Maximum rank distance codes and finite semifields John Sheekey - PowerPoint PPT Presentation

Maximum rank distance codes and finite semifields John Sheekey Universiteit Gent, Belgium Banff, January 2015

Rank metric codes A rank metric code is a set C ⊂ M n ( F ) of n × n matrices over a field F with the distance function d ( X , Y ) := rank ( X − Y ) . ◮ Mostly we will be concerned with F = F q . ◮ A code is F q 0 -linear if it is a subspace over F q 0 ≤ F q . ◮ Goals: ◮ Illustrate the link with semifields. ◮ Construct a new family of linear MRD-codes.

Rank metric codes A rank metric code is a set C ⊂ M n ( F ) of n × n matrices over a field F with the distance function d ( X , Y ) := rank ( X − Y ) . ◮ Mostly we will be concerned with F = F q . ◮ A code is F q 0 -linear if it is a subspace over F q 0 ≤ F q . ◮ Goals: ◮ Illustrate the link with semifields. ◮ Construct a new family of linear MRD-codes.

Rank metric codes A rank metric code is a set C ⊂ M n ( F ) of n × n matrices over a field F with the distance function d ( X , Y ) := rank ( X − Y ) . ◮ Mostly we will be concerned with F = F q . ◮ A code is F q 0 -linear if it is a subspace over F q 0 ≤ F q . ◮ Goals: ◮ Illustrate the link with semifields. ◮ Construct a new family of linear MRD-codes.

Rank metric codes A rank metric code is a set C ⊂ M n ( F ) of n × n matrices over a field F with the distance function d ( X , Y ) := rank ( X − Y ) . ◮ Mostly we will be concerned with F = F q . ◮ A code is F q 0 -linear if it is a subspace over F q 0 ≤ F q . ◮ Goals: ◮ Illustrate the link with semifields. ◮ Construct a new family of linear MRD-codes.

Rank metric codes Introduced by Delsarte (1978), who studied them via association schemes. Gabidulin (1985) provided more constructions and decoding algorithms. Have seen renewed interest in recent years, in part due to their connections to subspace codes and q -designs.

Rank metric codes Introduced by Delsarte (1978), who studied them via association schemes. Gabidulin (1985) provided more constructions and decoding algorithms. Have seen renewed interest in recent years, in part due to their connections to subspace codes and q -designs.

Rank metric codes Introduced by Delsarte (1978), who studied them via association schemes. Gabidulin (1985) provided more constructions and decoding algorithms. Have seen renewed interest in recent years, in part due to their connections to subspace codes and q -designs.

Equivalence of rank metric codes Two codes C 1 and C 2 are said to be equivalent if there exist invertible matrices A , B , a matrix D , and an automorphism ρ of F such that C 2 = { AX ρ B + D : X ∈ C 1 } Clearly operations of this form preserve rank distance. Can be viewed as codes in ( F q n ) n . Note: two notions of equivalence... one restricts A to a certain subgroup of GL ( n , F ) .

Equivalence of rank metric codes Two codes C 1 and C 2 are said to be equivalent if there exist invertible matrices A , B , a matrix D , and an automorphism ρ of F such that C 2 = { AX ρ B + D : X ∈ C 1 } Clearly operations of this form preserve rank distance. Can be viewed as codes in ( F q n ) n . Note: two notions of equivalence... one restricts A to a certain subgroup of GL ( n , F ) .

Equivalence of rank metric codes Two codes C 1 and C 2 are said to be equivalent if there exist invertible matrices A , B , a matrix D , and an automorphism ρ of F such that C 2 = { AX ρ B + D : X ∈ C 1 } Clearly operations of this form preserve rank distance. Can be viewed as codes in ( F q n ) n . Note: two notions of equivalence... one restricts A to a certain subgroup of GL ( n , F ) .

Equivalence of rank metric codes Two codes C 1 and C 2 are said to be equivalent if there exist invertible matrices A , B , a matrix D , and an automorphism ρ of F such that C 2 = { AX ρ B + D : X ∈ C 1 } Clearly operations of this form preserve rank distance. Can be viewed as codes in ( F q n ) n . Note: two notions of equivalence... one restricts A to a certain subgroup of GL ( n , F ) .

Easy upper bound (Singleton-like) Suppose C ⊂ M n ( F q ) is a rank metric code with minimum distance d . Then |C| ≤ q n ( n − d + 1 ) . Over any field, a linear rank metric code with minimum distance d can have dimension at most n ( n − d + 1 ) .

Easy upper bound (Singleton-like) Suppose C ⊂ M n ( F q ) is a rank metric code with minimum distance d . Then |C| ≤ q n ( n − d + 1 ) . Over any field, a linear rank metric code with minimum distance d can have dimension at most n ( n − d + 1 ) .

