some recent developments in the theory of linear mrd codes
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Some recent developments in the theory of linear MRD-codes Olga - PowerPoint PPT Presentation

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Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Some recent developments in the theory of linear MRD-codes Olga Polverino Universit degli Studi della Campania,

  1. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Equivalence of RD-codes C , C ′ ⊂ F m × n linear RD-codes q C , C ′ equivalent C ′ = { AC σ B : C ∈ C} A ∈ GL ( m , F q ) , B ∈ GL ( n , F q ) and σ ∈ Aut ( F q ) Aut ( C ) Automorphism group of C Aut ( C ) = { ( A , B , σ ) ∈ GL ( m , q ) × GL ( n , q ) × Aut ( F q ) : A C σ B = C}

  2. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Left and Right idealiser of RD-codes linear RD-code in F m × n C q

  3. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Left and Right idealiser of RD-codes linear RD-code in F m × n C q L ( C ) = { Y ∈ F m × m : YC ∈ C for all C ∈ C} q Left idealiser of C

  4. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Left and Right idealiser of RD-codes linear RD-code in F m × n C q L ( C ) = { Y ∈ F m × m : YC ∈ C for all C ∈ C} q Left idealiser of C R ( C ) = { Z ∈ F n × n : CZ ∈ C for all C ∈ C} q Right idealiser of C

  5. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Left and Right idealiser of RD-codes linear RD-code in F m × n C q L ( C ) = { Y ∈ F m × m : YC ∈ C for all C ∈ C} q Left idealiser of C R ( C ) = { Z ∈ F n × n : CZ ∈ C for all C ∈ C} q Right idealiser of C D. Liebhold, G. Nebe : Automorphism groups of Gabidulin-like codes. Arch. Math. (2016)

  6. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Left and Right idealiser of RD-codes linear RD-code in F m × n C q L ( C ) = { Y ∈ F m × m : YC ∈ C for all C ∈ C} q Left idealiser of C R ( C ) = { Z ∈ F n × n : CZ ∈ C for all C ∈ C} q Right idealiser of C L ( C ) ∗ × R ( C ) ∗ × { id } ⊆ Aut ( C ) D. Liebhold, G. Nebe : Automorphism groups of Gabidulin-like codes. Arch. Math. (2016)

  7. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Left and Right idealiser of MRD-codes Lunardon-Trombetti-Zhou 2017, JACO

  8. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Left and Right idealiser of MRD-codes Lunardon-Trombetti-Zhou 2017, JACO For two equivalent linear metric codes their right (resp. left) idealisers are also equivalent.

  9. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Left and Right idealiser of MRD-codes Lunardon-Trombetti-Zhou 2017, JACO For two equivalent linear metric codes their right (resp. left) idealisers are also equivalent. Let C be a linear MRD code in F m × n with d > 1. q If m ≤ n then L ( C ) is a finite field. Hence | L ( C ) | ≤ q m . If n ≤ m then R ( C ) is a finite field. Hence | R ( C ) | ≤ q n .

  10. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Left and Right idealiser of MRD-codes Lunardon-Trombetti-Zhou 2017, JACO For two equivalent linear metric codes their right (resp. left) idealisers are also equivalent. Let C be a linear MRD code in F m × n with d > 1. q If m ≤ n then L ( C ) is a finite field. Hence | L ( C ) | ≤ q m . If n ≤ m then R ( C ) is a finite field. Hence | R ( C ) | ≤ q n . If C is a linear MRD code in F n × n with d > 1 then the left q and the right idealisers of C are both finite fields.

  11. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Left and Right idealiser of MRD-codes Lunardon-Trombetti-Zhou 2017, JACO For two equivalent linear metric codes their right (resp. left) idealisers are also equivalent. Let C be a linear MRD code in F m × n with d > 1. q If m ≤ n then L ( C ) is a finite field. Hence | L ( C ) | ≤ q m . If n ≤ m then R ( C ) is a finite field. Hence | R ( C ) | ≤ q n . If C is a linear MRD code in F n × n with d > 1 then the left q and the right idealisers of C are both finite fields.

