m .

Duality There is also a notion of duality, which preserves the MRD property: b ( X , Y ) = Tr ( XY T ) .

Duality There is also a notion of duality, which preserves the MRD property: b ( X , Y ) = Tr ( XY T ) . Delsarte showed that if C is MRD with minimum distance d , then C ⊥ is MRD with minimum distance m − d + 2.

Duality There is also a notion of duality, which preserves the MRD property: b ( X , Y ) = Tr ( XY T ) . Delsarte showed that if C is MRD with minimum distance d , then C ⊥ is MRD with minimum distance m − d + 2. See paper of Ravagnani (2015) for an elementary proof.

Linearized polynomials A linearized polynomial is a polynomial in F q n [ x ] of the form f ( x ) = f 0 x + f 1 x q + · · · + f n − 1 x q n − 1 .

Linearized polynomials A linearized polynomial is a polynomial in F q n [ x ] of the form f ( x ) = f 0 x + f 1 x q + · · · + f n − 1 x q n − 1 . Each such polynomial is an F q -linear map from F q n to itself.

Linearized polynomials A linearized polynomial is a polynomial in F q n [ x ] of the form f ( x ) = f 0 x + f 1 x q + · · · + f n − 1 x q n − 1 . Each such polynomial is an F q -linear map from F q n to itself. Linearized polynomials ⇔ M n ( F q ) Composition mod x q n − x ⇔ Matrix multiplication

Linearized polynomials A linearized polynomial is a polynomial in F q n [ x ] of the form f ( x ) = f 0 x + f 1 x q + · · · + f n − 1 x q n − 1 . Each such polynomial is an F q -linear map from F q n to itself. Linearized polynomials ⇔ M n ( F q ) Composition mod x q n − x ⇔ Matrix multiplication With this representation, we can talk of a code being F q n -linear.

Linearized polynomials A linearized polynomial is a polynomial in F q n [ x ] of the form f ( x ) = f 0 x + f 1 x q + · · · + f n − 1 x q n − 1 . Each such polynomial is an F q -linear map from F q n to itself. Linearized polynomials ⇔ M n ( F q ) Composition mod x q n − x ⇔ Matrix multiplication With this representation, we can talk of a code being F q n -linear. In this setting, the dual operation becomes very easy to use.

Delsarte/Gabidulin codes The Delsarte/Gabidulin code G k is a the set of linearized polynomials of q -degree at most k − 1, i.e. G k := { f 0 x + f 1 x q + · · · + f k − 1 x q k − 1 : f i ∈ F q n } .

Delsarte/Gabidulin codes The Delsarte/Gabidulin code G k is a the set of linearized polynomials of q -degree at most k − 1, i.e. G k := { f 0 x + f 1 x q + · · · + f k − 1 x q k − 1 : f i ∈ F q n } . Clearly each element of G k has at most q k − 1 roots, and hence rank at least n − k + 1 =: d .

Delsarte/Gabidulin codes The Delsarte/Gabidulin code G k is a the set of linearized polynomials of q -degree at most k − 1, i.e. G k := { f 0 x + f 1 x q + · · · + f k − 1 x q k − 1 : f i ∈ F q n } . Clearly each element of G k has at most q k − 1 roots, and hence rank at least n − k + 1 =: d . G k has dimension nk = n ( n − d + 1 ) over F q .

Delsarte/Gabidulin codes The Delsarte/Gabidulin code G k is a the set of linearized polynomials of q -degree at most k − 1, i.e. G k := { f 0 x + f 1 x q + · · · + f k − 1 x q k − 1 : f i ∈ F q n } . Clearly each element of G k has at most q k − 1 roots, and hence rank at least n − k + 1 =: d . G k has dimension nk = n ( n − d + 1 ) over F q . In fact, it is linear over F q n .

Delsarte/Gabidulin codes The Delsarte/Gabidulin code G k is a the set of linearized polynomials of q -degree at most k − 1, i.e. G k := { f 0 x + f 1 x q + · · · + f k − 1 x q k − 1 : f i ∈ F q n } . Clearly each element of G k has at most q k − 1 roots, and hence rank at least n − k + 1 =: d . G k has dimension nk = n ( n − d + 1 ) over F q . In fact, it is linear over F q n . Hence G k is an F q n -linear MRD-code with parameters [ n × n , nk , n − k + 1 ] q .

Delsarte/Gabidulin codes The Delsarte/Gabidulin code G k is a the set of linearized polynomials of q -degree at most k − 1, i.e. G k := { f 0 x + f 1 x q + · · · + f k − 1 x q k − 1 : f i ∈ F q n } . Clearly each element of G k has at most q k − 1 roots, and hence rank at least n − k + 1 =: d . G k has dimension nk = n ( n − d + 1 ) over F q . In fact, it is linear over F q n . Hence G k is an F q n -linear MRD-code with parameters [ n × n , nk , n − k + 1 ] q . Can replace x q i with x q si for any s with ( n , s ) = 1, and define G k , s (generalised Gabidulin codes).

Non-square case We can produce MRD-codes in M m × n from an MRD code in M m × n by either shortening or puncturing; C U = { f : f ∈ C | f ( U ) = 0 } , dim ( U ) = n − m ; C h := { f ◦ h : f ∈ C} , rank ( h ) = m .

Non-square case We can produce MRD-codes in M m × n from an MRD code in M m × n by either shortening or puncturing; C U = { f : f ∈ C | f ( U ) = 0 } , dim ( U ) = n − m ; C h := { f ◦ h : f ∈ C} , rank ( h ) = m . ◮ When are different codes obtained by puncturing/shortening an MRD code equivalent? ◮ Are there examples of MRD codes which cannot be obtained by puncturing/shortening?

Alternative formulation We can also think of Rank-Metric codes as sets of vectors in ( F q n ) m .

Alternative formulation We can also think of Rank-Metric codes as sets of vectors in ( F q n ) m . The rank weight of a vector ( v 0 , . . . , v m − 1 ) is given by dim � v 0 , . . . , v m − 1 � F q .

Alternative formulation We can also think of Rank-Metric codes as sets of vectors in ( F q n ) m . The rank weight of a vector ( v 0 , . . . , v m − 1 ) is given by dim � v 0 , . . . , v m − 1 � F q . We go from one setting to the other by fixing an F q -basis for F q n (say, { e 0 , . . . e n − 1 } ), and identifying f ↔ ( f ( e 0 ) , . . . , f ( e m − 1 )) .

Alternative formulation We can also think of Rank-Metric codes as sets of vectors in ( F q n ) m . The rank weight of a vector ( v 0 , . . . , v m − 1 ) is given by dim � v 0 , . . . , v m − 1 � F q . We go from one setting to the other by fixing an F q -basis for F q n (say, { e 0 , . . . e n − 1 } ), and identifying f ↔ ( f ( e 0 ) , . . . , f ( e m − 1 )) . Using this representation, Horlemann-Trautmann and Marshall gave very useful criteria for identifying when a code is equivalent to a (generalised) Gabidulin code.