MRD codes 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 dimension nk and minimum distance n − k + 1, we say it has parameters [ n 2 , nk , n − k + 1 ] . Duality: C ⊥ the orthogonal space with respect to e.g. b ( X , Y ) := tr ( Tr ( XY T )) . Delsarte: C MRD ⇒ C ⊥ MRD; parameters [ n 2 , n ( n − k ) , k + 1 ] Gabidulin constructed examples for all parameters using linearized polynomials.

MRD codes 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 dimension nk and minimum distance n − k + 1, we say it has parameters [ n 2 , nk , n − k + 1 ] . Duality: C ⊥ the orthogonal space with respect to e.g. b ( X , Y ) := tr ( Tr ( XY T )) . Delsarte: C MRD ⇒ C ⊥ MRD; parameters [ n 2 , n ( n − k ) , k + 1 ] Gabidulin constructed examples for all parameters using linearized polynomials.

MRD codes 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 dimension nk and minimum distance n − k + 1, we say it has parameters [ n 2 , nk , n − k + 1 ] . Duality: C ⊥ the orthogonal space with respect to e.g. b ( X , Y ) := tr ( Tr ( XY T )) . Delsarte: C MRD ⇒ C ⊥ MRD; parameters [ n 2 , n ( n − k ) , k + 1 ] Gabidulin constructed examples for all parameters using linearized polynomials.

MRD codes 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 dimension nk and minimum distance n − k + 1, we say it has parameters [ n 2 , nk , n − k + 1 ] . Duality: C ⊥ the orthogonal space with respect to e.g. b ( X , Y ) := tr ( Tr ( XY T )) . Delsarte: C MRD ⇒ C ⊥ MRD; parameters [ n 2 , n ( n − k ) , k + 1 ] Gabidulin constructed examples for all parameters using linearized polynomials.

MRD codes 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 dimension nk and minimum distance n − k + 1, we say it has parameters [ n 2 , nk , n − k + 1 ] . Duality: C ⊥ the orthogonal space with respect to e.g. b ( X , Y ) := tr ( Tr ( XY T )) . Delsarte: C MRD ⇒ C ⊥ MRD; parameters [ n 2 , n ( n − k ) , k + 1 ] Gabidulin constructed examples for all parameters using linearized polynomials.

Subspace codes A subspace code is a set of subspaces of V ( N , F ) with the distance function d s ( U , W ) = dim ( U ) + dim ( W ) − 2 dim ( U ∩ W ) . Recently have received a lot of attention due to important applications in random network coding: Koetter-Kschichang (2008).

Subspace codes A subspace code is a set of subspaces of V ( N , F ) with the distance function d s ( U , W ) = dim ( U ) + dim ( W ) − 2 dim ( U ∩ W ) . Recently have received a lot of attention due to important applications in random network coding: Koetter-Kschichang (2008).

Subspace codes from rank metric codes Given a matrix X in M n ( F ) , define a subspace S X in V ( 2 n , q ) ≃ F 2 n by S X = { ( v , vX ) : v ∈ F n } = rowspace ( I | X ) . Then d s ( S X , S Y ) = 2 d ( X , Y ) . Hence good rank metric codes give good (constant-dimension) subspace codes. Note: converse not necessarily true.

Subspace codes from rank metric codes Given a matrix X in M n ( F ) , define a subspace S X in V ( 2 n , q ) ≃ F 2 n by S X = { ( v , vX ) : v ∈ F n } = rowspace ( I | X ) . Then d s ( S X , S Y ) = 2 d ( X , Y ) . Hence good rank metric codes give good (constant-dimension) subspace codes. Note: converse not necessarily true.

Subspace codes from rank metric codes Given a matrix X in M n ( F ) , define a subspace S X in V ( 2 n , q ) ≃ F 2 n by S X = { ( v , vX ) : v ∈ F n } = rowspace ( I | X ) . Then d s ( S X , S Y ) = 2 d ( X , Y ) . Hence good rank metric codes give good (constant-dimension) subspace codes. Note: converse not necessarily true.

Subspace codes from rank metric codes Given a matrix X in M n ( F ) , define a subspace S X in V ( 2 n , q ) ≃ F 2 n by S X = { ( v , vX ) : v ∈ F n } = rowspace ( I | X ) . Then d s ( S X , S Y ) = 2 d ( X , Y ) . Hence good rank metric codes give good (constant-dimension) subspace codes. Note: converse not necessarily true.

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. In fact, every F q -linear map on F q n can be uniquely realised as a linearized polynomial of degree at most q n − 1 ( q -degree at most n − 1). Linearized polynomials ⇔ M n ( F q ) Composition mod x q n − x ⇔ Matrix multiplication