  12. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Left and Right idealiser of MRD-codes Lunardon-Trombetti-Zhou 2017, JACO For two equivalent linear metric codes their right (resp. left) idealisers are also equivalent. Let C be a linear MRD code in F m × n with d > 1. q If m ≤ n then L ( C ) is a finite field. Hence | L ( C ) | ≤ q m . If n ≤ m then R ( C ) is a finite field. Hence | R ( C ) | ≤ q n . If C is a linear MRD code in F n × n with d > 1 then the left q and the right idealisers of C are both finite fields. dim L ( C ) ( C ) dim R ( C ) ( C )

  13. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim First Examples of linear MRD-codes P. Delsarte : Bilinear forms over a finite field, with applications to coding theory, J. Comb. Theory Ser. A (1978)

  14. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim First Examples of linear MRD-codes P. Delsarte : Bilinear forms over a finite field, with applications to coding theory, J. Comb. Theory Ser. A (1978) E. Gabidulin : Theory of codes with maximum rank distance, Probl. Inf. Transm. (1985)

  15. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim First Examples of linear MRD-codes P. Delsarte : Bilinear forms over a finite field, with applications to coding theory, J. Comb. Theory Ser. A (1978) E. Gabidulin : Theory of codes with maximum rank distance, Probl. Inf. Transm. (1985) A. Kshevetskiy, E. Gabidulin : The new construction of rank codes, Proceedings ISIT, (2005)

  16. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Gabidulin MRD-codes G k = { f ( x ) = a 0 x + a 1 x q + . . . a k − 1 x q ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k ∈ Z + k < n

  17. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Gabidulin MRD-codes G k = { f ( x ) = a 0 x + a 1 x q + . . . a k − 1 x q ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k ∈ Z + k < n dim F q Ker f ≤ k − 1

  18. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Gabidulin MRD-codes G k = { f ( x ) = a 0 x + a 1 x q + . . . a k − 1 x q ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k ∈ Z + k < n dim F q Ker f ≤ k − 1 ⇒ dim F q Im f ≥ n − k + 1

  19. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Gabidulin MRD-codes G k = { f ( x ) = a 0 x + a 1 x q + . . . a k − 1 x q ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k ∈ Z + k < n dim F q Ker f ≤ k − 1 ⇒ dim F q Im f ≥ n − k + 1 M f ∈ F n × n B → F q -basis of F q n q matrix associated with f w.r.t. B

  20. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Gabidulin MRD-codes G k = { f ( x ) = a 0 x + a 1 x q + . . . a k − 1 x q ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k ∈ Z + k < n dim F q Ker f ≤ k − 1 ⇒ dim F q Im f ≥ n − k + 1 M f ∈ F n × n B → F q -basis of F q n q matrix associated with f w.r.t. B C G k = { M f : f ∈ G k } ⊆ F n × n , q

  21. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Gabidulin MRD-codes G k = { f ( x ) = a 0 x + a 1 x q + . . . a k − 1 x q ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k ∈ Z + k < n dim F q Ker f ≤ k − 1 ⇒ dim F q Im f ≥ n − k + 1 M f ∈ F n × n B → F q -basis of F q n q matrix associated with f w.r.t. B C G k = { M f : f ∈ G k } ⊆ F n × n , q C G k Gabidulin MRD-code [ n × n , kn , n − k + 1 ] F q

  22. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Gabidulin MRD-codes G k = { f ( x ) = a 0 x + a 1 x q + · · · + a k − 1 x q ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k ∈ Z + k < n

  23. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Gabidulin MRD-codes G k = { f ( x ) = a 0 x + a 1 x q + · · · + a k − 1 x q ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k ∈ Z + k < n U ⊆ F q n , dim F q U = m ≤ n , B U basis of U

  24. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Gabidulin MRD-codes G k = { f ( x ) = a 0 x + a 1 x q + · · · + a k − 1 x q ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k ∈ Z + k < n U ⊆ F q n , dim F q U = m ≤ n , B U basis of U C G k ( U ) = { M f | U : f ∈ G k } ⊆ F n × m , q

