MDS Conjecture and the Projective Line Iwan Duursma Department of - PowerPoint PPT Presentation

MDS Conjecture and the Projective Line Iwan Duursma Department of Mathematics and Coordinated Science Laboratory University of Illinois at Urbana-Champaign Dagstuhl 2016

This talk MDS codes over finite alphabet of size q = p h Uniform k out of n reconstruction threshold, attractive for Secret Sharing, Network Coding, DSS, LRC, . . . Conjectured upper bound n ≤ q + 1 for the code length [Segre 1955] Upper bound holds for k ≤ p [B1, Ball 2012] and for p ≤ k ≤ 2 p − 2 [B2, Ball and de Beule 2012] Preliminary part of the proof: analyse 2-dimensional subcodes [B1] Main part of the proof: Obtain contradictions from glueing several 2-dimensional subcodes [B1], [B2] We simplify both parts of the proof

MDS codes A k × n matrix is MDS (generates a MDS code) if all its k × k minors are invertible. 1 The columns of an MDS matrix form an arc (a set of points in projective space such that no hyperplane contains more than the expected number of points) A MDS matrix (or arc) is of normal rational type if (possibly after row operations) each of the n columns is of the form ( x k − 1 , x k − 2 y , . . . , xy k − 2 , y k − 1 ) , for n distinct projective points ( x , y ) = ( x 1 , y 1 ) , . . . , ( x n , y n ) . For n = q + 1, a matrix of normal rational type generates a doubly-extended Reed-Solomon code 1 We assume throughout 2 ≤ k ≤ n − 2.

Plücker relations The minors of a matrix satisfy Plücker relations. � x 1 � � � x 2 x 3 x 4 x i x j � � For M = , let p i , j = � = x i y j − y i x j . � � y 1 y 2 y 3 y 4 y i y j � Then ( p 1 , 4 , p 2 , 4 , p 3 , 4 , 0 ) ∈ row M and � � p 1 , 4 p 2 , 4 p 3 , 4 � � � � 0 = x 1 x 2 x 3 = p 1 , 4 p 2 , 3 − p 2 , 4 p 1 , 3 + p 3 , 4 p 1 , 2 . � � � � y 1 y 2 y 3 � � The relation excludes a 2 × 4 MDS matrix over F 2 since 0 � = 1 · 1 − 1 · 1 + 1 · 1 .

Simeon Ball 2012 [B1] Simeon Ball, On sets of vectors of a finite vector space in which every subset of basis size is a basis, JEMS 2012. Theorem (a) Let M be a k × n MDS matrix over a field K of size q = p h . Then n ≤ q + 1 for k ≤ p . 2 (b) Moreover, for k ≤ p , equality n = q + 1 holds if and only if M is of normal rational type. 2 And n ≤ q + 1 + k − p for k ≥ p

Simeon Ball and Jan de Beule 2012 [B2] Simeon Ball and Jan de Beule, On sets of vectors of a finite vector space in which every subset of basis size is a basis II, DCC 2012. Theorem (c) Let M be a k × n MDS matrix over a field K of size q = p h . Then n ≤ q + 1 for p ≤ k ≤ 2 p − 2.

References [S] Beniamino Segre, Curve razionali normali e k-archi negli spazi finiti, Ann. Mat. Pura Appl. 1955. [RS] Ron Roth and Gadiel Seroussi, On Generator Matrices of MDS Codes, IEEE-IT 1985. [RL] Ron Roth and Abraham Lempel, A Construction of Non-Reed-Solomon Type MDS Codes, IEEE-IT 1989. [B1, Simeon Ball], [B2, Simeon Ball and Jan De Beule] [B] Simeon Ball, Finite geometry and combinatorial applications, LMS Student Text 82, Cambridge 2015. [C] Ameera Chowdhury, Inclusion Matrices and the MDS Conjecture, arXiv 2015. [B3] Simeon Ball, Extending small arcs to large arcs, arXiv 2016 [B4] Simeon Ball and Jan De Beule, On subsets of the normal rational curve, arXiv 2016

Interpolation (used in [B1], [B2], [C]) The Plücker relation 0 = p 1 , 4 p 2 , 3 − p 2 , 4 p 1 , 3 + p 3 , 4 p 1 , 2 , interpolates the linear function p i , 4 = y 4 x i − x 4 y i in the three points ( x i , y i ) , i = 1 , 2 , 3. After division p 1 , 4 p 2 , 4 p 3 , 4 0 = + + . p 1 , 2 p 1 , 3 p 2 , 1 p 2 , 3 p 3 , 1 p 3 , 2 The formula generalizes, for nonzero Plücker coordinates, to interpolation formulas for polynomials of higher degree. 3 3 Use Cramer’s formula, details included in [B1],[B], [C]

Segre’s Tangent Lemma (used in [B1], [B2], [C]) A relation among three 2-dimensional subcodes. We give the coordinate free version of [B1]. Lemma Let M be a 3 × n MDS matrix. For distinct i , j ∈ [ n ] , let T i , j = { c ∈ row M : c i = 0 , c j = 1 , c k � = 0 for k � = i , j } . Then, for distinct i , j , k ∈ [ n ] , � � � c j = ( − 1 ) t + 1 c k c i c ∈ T i , j c ∈ T j , k c ∈ T k , i where t = | T i , j | = | T j , k | = | T k , i | = q + 2 − n .

