G ENERALIZED R EED -S OLOMON CODES (GRS CODES ) A CHARACTERIZATION OF - PowerPoint PPT Presentation

A CHARACTERIZATION OF MDS CODES THAT HAVE AN ERROR CORRECTING PAIR I NTRODUCTION TO C ODING T HEORY MDS CODES A CHARACTERIZATION OF MDS CODES THAT GRS CODES HAVE AN ERROR CORRECTING PAIR ECP M OTIVATION O UR GOAL ARQUEZ -C ORBELLA 1 R. P ELLIKAAN 2 I. M ´ 1Department of Algebra, Geometry and Topology, University of Valladolid. Supported by a FPU grant AP2008-01598 by Spanish MEC. 2Department of Mathematics and Computing Science, Eindhoven University of Technology. Code-based Cryptography Workshop (CBC) 2012

I NTRODUCTION TO C ODING T HEORY A CHARACTERIZATION OF MDS CODES THAT HAVE AN An [ n , k ] linear code C over F q is a k -dimensional subspace of F n q . ERROR CORRECTING PAIR Its size is M = q k , the information rate is R = k n and the redundancy is n − k . I NTRODUCTION TO C ODING The generator matrix of C is a k × n matrix G whose rows form a basis of C , T HEORY i.e. MDS CODES � � x G | x ∈ F k C = . q GRS CODES The parity-check matrix of C is an ( n − k ) × n matrix H whose nullspace is ECP generated by the codewords of C , i.e. M OTIVATION O UR GOAL � q | H y T = 0 � y ∈ F n C = . The hamming distance between x , y ∈ F n q is d H ( x , y ) = |{ i | x i � = y i }| . The minimum distance of C is d ( C ) = min { d H ( c 1 , c 2 ) | c 1 , c 2 ∈ C and c 1 � = c 2 } . y y x 1 x 2 x 1 x 2 F IGURE : If d ( C ) = 3 F IGURE : If d ( C ) = 4

One of the most fascinating chapters in MDS CODES all of coding theory A CHARACTERIZATION OF MDS CODES THAT HAVE AN ERROR CORRECTING PAIR Let C be a linear code over F q , we will denote: I NTRODUCTION TO C ODING ➜ Its length by n ( C ) ➜ Its dimension by k ( C ) ➜ Its minimum distance by d ( C ) T HEORY MDS CODES GRS CODES S INGLETON B OUND ECP d ( C ) ≤ n ( C ) − k ( C ) + 1 M OTIVATION If the equality holds = ⇒ C is an MDS code . O UR GOAL E XAMPLES The zero code of length n (i.e. the [ n , 0 , n + 1 ] linear code) 1 and its dual (i.e. F n q which has parameters [ n , n , 1 ] ). The [ n , 1 , n ] repetition code over F q . 2 The (Extended / Generalized) Reed-Solomon codes . 3 F. J. MacWilliams, N. J. A. Sloane The theory of error-correcting codes II . North-Holland Mathematical Library, Vol 16.

MDS CODES A CHARACTERIZATION OF MDS CODES THAT HAVE AN ERROR CORRECTING PAIR I NTRODUCTION TO C ODING T HEORY MDS CODES A collection of some properties characterizing MDS codes: GRS CODES T HEOREM : P ROPERTIES OF MDS CODES ECP Let C be an [ n , k ] code over F q . The following are equivalent: M OTIVATION C is MDS. O UR GOAL 1 C ⊥ is MDS. 2 Every k -tuple of columns of a generator matrix of C is independent. 3 Every set of k coordinates form an information set. 4 Every n − k -tuple of columns of a parity check matrix of C is independent. 5

M ODIFYING C ODES A CHARACTERIZATION OF ➜ Let C be a linear [ n , k ] code over F q and ( J , J ) be a partition of { 1 , . . . , n } MDS CODES THAT HAVE AN where J = { i 1 , . . . , i m } ⊆ { 1 , . . . , n } has m elements. ERROR CORRECTING PAIR � � the restriction of any vector x ∈ F n ➜ We denote by x J = x i 1 , . . . , x im q to the I NTRODUCTION TO C ODING coordinates indexed by J . T HEORY ➜ Via the operation of puncturing and shortening we can obtained codes of MDS CODES shorter lenght from C . GRS CODES P UNCTURING A CODE ( C J ) ECP M OTIVATION We can punctured C by deleting columns from a generator matrix of C i.e. O UR GOAL C J = � c J | c ∈ C � = ⇒ C J is an [ n ( C ) − m , k ( C J ) , d ( C J )] code with d ( C ) − m ≤ d ( C J ) ≤ d ( C ) and k ( C ) − m ≤ k ( C J ) ≤ k ( C ) ➜ Moreover if m < d ( C ) then k ( C J ) = k ( C ) . � C J � S HORTENING A CODE We can shorten C by deleting columns from a parity check matrix of C . Thus the words of C J are codewords of the initial code that have a zero in the J -location, i.e. C J = C J is an [ n ( C ) − m , k ( C J ) , d ( C J )] code with � � c J | c ∈ C and c J = 0 ⇒ d ( C ) ≤ d ( C J ) k ( C ) − m ≤ k ( C J ) ≤ k ( C ) and

M ODIFYING C ODES A CHARACTERIZATION OF MDS CODES THAT HAVE AN ERROR CORRECTING PAIR I NTRODUCTION TO C ODING T HEORY S OME PROPERTIES OF THESE OPERATIONS MDS CODES C J ⊆ C J . GRS CODES 1 ECP dim ( C J ) + dim � C J � = dim ( C ) . 2 M OTIVATION ( C J ) ⊥ = ( C ) J and ( C J ) ⊥ = ( C ⊥ ) J . 3 O UR GOAL L EMMA 1 Let C be an MDS code. If n ( C ) − m ≥ k ( C ) , then C J and C J are MDS codes with parameters: [ n ( C ) − m , k ( C )] and [ n ( C ) − m , k ( C ) − m ] , respectively.