  25. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Gabidulin MRD-codes G k = { f ( x ) = a 0 x + a 1 x q + · · · + a k − 1 x q ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k ∈ Z + k < n U ⊆ F q n , dim F q U = m ≤ n , B U basis of U C G k ( U ) = { M f | U : f ∈ G k } ⊆ F n × m , q C G k ( U ) Gabidulin MRD-code [ n × m , kn , n − k + 1 ] F q

  26. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Gabidulin MRD-codes G k = { f ( x ) = a 0 x + a 1 x q + · · · + a k − 1 x q ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k ∈ Z + k < n U ⊆ F q n , dim F q U = m ≤ n , B U basis of U C G k ( U ) = { M f | U : f ∈ G k } ⊆ F n × m , q C G k ( U ) Gabidulin MRD-code [ n × m , kn , n − k + 1 ] F q C G k ( U ) T Gabidulin MRD-code [ m × n , kn , n − k + 1 ] F q

  27. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalised Gabidulin MRD-codes G k , s = { f ( x ) = a 0 x + a 1 x q s + · · · + a k − 1 x q s ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k , s ∈ Z + , k < n , gcd ( n , s ) = 1

  28. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalised Gabidulin MRD-codes G k , s = { f ( x ) = a 0 x + a 1 x q s + · · · + a k − 1 x q s ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k , s ∈ Z + , k < n , gcd ( n , s ) = 1 C G k , s = { M f : f ∈ G k } ⊆ F n × n , q

  29. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalised Gabidulin MRD-codes G k , s = { f ( x ) = a 0 x + a 1 x q s + · · · + a k − 1 x q s ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k , s ∈ Z + , k < n , gcd ( n , s ) = 1 C G k , s = { M f : f ∈ G k } ⊆ F n × n , q C G k , s Generalised Gabidulin MRD-code [ n × n , kn , n − k + 1 ] F q

  30. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalised Gabidulin MRD-codes G k , s = { f ( x ) = a 0 x + a 1 x q s + · · · + a k − 1 x q s ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } , n , k , s ∈ Z + , k < n , gcd ( n , s ) = 1 C G k , s = { M f : f ∈ G k } ⊆ F n × n , q C G k , s Generalised Gabidulin MRD-code [ n × n , kn , n − k + 1 ] F q C G k , s ( U ) Generalised Gabidulin MRD-code [ n × m , kn , n − k + 1 ] F q C G k , s ( U ) T Generalised Gabidulin MRD-code [ m × n , kn , n − k + 1 ] F q

  31. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials F n × n q

  32. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials F n × n q ↓ End ( F q n , F q )

  33. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials F n × n q ↓ End ( F q n , F q ) ↓ L n , q = { f ( x ) = a 0 x + a 1 x q + . . . a n − 1 x q ( n − 1 ) : a 0 , a 1 , . . . , a n − 1 ∈ F q n }

  34. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials F n × n q ↓ End ( F q n , F q ) ↓ L n , q = { f ( x ) = a 0 x + a 1 x q + . . . a n − 1 x q ( n − 1 ) : a 0 , a 1 , . . . , a n − 1 ∈ F q n } ( F n × n ( L n , q , + , ◦ ) ≃ , + , · ) q

  35. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials F n × n q ↓ End ( F q n , F q ) ↓ L n , q = { f ( x ) = a 0 x + a 1 x q + . . . a n − 1 x q ( n − 1 ) : a 0 , a 1 , . . . , a n − 1 ∈ F q n } ( F n × n ( L n , q , + , ◦ ) ≃ , + , · ) q composition modulo x q n − x ◦ →

  36. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials C , C ′ ⊆ L n , q equivalent MRD-codes

  37. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials C , C ′ ⊆ L n , q equivalent MRD-codes C ′ = { g ◦ f σ ◦ h : f ∈ C} f , g permutation q -polynomials, σ ∈ Aut ( F q )

  38. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials C , C ′ ⊆ L n , q equivalent MRD-codes C ′ = { g ◦ f σ ◦ h : f ∈ C} f , g permutation q -polynomials, σ ∈ Aut ( F q ) C ⊆ L n , q MRD-code

