Rank Metric Codes and related Structures Yue Zhou July 5, 2017 The - PowerPoint PPT Presentation

Gabidulin codes Definition A linearized polynomial ( q -polynomial) is in F q n [ X ] of the form a 0 X + a 1 X q + · · · + a i X q i + · · · . Let L ( n , q ) [ X ] denote all linearized polynomials in F q n [ X ]. • L ( n , q ) [ X ] / ( X q n − X ) ∼ = End F q ( F q n ). • Gabidulin codes ( k = n − d + 1, m = n ) G = { a 0 X + a 1 X q + . . . a k − 1 X q k − 1 : a 0 , a 1 , . . . , a k − 1 ∈ F q n } . 6/34

Gabidulin codes Definition A linearized polynomial ( q -polynomial) is in F q n [ X ] of the form a 0 X + a 1 X q + · · · + a i X q i + · · · . Let L ( n , q ) [ X ] denote all linearized polynomials in F q n [ X ]. • L ( n , q ) [ X ] / ( X q n − X ) ∼ = End F q ( F q n ). • Gabidulin codes ( k = n − d + 1, m = n ) G = { a 0 X + a 1 X q + . . . a k − 1 X q k − 1 : a 0 , a 1 , . . . , a k − 1 ∈ F q n } . ⊲ For each f ∈ G , f has at most q k − 1 roots. 6/34

Gabidulin codes Definition A linearized polynomial ( q -polynomial) is in F q n [ X ] of the form a 0 X + a 1 X q + · · · + a i X q i + · · · . Let L ( n , q ) [ X ] denote all linearized polynomials in F q n [ X ]. • L ( n , q ) [ X ] / ( X q n − X ) ∼ = End F q ( F q n ). • Gabidulin codes ( k = n − d + 1, m = n ) G = { a 0 X + a 1 X q + . . . a k − 1 X q k − 1 : a 0 , a 1 , . . . , a k − 1 ∈ F q n } . ⊲ For each f ∈ G , f has at most q k − 1 roots. ⊲ # G = q nk = q n ( m − d +1) with d = m − k + 1. 6/34

Gabidulin codes Definition A linearized polynomial ( q -polynomial) is in F q n [ X ] of the form a 0 X + a 1 X q + · · · + a i X q i + · · · . Let L ( n , q ) [ X ] denote all linearized polynomials in F q n [ X ]. • L ( n , q ) [ X ] / ( X q n − X ) ∼ = End F q ( F q n ). • Gabidulin codes ( k = n − d + 1, m = n ) G = { a 0 X + a 1 X q + . . . a k − 1 X q k − 1 : a 0 , a 1 , . . . , a k − 1 ∈ F q n } . ⊲ For each f ∈ G , f has at most q k − 1 roots. ⊲ # G = q nk = q n ( m − d +1) with d = m − k + 1. ⊲ Gabidulin codes are F q n -linear MRD codes. 6/34

Known families of MRD codes ( d = m = n ) When m = n = d ( k = 1), G = { a 0 ∗ X : a 0 ∈ F q n } . 7/34

Known families of MRD codes ( d = m = n ) When m = n = d ( k = 1), G = { a 0 ∗ X : a 0 ∈ F q n } . MRD codes C and the following algebraic/geometric objects are equivalent. 7/34

Known families of MRD codes ( d = m = n ) When m = n = d ( k = 1), G = { a 0 ∗ X : a 0 ∈ F q n } . MRD codes C and the following algebraic/geometric objects are equivalent. • (Pre)quasifield Q ; 7/34

Known families of MRD codes ( d = m = n ) When m = n = d ( k = 1), G = { a 0 ∗ X : a 0 ∈ F q n } . MRD codes C and the following algebraic/geometric objects are equivalent. • (Pre)quasifield Q ; ⊲ When C is F q -linear, Q is a (pre)semifield. 7/34

Known families of MRD codes ( d = m = n ) When m = n = d ( k = 1), G = { a 0 ∗ X : a 0 ∈ F q n } . MRD codes C and the following algebraic/geometric objects are equivalent. • (Pre)quasifield Q ; ⊲ When C is F q -linear, Q is a (pre)semifield. • Spreads. 7/34