G ENERALIZED R EED -S OLOMON CODES (GRS CODES ) A CHARACTERIZATION OF MDS CODES THAT HAVE AN ERROR CORRECTING PAIR Let a = ( a 1 , . . . , a n ) be an n -tuple of mutually distinct elements of P 1 ( F q ) . I NTRODUCTION TO C ODING b = ( b 1 , . . . , b n ) be an n -tuple of nonzero elements of F q . T HEORY MDS CODES The GRS code GRS k ( a , b ) is defined by: GRS CODES ECP GRS k ( a , b ) = { ( f ( a 1 ) b 1 , . . . , f ( a n ) b n ) | f ∈ F q [ X ] and deg ( f ) < k } M OTIVATION O UR GOAL T HEOREM : P ARAMETERS OF GRS k ( a , b ) ➜ The GRS k ( a , b ) is an MDS code with parameters [ n , k , n − k + 1 ] . ➜ Furthermore a generator matrix of GRS k ( a , b ) is given by b 1 bn − 1 0   . . . b 1 bn   b 1 a 1 bn − 1 an − 1 0 . . . . . .   b 1 a 1 bnan     . . . . . .      . . .   . .  G a , b = . . . or     . . . . .     . . . . .     b 1 ak − 2 bn − 1 ak − 2  0    b 1 ak − 1 bnak − 1  1 . . . n − 1    1 n b 1 ak − 1 bn − 1 ak − 1 . . . 1 1 . . . n − 1 if a n = ∞ .

G ENERALIZED R EED -S OLOMON CODES (GRS CODES ) A CHARACTERIZATION OF MDS CODES THAT HAVE AN ERROR CORRECTING PAIR I NTRODUCTION TO C ODING P ROPOSITION GRS T HEORY MDS CODES We have GRS k ( a , b ) ⊥ = GRS n − k ( a , s ) GRS CODES where s = ( s 1 , . . . , s n ) with s − 1 � = b i j � = i ( a i − a j ) . ECP i M OTIVATION O UR GOAL P ROPOSITION If 2 ≤ k ≤ n − 2 then a representation of a GRS code is unique up to a fractional map of the projective line that induces an automorphism of the code, i.e. ➜ Different values of a and b gives rise to the same GRS code. ➜ But... the pair ( a , b ) is unique up to the action of fractional transformations.

N OTATION A CHARACTERIZATION OF MDS CODES THAT HAVE AN ERROR CORRECTING PAIR I NTRODUCTION TO C ODING T HEORY ➜ For all a , b ∈ F n q we define: MDS CODES GRS CODES Star Multiplication: a ∗ b = ( a 1 b 1 , . . . , a n b n ) ∈ F n q . ECP M OTIVATION Standard Inner Multiplication: a · b = � n i = 1 a i b i . O UR GOAL ➜ For all subsets A , B ⊆ F n q we define: A ∗ B = { a ∗ b | a ∈ A and b ∈ B } . A ⊥ B ⇐ ⇒ a · b = 0 ∀ a ∈ A and b ∈ B .

E RROR - CORRECTING PAIRS (ECP) A CHARACTERIZATION OF MDS CODES THAT HAVE AN ERROR CORRECTING PAIR E RROR - CORRECTING PAIRS (ECP) I NTRODUCTION TO C ODING Let C be an F q linear code of length n . The pair ( A , B ) of F qN -linear codes of length n T HEORY is a t -ECP for C over F qN if the following properties hold: MDS CODES GRS CODES ECP E.1 ( A ∗ B ) ⊥ C . An [ n , k ] code which has a t -ECP over F qN M OTIVATION E.2 k ( A ) > t . has a decoding algorithm with complexity O UR GOAL � ( nN ) 3 � E.3 d ( B ⊥ ) > t . O . E.4 d ( A )+ d ( C ) > n . R. Pellikaan On decoding by error location and dependent sets of error positions . Discrete Math., 106–107: 369–381 (1992). R. K¨ otter. A unified description of an error locating procedure for linear codes . In Proceedings of Algebraic and Combinatorial Coding Theory, 113–117. Voneshta Voda (1992).

E XAMPLES OF THE EXISTENCE OF ECP A CHARACTERIZATION OF MDS CODES THAT HAVE AN 1. GRS CODES ERROR CORRECTING PAIR Let C = GRS 2 t ( a , b 1 ∗ b 2 ) ⊥ I NTRODUCTION TO C ODING A = GRS t + 1 ( a , b 1 ) , B = GRS t ( a , b 2 ) and T HEORY then ( A , B ) is a t -ECP for C . MDS CODES GRS CODES Conversely, let C = GRS k ( a , b ) then ECP M OTIVATION A = GRS t + 1 ( a , b ′ ) and B = GRS t ( a , 1 ) O UR GOAL and b ′ ∈ ( F q \ { 0 } ) n verifies that � � n − k is a t -ECP for C where t = 2 GRS k ( a , b ) ⊥ = GRS n − k ( a , b ′ ) . 2. C YCLIC - CODES I. Duursma Decoding codes from curves and cyclic codes . Ph.D thesis, Eindhoven University of Technology R. K¨ otter. (1993) On algebraic decoding of algebraic-geometric and cyclic codes . I. Duursma, R. K¨ otter. Ph.D thesis, Link¨ oping University of Technology (1996). Error-locating pairs for cyclic codes . IEEE Trans. Inform. Theory, Vol.40, 1108–1121 (1994)