  39. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials C , C ′ ⊆ L n , q equivalent MRD-codes C ′ = { g ◦ f σ ◦ h : f ∈ C} f , g permutation q -polynomials, σ ∈ Aut ( F q ) C ⊆ L n , q MRD-code L ( C ) = { g ∈ L n , q : g ◦ f ∈ C for all f ∈ C} Left idealiser of C

  40. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials C , C ′ ⊆ L n , q equivalent MRD-codes C ′ = { g ◦ f σ ◦ h : f ∈ C} f , g permutation q -polynomials, σ ∈ Aut ( F q ) C ⊆ L n , q MRD-code L ( C ) = { g ∈ L n , q : g ◦ f ∈ C for all f ∈ C} Left idealiser of C R ( C ) = { h ∈ L n , q : f ◦ h ∈ C for all f ∈ C} Right idealiser of C

  41. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials G k , s = { f ( x ) = a 0 x + a 1 x q s + . . . a k − 1 x q s ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } n , k , s ∈ Z + , k < n , gcd ( n , s ) = 1 G k , s Generalised Gabidulin MRD-code [ n × n , kn , n − k + 1 ] F q

  42. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials G k , s = { f ( x ) = a 0 x + a 1 x q s + . . . a k − 1 x q s ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } n , k , s ∈ Z + , k < n , gcd ( n , s ) = 1 G k , s Generalised Gabidulin MRD-code [ n × n , kn , n − k + 1 ] F q

  43. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials G k , s = { f ( x ) = a 0 x + a 1 x q s + . . . a k − 1 x q s ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } n , k , s ∈ Z + , k < n , gcd ( n , s ) = 1 G k , s Generalised Gabidulin MRD-code [ n × n , kn , n − k + 1 ] F q F q n = { τ α ( x ) = α x : α ∈ F q n }

  44. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials G k , s = { f ( x ) = a 0 x + a 1 x q s + . . . a k − 1 x q s ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } n , k , s ∈ Z + , k < n , gcd ( n , s ) = 1 G k , s Generalised Gabidulin MRD-code [ n × n , kn , n − k + 1 ] F q F q n = { τ α ( x ) = α x : α ∈ F q n } f ◦ τ α ∈ G k , s , τ α ◦ f ∈ G k , s for all f ∈ G k , s , τ α ∈ F q n

  45. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials G k , s = { f ( x ) = a 0 x + a 1 x q s + . . . a k − 1 x q s ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } n , k , s ∈ Z + , k < n , gcd ( n , s ) = 1 G k , s Generalised Gabidulin MRD-code [ n × n , kn , n − k + 1 ] F q F q n = { τ α ( x ) = α x : α ∈ F q n } f ◦ τ α ∈ G k , s , τ α ◦ f ∈ G k , s for all f ∈ G k , s , τ α ∈ F q n L ( G k , s ) = F q n R ( G k , s ) = F q n

  46. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials G k , s = { f ( x ) = a 0 x + a 1 x q s + . . . a k − 1 x q s ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } n , k , s ∈ Z + , k < n , gcd ( n , s ) = 1 G k , s Generalised Gabidulin MRD-code [ n × n , kn , n − k + 1 ] F q F q n = { τ α ( x ) = α x : α ∈ F q n } f ◦ τ α ∈ G k , s , τ α ◦ f ∈ G k , s for all f ∈ G k , s , τ α ∈ F q n L ( G k , s ) = F q n R ( G k , s ) = F q n

  47. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Linearized polynomials G k , s = { f ( x ) = a 0 x + a 1 x q s + . . . a k − 1 x q s ( k − 1 ) : a 0 , a 1 , . . . , a k − 1 ∈ F q n } n , k , s ∈ Z + , k < n , gcd ( n , s ) = 1 G k , s Generalised Gabidulin MRD-code [ n × n , kn , n − k + 1 ] F q F q n = { τ α ( x ) = α x : α ∈ F q n } f ◦ τ α ∈ G k , s , τ α ◦ f ∈ G k , s for all f ∈ G k , s , τ α ∈ F q n L ( G k , s ) = F q n R ( G k , s ) = F q n G k , s F q n -linear MRD-code