Known families of MRD codes ( d = m = n ) When m = n = d ( k = 1), G = { a 0 ∗ X : a 0 ∈ F q n } . MRD codes C and the following algebraic/geometric objects are equivalent. • (Pre)quasifield Q ; ⊲ When C is F q -linear, Q is a (pre)semifield. • Spreads. There are a considerable amount of inequivalent quasifields and semifields. 7/34

Known families of MRD codes ( d = m = n ) When m = n = d ( k = 1), G = { a 0 ∗ X : a 0 ∈ F q n } . MRD codes C and the following algebraic/geometric objects are equivalent. • (Pre)quasifield Q ; ⊲ When C is F q -linear, Q is a (pre)semifield. • Spreads. There are a considerable amount of inequivalent quasifields and semifields. In particular, for q = 2 m , there are exponentially many inequivalent ones (Kantor). 7/34

Known families of F q -linear MRD codes ( d ≤ m = n ) Let m , n , k , s ∈ Z + , gcd( n , s ) = 1, k < m and q a power of prime. 8/34

Known families of F q -linear MRD codes ( d ≤ m = n ) Let m , n , k , s ∈ Z + , gcd( n , s ) = 1, k < m and q a power of prime. (generalized) twisted Gabidulin codes [Sheekey 2016]: 0 X q sk : a 0 , . . . , a k − 1 ∈ F q n } , H k , s ( η, h ) = { a 0 X + · · · + a k − 1 X q s ( k − 1) + η a q h where h ∈ Z + and η ∈ F q n is such that N q sn / q s ( η ) � = ( − 1) nk . 8/34

Known families of F q -linear MRD codes ( d ≤ m = n ) Let m , n , k , s ∈ Z + , gcd( n , s ) = 1, k < m and q a power of prime. (generalized) twisted Gabidulin codes [Sheekey 2016]: 0 X q sk : a 0 , . . . , a k − 1 ∈ F q n } , H k , s ( η, h ) = { a 0 X + · · · + a k − 1 X q s ( k − 1) + η a q h where h ∈ Z + and η ∈ F q n is such that N q sn / q s ( η ) � = ( − 1) nk . • H k , s (0 , � ) is a Gabidulin code [Delsarte 1978], [Gabidulin 1985], [Kshevetskiy and Gabidulin 2005]. 8/34

Known families of F q -linear MRD codes ( d ≤ m = n ) Let m , n , k , s ∈ Z + , gcd( n , s ) = 1, k < m and q a power of prime. (generalized) twisted Gabidulin codes [Sheekey 2016]: 0 X q sk : a 0 , . . . , a k − 1 ∈ F q n } , H k , s ( η, h ) = { a 0 X + · · · + a k − 1 X q s ( k − 1) + η a q h where h ∈ Z + and η ∈ F q n is such that N q sn / q s ( η ) � = ( − 1) nk . • H k , s (0 , � ) is a Gabidulin code [Delsarte 1978], [Gabidulin 1985], [Kshevetskiy and Gabidulin 2005]. • When q = 2, η must be 0. 8/34

Known families of F q -linear MRD codes ( d ≤ m = n ) Let m , n , k , s ∈ Z + , gcd( n , s ) = 1, k < m and q a power of prime. (generalized) twisted Gabidulin codes [Sheekey 2016]: 0 X q sk : a 0 , . . . , a k − 1 ∈ F q n } , H k , s ( η, h ) = { a 0 X + · · · + a k − 1 X q s ( k − 1) + η a q h where h ∈ Z + and η ∈ F q n is such that N q sn / q s ( η ) � = ( − 1) nk . • H k , s (0 , � ) is a Gabidulin code [Delsarte 1978], [Gabidulin 1985], [Kshevetskiy and Gabidulin 2005]. • When q = 2, η must be 0. • The equivalence between different members and the automorphism groups can be completely determined (Lunardon, Trombetti, Z) 8/34

Known families of MRD codes ( d ≤ m = n ) Nonlinear families: 9/34

Known families of MRD codes ( d ≤ m = n ) Nonlinear families: 1. Size q 2 n [Cossidente, Marino, Pavese 2016] [Durante, Siciliano]. 9/34

Known families of MRD codes ( d ≤ m = n ) Nonlinear families: 1. Size q 2 n [Cossidente, Marino, Pavese 2016] [Durante, Siciliano]. 2. Slight modifications of twisted Gabidulin codes [Otal and ¨ Ozbudak 2016]. 9/34