  48. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Finite presemifields J. De La Cruz, Kiermaier, Wassermann, Willem : Algebraic structures of MRD codes, Adv. Math. Commun. (2016)

  49. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Finite presemifields J. De La Cruz, Kiermaier, Wassermann, Willem : Algebraic structures of MRD codes, Adv. Math. Commun. (2016) J. Sheekey: A new family of linear maximum rank distance codes, Adv. Math. Commun. (2016)

  50. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Finite presemifields J. De La Cruz, Kiermaier, Wassermann, Willem : Algebraic structures of MRD codes, Adv. Math. Commun. (2016) J. Sheekey: A new family of linear maximum rank distance codes, Adv. Math. Commun. (2016) C F q -linear MRD-code, [ n × n , n , n ] F q � S C presemifield of order q n with left nucleus containing F q

  51. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Finite presemifields J. De La Cruz, Kiermaier, Wassermann, Willem : Algebraic structures of MRD codes, Adv. Math. Commun. (2016) J. Sheekey: A new family of linear maximum rank distance codes, Adv. Math. Commun. (2016) C F q -linear MRD-code, [ n × n , n , n ] F q � π S C semifield plane of order q n

  52. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Finite presemifields J. De La Cruz, Kiermaier, Wassermann, Willem : Algebraic structures of MRD codes, Adv. Math. Commun. (2016) J. Sheekey: A new family of linear maximum rank distance codes, Adv. Math. Commun. (2016) C F q -linear MRD-code, [ n × n , n , n ] F q � π S C semifield plane of order q n G 1 , s → S G 1 , s isotopic to F q n

  53. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Finite presemifields J. De La Cruz, Kiermaier, Wassermann, Willem : Algebraic structures of MRD codes, Adv. Math. Commun. (2016) J. Sheekey: A new family of linear maximum rank distance codes, Adv. Math. Commun. (2016) C F q -linear MRD-code, [ n × n , n , n ] F q � π S C semifield plane of order q n G 1 , s → π S G 1 , s Desarguesian plane

  54. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalized Twisted Gabidulin Codes Sheekey, 2016 H k , s ( η, h ) = { f ( x ) = a 0 x + a 1 x q s + · · · + a k − 1 x q s ( k − 1 ) + η a q h 0 x q sk : a i ∈ F q n } n , k , s ∈ Z + , N q ( η ) � = ( − 1 ) nk k < n , gcd ( n , s ) = 1,

  55. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalized Twisted Gabidulin Codes Sheekey, 2016 H k , s ( η, h ) = { f ( x ) = a 0 x + a 1 x q s + · · · + a k − 1 x q s ( k − 1 ) + η a q h 0 x q sk : a i ∈ F q n } n , k , s ∈ Z + , N q ( η ) � = ( − 1 ) nk k < n , gcd ( n , s ) = 1,

  56. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalized Twisted Gabidulin Codes Sheekey, 2016 H k , s ( η, h ) = { f ( x ) = a 0 x + a 1 x q s + · · · + a k − 1 x q s ( k − 1 ) + η a q h 0 x q sk : a i ∈ F q n } n , k , s ∈ Z + , N q ( η ) � = ( − 1 ) nk k < n , gcd ( n , s ) = 1, H k , s ( η, h ) MRD-code [ n × n , kn , n − k + 1 ] F q

  57. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalized Twisted Gabidulin Codes Sheekey, 2016 H k , s ( η, h ) = { f ( x ) = a 0 x + a 1 x q s + · · · + a k − 1 x q s ( k − 1 ) + η a q h 0 x q sk : a i ∈ F q n } n , k , s ∈ Z + , N q ( η ) � = ( − 1 ) nk k < n , gcd ( n , s ) = 1, H k , s ( η, h ) MRD-code [ n × n , kn , n − k + 1 ] F q H k , s ( 0 , h ) = G k , s

  58. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalized Twisted Gabidulin Codes Sheekey, 2016 H k , s ( η, h ) = { f ( x ) = a 0 x + a 1 x q s + · · · + a k − 1 x q s ( k − 1 ) + η a q h 0 x q sk : a i ∈ F q n } n , k , s ∈ Z + , N q ( η ) � = ( − 1 ) nk k < n , gcd ( n , s ) = 1, H k , s ( η, h ) MRD-code [ n × n , kn , n − k + 1 ] F q H k , s ( 0 , h ) = G k , s a 0 x + η a q h 0 x q s H 1 , s ( η, h ) → Generalized twisted field