Known families of MRD codes ( d ≤ m = n ) Nonlinear families: 1. Size q 2 n [Cossidente, Marino, Pavese 2016] [Durante, Siciliano]. 2. Slight modifications of twisted Gabidulin codes [Otal and ¨ Ozbudak 2016]. Question Find more new MRD codes for d ≤ m = n . 9/34

Known families of MRD codes ( d ≤ m < n ) 1. Puncturing n × n MRD codes F : 10/34

Known families of MRD codes ( d ≤ m < n ) 1. Puncturing n × n MRD codes F : Take F q -linearly independent elements α 1 , . . . , α m ∈ F q n . Then C = { ( f ( α 1 ) , · · · , f ( α m )) T : f ∈ F} 10/34

Known families of MRD codes ( d ≤ m < n ) 1. Puncturing n × n MRD codes F : Take F q -linearly independent elements α 1 , . . . , α m ∈ F q n . Then C = { ( f ( α 1 ) , · · · , f ( α m )) T : f ∈ F} 2. For k = m − d + 1, randomly generate MRD codes [Neri,Trautmann,Randrianarisoa,Rosenthal,2016]. Pr > 1 − kq km − n . 10/34

Known families of MRD codes ( d ≤ m < n ) 1. Puncturing n × n MRD codes F : Take F q -linearly independent elements α 1 , . . . , α m ∈ F q n . Then C = { ( f ( α 1 ) , · · · , f ( α m )) T : f ∈ F} 2. For k = m − d + 1, randomly generate MRD codes [Neri,Trautmann,Randrianarisoa,Rosenthal,2016]. Pr > 1 − kq km − n . 3. Twisting construction using chains of subfields [Puchinger, Nielsen, Sheekey]. 10/34

Known families of MRD codes ( d ≤ m < n ) 1. Puncturing n × n MRD codes F : Take F q -linearly independent elements α 1 , . . . , α m ∈ F q n . Then C = { ( f ( α 1 ) , · · · , f ( α m )) T : f ∈ F} 2. For k = m − d + 1, randomly generate MRD codes [Neri,Trautmann,Randrianarisoa,Rosenthal,2016]. Pr > 1 − kq km − n . 3. Twisting construction using chains of subfields [Puchinger, Nielsen, Sheekey]. 4. Using maximum scattered linear sets [Csajb´ ok, Marino, Polverino, Zullo]. 10/34

Known families of MRD codes ( d ≤ m < n ) 1. Puncturing n × n MRD codes F : Take F q -linearly independent elements α 1 , . . . , α m ∈ F q n . Then C = { ( f ( α 1 ) , · · · , f ( α m )) T : f ∈ F} 2. For k = m − d + 1, randomly generate MRD codes [Neri,Trautmann,Randrianarisoa,Rosenthal,2016]. Pr > 1 − kq km − n . 3. Twisting construction using chains of subfields [Puchinger, Nielsen, Sheekey]. 4. Using maximum scattered linear sets [Csajb´ ok, Marino, Polverino, Zullo]. 5. Other constructions [Trautmann, Marshall 2016]. 10/34

Known families of MRD codes ( d ≤ m < n ) How many inequivalent MRD codes are there in F m × n ? q 11/34

Known families of MRD codes ( d ≤ m < n ) How many inequivalent MRD codes are there in F m × n ? q • By looking at Gabidulin codes for different U = � α 1 , · · · , α m � , we [Schmidt, Z] can show that this number ( q − 1) [ n m ] q ≥ n ( q n − 1) . 11/34

Known families of MRD codes ( d ≤ m < n ) How many inequivalent MRD codes are there in F m × n ? q • By looking at Gabidulin codes for different U = � α 1 , · · · , α m � , we [Schmidt, Z] can show that this number ( q − 1) [ n m ] q ≥ n ( q n − 1) . • Proved by investigating their right nuclei and middle nuclei. 11/34

Nuclei of rank metric codes Definition For rank metric codes in K m × n : Right nucleus: N r ( C ) = { Y ∈ K n × n : CY ∈ C for all C ∈ C} . 12/34

Nuclei of rank metric codes Definition For rank metric codes in K m × n : Right nucleus: N r ( C ) = { Y ∈ K n × n : CY ∈ C for all C ∈ C} . Middle nucleus: N m ( C ) = { Z ∈ K m × m : ZC ∈ C for all C ∈ C} . 12/34

Nuclei of rank metric codes Definition For rank metric codes in K m × n : Right nucleus: N r ( C ) = { Y ∈ K n × n : CY ∈ C for all C ∈ C} . Middle nucleus: N m ( C ) = { Z ∈ K m × m : ZC ∈ C for all C ∈ C} . • When C is a spreadset defining a semifield S , then N m ( C ) and N r ( C ) correspond to the middle nucleus and the right nucleus of S respectively. 12/34

Nuclei of rank metric codes Definition For rank metric codes in K m × n : Right nucleus: N r ( C ) = { Y ∈ K n × n : CY ∈ C for all C ∈ C} . Middle nucleus: N m ( C ) = { Z ∈ K m × m : ZC ∈ C for all C ∈ C} . • When C is a spreadset defining a semifield S , then N m ( C ) and N r ( C ) correspond to the middle nucleus and the right nucleus of S respectively. • For MRD codes with d < m , we can also define the left nucleus which is always K . 12/34

Nuclei of rank metric codes Definition For rank metric codes in K m × n : Right nucleus: N r ( C ) = { Y ∈ K n × n : CY ∈ C for all C ∈ C} . Middle nucleus: N m ( C ) = { Z ∈ K m × m : ZC ∈ C for all C ∈ C} . • When C is a spreadset defining a semifield S , then N m ( C ) and N r ( C ) correspond to the middle nucleus and the right nucleus of S respectively. • For MRD codes with d < m , we can also define the left nucleus which is always K . • Not invariant for nonlinear rank metric codes. 12/34

Nuclei of rank metric codes • For two equivalent linear rank metric codes C 1 and C 2 in K m × n , their right (resp. middle) nuclei are also equivalent. 13/34

Nuclei of rank metric codes • For two equivalent linear rank metric codes C 1 and C 2 in K m × n , their right (resp. middle) nuclei are also equivalent. C 2 = { AX γ B : X ∈ C 1 } ⇒ Z ∈ N m ( C 1 ) iff AZ γ A − 1 ∈ N m ( C 2 ) 13/34

Nuclei of rank metric codes • For two equivalent linear rank metric codes C 1 and C 2 in K m × n , their right (resp. middle) nuclei are also equivalent. C 2 = { AX γ B : X ∈ C 1 } ⇒ Z ∈ N m ( C 1 ) iff AZ γ A − 1 ∈ N m ( C 2 ) If γ = id and C 1 = C 2 , then A ∈ N GL ( m , q ) ( N m ( C )). 13/34

Nuclei of rank metric codes • For two equivalent linear rank metric codes C 1 and C 2 in K m × n , their right (resp. middle) nuclei are also equivalent. C 2 = { AX γ B : X ∈ C 1 } ⇒ Z ∈ N m ( C 1 ) iff AZ γ A − 1 ∈ N m ( C 2 ) If γ = id and C 1 = C 2 , then A ∈ N GL ( m , q ) ( N m ( C )). • For (generalized) Gabidulin codes G s = { a 0 X + a 1 X q s + . . . a k − 1 X q s ( k − 1) : a 0 , . . . , a k − 1 ∈ F q n } , N r ( G s ) = { g : g ◦ f ∈ G s for all f ∈ G s } ∼ = F q n , N m ( G s ) = { g : f ◦ g ∈ G s for all f ∈ G s } ∼ = F q n . 13/34

Quadratic bent-Negabent functions

Maximum rank metric codes with restrictions • Restrictions: Symmetric, symplectic, hermitian... 14/34

Maximum rank metric codes with restrictions • Restrictions: Symmetric, symplectic, hermitian... • Given minimum distance d , the upper bound of C is not completely clear. 14/34

Maximum rank metric codes with restrictions • Restrictions: Symmetric, symplectic, hermitian... • Given minimum distance d , the upper bound of C is not completely clear. For instance: 14/34