  59. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalized Twisted Gabidulin Codes Sheekey, 2016 H k , s ( η, h ) = { f ( x ) = a 0 x + a 1 x q s + · · · + a k − 1 x q s ( k − 1 ) + η a q h 0 x q sk : a i ∈ F q n } n , k , s ∈ Z + , N q ( η ) � = ( − 1 ) nk k < n , gcd ( n , s ) = 1, H k , s ( η, h ) MRD-code [ n × n , kn , n − k + 1 ] F q H k , s ( 0 , h ) = G k , s a 0 x + η a q h 0 x q s H 1 , s ( η, h ) → Generalized twisted field H 2 ( η, 1 ) K. Otal, F. Özbudak : Explicit Construction of Some Non-Gabidulin Linear Maximum Rank Distance Codes, Adv. Math. Commun. (2016)

  60. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalized Twisted Gabidulin Codes Sheekey 2016/ Lunardon-Trombetti-Zhou, ArXiv (2015)

  61. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalized Twisted Gabidulin Codes Sheekey 2016/ Lunardon-Trombetti-Zhou, ArXiv (2015) If η � = 0, then H k , s ( η, h ) �≃ G k , s unless k ∈ { 1 , n − 1 } and h ∈ { 0 , 1 } .

  62. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalized Twisted Gabidulin Codes Sheekey 2016/ Lunardon-Trombetti-Zhou, ArXiv (2015) If η � = 0, then H k , s ( η, h ) �≃ G k , s unless k ∈ { 1 , n − 1 } and h ∈ { 0 , 1 } . Lunardon-Trombetti-Zhou (2017),JACO

  63. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalized Twisted Gabidulin Codes Sheekey 2016/ Lunardon-Trombetti-Zhou, ArXiv (2015) If η � = 0, then H k , s ( η, h ) �≃ G k , s unless k ∈ { 1 , n − 1 } and h ∈ { 0 , 1 } . Lunardon-Trombetti-Zhou (2017),JACO H k , s ( η, h )

  64. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalized Twisted Gabidulin Codes Sheekey 2016/ Lunardon-Trombetti-Zhou, ArXiv (2015) If η � = 0, then H k , s ( η, h ) �≃ G k , s unless k ∈ { 1 , n − 1 } and h ∈ { 0 , 1 } . Lunardon-Trombetti-Zhou (2017),JACO H k , s ( η, h ) If η = 0, then H k , s ( 0 , h ) = G k , s and L ( G k , s ) = R ( G k , s ) ≃ F q n .

  65. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Generalized Twisted Gabidulin Codes Sheekey 2016/ Lunardon-Trombetti-Zhou, ArXiv (2015) If η � = 0, then H k , s ( η, h ) �≃ G k , s unless k ∈ { 1 , n − 1 } and h ∈ { 0 , 1 } . Lunardon-Trombetti-Zhou (2017),JACO H k , s ( η, h ) If η = 0, then H k , s ( 0 , h ) = G k , s and L ( G k , s ) = R ( G k , s ) ≃ F q n . If η � = 0 and 1 < k < n − 1, then L ( H k , s ( η, h )) ≃ F q gcd ( n , h ) and R ( H k , s ( η, h )) ≃ F q gcd ( n , sk − h ) .

  66. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Subfamilies of H H = {H k , s ( η, h ) ⊆ L n , q }

  67. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Subfamilies of H H = {H k , s ( η, h ) ⊆ L n , q } G = {G k , s ⊆ L n , q }

  68. Linear MRD-codes Generalized Gabidulin Codes and linearized polynomials Generalized Twisted Gabidulin Codes MRD-codes-Maxim Subfamilies of H H = {H k , s ( η, h ) ⊆ L n , q } G = {G k , s ⊆ L n , q } H L = {C ∈ H : L ( C ) ≃ F q n } H R = {C ∈ H : R ( C ) ≃ F q n }

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