Maximum rank metric codes with restrictions • Restrictions: Symmetric, symplectic, hermitian... • Given minimum distance d , the upper bound of C is not completely clear. For instance: • Let C be an additive d -code consisting of m × m symmetric matrix over F q . If 2 ∤ q (2 | q and 2 ∤ d or d = m ), then � q m ( m − d +2) / 2 , if m − d is even; # C ≤ q ( m +1)( m − d +1) / 2 , if m − d is odd. 14/34

Maximum rank metric codes with restrictions • Restrictions: Symmetric, symplectic, hermitian... • Given minimum distance d , the upper bound of C is not completely clear. For instance: • Let C be an additive d -code consisting of m × m symmetric matrix over F q . If 2 ∤ q (2 | q and 2 ∤ d or d = m ), then � q m ( m − d +2) / 2 , if m − d is even; # C ≤ q ( m +1)( m − d +1) / 2 , if m − d is odd. • Proved by using association schemes. The upper bound is tight. (Schmidt 2010, 2015) 14/34

• Quadratic APN functions, AB functions, (vectorial) bent functions... can be considered as rank metric codes with special properties. 15/34

• Quadratic APN functions, AB functions, (vectorial) bent functions... can be considered as rank metric codes with special properties. • f : F n p → F m p is quadratic if δ f , a : x �→ f ( x + a ) − f ( x ) − f ( a ) is F p -linear for all a . 15/34

• Quadratic APN functions, AB functions, (vectorial) bent functions... can be considered as rank metric codes with special properties. • f : F n p → F m p is quadratic if δ f , a : x �→ f ( x + a ) − f ( x ) − f ( a ) is F p -linear for all a . • Quadratic APN: kernel of δ f , a is of dimension 1 for a ∈ F ∗ 2 n . 15/34

• Quadratic APN functions, AB functions, (vectorial) bent functions... can be considered as rank metric codes with special properties. • f : F n p → F m p is quadratic if δ f , a : x �→ f ( x + a ) − f ( x ) − f ( a ) is F p -linear for all a . • Quadratic APN: kernel of δ f , a is of dimension 1 for a ∈ F ∗ 2 n . • { δ f , a : a ∈ F 2 n } is a subspace of binary n × n matrices of rank n − 1. 15/34

• Quadratic APN functions, AB functions, (vectorial) bent functions... can be considered as rank metric codes with special properties. • f : F n p → F m p is quadratic if δ f , a : x �→ f ( x + a ) − f ( x ) − f ( a ) is F p -linear for all a . • Quadratic APN: kernel of δ f , a is of dimension 1 for a ∈ F ∗ 2 n . • { δ f , a : a ∈ F 2 n } is a subspace of binary n × n matrices of rank n − 1. • Quadratic AB: the set of alternating bilinear forms { Tr ( c ( f ( x + y ) − f ( x ) − f ( y ))) : c ∈ F ∗ 2 n } defines a subspace of alternating binary n × n matrices of rank n − 1. 15/34

• Quadratic APN functions, AB functions, (vectorial) bent functions... can be considered as rank metric codes with special properties. • f : F n p → F m p is quadratic if δ f , a : x �→ f ( x + a ) − f ( x ) − f ( a ) is F p -linear for all a . • Quadratic APN: kernel of δ f , a is of dimension 1 for a ∈ F ∗ 2 n . • { δ f , a : a ∈ F 2 n } is a subspace of binary n × n matrices of rank n − 1. • Quadratic AB: the set of alternating bilinear forms { Tr ( c ( f ( x + y ) − f ( x ) − f ( y ))) : c ∈ F ∗ 2 n } defines a subspace of alternating binary n × n matrices of rank n − 1. • See Edel and Dempwolff’s work: Nuclei, dimensional dual hyperovals . . . 15/34

Quadratic bent functions For f : F n 2 → F 2 , 16/34

Quadratic bent functions For f : F n 2 → F 2 , • it is bent if x �→ f ( x + a ) − f ( x ) is balanced for all nonzero a ( n has to be even). 16/34

Quadratic bent functions For f : F n 2 → F 2 , • it is bent if x �→ f ( x + a ) − f ( x ) is balanced for all nonzero a ( n has to be even). • it is quadratic bent if the alternating matrix associated with f ( x + y ) − f ( x ) − f ( y ) is nonsingular. 16/34

Quadratic bent functions For f : F n 2 → F 2 , • it is bent if x �→ f ( x + a ) − f ( x ) is balanced for all nonzero a ( n has to be even). • it is quadratic bent if the alternating matrix associated with f ( x + y ) − f ( x ) − f ( y ) is nonsingular. • all quadratic bent functions are (extended affine) equivalent to f ( x 1 , · · · , x 2 m ) = x 1 x 2 + x 3 x 4 + · · · + x 2 m − 1 x 2 m .   0 1 . . . 0 0   1 0 . . . 0 0     . . . . ...   . . . . . . . .      0 0 . . . 0 1  0 0 . . . 1 0 16/34

Quadratic bent-Negabent functions For f : F n 2 → F 2 , • it is quadratic negabent if the associated alternating matrix M is such that M + I is nonsingular. 17/34

Quadratic bent-Negabent functions For f : F n 2 → F 2 , • it is quadratic negabent if the associated alternating matrix M is such that M + I is nonsingular. • How many quadratic bent-negabent functions? (Pott, Parker 2008) 17/34

Quadratic bent-Negabent functions For f : F n 2 → F 2 , • it is quadratic negabent if the associated alternating matrix M is such that M + I is nonsingular. • How many quadratic bent-negabent functions? (Pott, Parker 2008) • The number of bent-negabent quadratic forms on F 2 m is 2 � m � m m − i � � 1 ( − 1) i 2 i ( i − 1) (2 2 k − 1 − 1) 2 . 2 m i 4 i =0 k =1 (Pott, Schmidt, Z 2016) 17/34

Quadratic bent-Negabent functions Let X j stand for the n × n alternating matrices of rank j over F q and X = � X j = F n × n . q 18/34

Quadratic bent-Negabent functions Let X j stand for the n × n alternating matrices of rank j over F q and X = � X j = F n × n . q • f is bent-negabent if and only if M and M + I + J are both nonsingular (Pott, Parker 2008). 18/34

Quadratic bent-Negabent functions Let X j stand for the n × n alternating matrices of rank j over F q and X = � X j = F n × n . q • f is bent-negabent if and only if M and M + I + J are both nonsingular (Pott, Parker 2008). • M and M + I + J are both alternating. 18/34

Quadratic bent-Negabent functions Let X j stand for the n × n alternating matrices of rank j over F q and X = � X j = F n × n . q • f is bent-negabent if and only if M and M + I + J are both nonsingular (Pott, Parker 2008). • M and M + I + J are both alternating. � � � { ( A , B ) ∈ X r × X s : A + B ∈ X k } � . • We count N X ( r , s , k ) = 18/34

Quadratic bent-Negabent functions Let X j stand for the n × n alternating matrices of rank j over F q and X = � X j = F n × n . q • f is bent-negabent if and only if M and M + I + J are both nonsingular (Pott, Parker 2008). • M and M + I + J are both alternating. � � � { ( A , B ) ∈ X r × X s : A + B ∈ X k } � . • We count N X ( r , s , k ) = • # quadratic bent-negabent functions = N X ( n , n , n ) . | X n | 18/34

Quadratic bent-Negabent functions • � � � { ( A , B ) ∈ X r × X s : A + B ∈ X k } � N X ( r , s , k ) = � � � � 1 = φ ( A ) φ ( B ) φ ( C ) . | X | A ∈ X r B ∈ X s C ∈ X k φ ∈ � X 19/34

Quadratic bent-Negabent functions • � � � { ( A , B ) ∈ X r × X s : A + B ∈ X k } � N X ( r , s , k ) = � � � � 1 = φ ( A ) φ ( B ) φ ( C ) . | X | A ∈ X r B ∈ X s C ∈ X k φ ∈ � X • All X 0 , X 1 , · · · , X n form a partition of F n × n and it is a q translation scheme. 19/34

Quadratic bent-Negabent functions • � � � { ( A , B ) ∈ X r × X s : A + B ∈ X k } � N X ( r , s , k ) = � � � � 1 = φ ( A ) φ ( B ) φ ( C ) . | X | A ∈ X r B ∈ X s C ∈ X k φ ∈ � X • All X 0 , X 1 , · · · , X n form a partition of F n × n and it is a q translation scheme. • � m 1 | � N X ( r , s , k ) = X i | P r ( i ) P s ( i ) P k ( i ) . | X | i =0 19/34