Different preliminaries [D, MDS codes and the projective line, preprint] 4 Lemma Let F ( x ) and G ( x ) be polynomials over K dividing x q − x and let E ( x ) = gcd ( F ( x ) , G ( x )) . If deg F + deg G ≥ q + 2 then 1 1 � G ′ ( α ) = 0 . F ′ ( α ) E ( α )= 0 The lemma is formulated for the affine line. A modified version holds for the projective line. 4 Reference for all that follows

Proof for the lemma Proof. Let Ff = Gg = x q − x . For α with F ( α ) = G ( α ) = 0, F ′ ( α ) f ( α ) = G ′ ( α ) g ( α ) = − 1 , and thus 1 1 G ′ ( α ) = f ( α ) g ( α ) . F ′ ( α ) For α ∈ K , F ( α ) = G ( α ) = 0 if and only if f ( α ) g ( α ) � = 0 . Thus the sum becomes � α ∈ K f ( α ) g ( α ) and this equals zero for 0 ≤ deg ( fg ) ≤ q − 2 .

Relations between two arcs F and G For an arc F of rank k and for C ⊂ F of size k − 1, define the norm � det ( x ′ , C ) . N F ( C ) = x ′ ∈ F \ C Lemma Let F , G ⊂ P k − 1 be arcs of rank k with | F | + | G | = q − 1 + 2 k, and let F ∩ G = A ∪ E be a partition with | A | = k − 2 . Then � N F ( C ) − 1 N G ( C ) − 1 . 0 = ( C = A ∪ x ) x ∈ E

Remarks The lemma holds for arcs F and G with sufficiently large intersection, it is not necessary that G ⊂ F . Since ( − 1 ) q + 1 = 1, the product N F ( C ) − 1 N G ( C ) − 1 does not depend on the ordering of the elements in C .

The proofs for (a), (b), (c) To prove each of (a) Let M be a k × n MDS matrix over a field K of size q = p h . Then n ≤ q + 1 for k ≤ p . (b) Moreover, for k ≤ p , equality n = q + 1 holds if and only if M is of normal rational type. (c) Let M be a k × n MDS matrix over a field K of size q = p h . Then n ≤ q + 1 for p ≤ k ≤ 2 p − 2. We need three further relations similar to � N F ( C ) − 1 N G ( C ) − 1 . 0 = x ∈ E

(a) The case k ≤ p Proposition Let F , G be arcs of rank k with | F | + | G | = q − 1 + 2 k, and let F ∩ G = A ∪ D ∪ E be a partition with | A | = k − 2 − r and | D | = r for some 0 ≤ r ≤ k − 2 . Then � N F ( C ) − 1 N G ( C ) − 1 . 0 = ( r + 1 )! ( C = A ∪ X ) X ∈ ( E r + 1 ) ∼ [B1, Lemma 4.1].

(c) The case p ≤ k ≤ 2 p − 2 Proposition Let F , G be arcs of rank k with | F | + | G | = q − 1 + 2 k, and let F ∩ G = A ∪ Y ∪ Z ∪ D ∪ E be a partition with | A | = k − 2 − m − r, | Y | = | Z | = m, and | D | = r, for some m ≥ 0 and 0 ≤ r ≤ k − 2 − m. Then � � ( − 1 ) | τ | N F ( C ) − 1 N G ( C ) − 1 . 0 = ( r + 1 )! ( C = A τ ∪ X ) X ∈ ( E τ ⊂ [ m ] r + 1 ) For a subset τ ⊂ [ m ] = { 1 , 2 , . . . , m } , A τ = A ∪ Y τ ∪ Z ¯ τ , where Y τ ⊂ Y is the subset of elements of Y with index in τ and Z ¯ τ ⊂ Z is the subset of elements of Z with index in the complement ¯ τ = [ m ] \ τ . ∼ [B2, Lemma 4.1]

(b) Arcs of length n = q + 1 are of normal rational type Proposition Let F be an arc of length q + 1 . The arc F is equivalent to a set of points on a normal rational curve if and only if for every subarc G ⊂ F of size 2 k − 2 and for every partition G = A ∪ E with | A | = k − 2 , | E | = k, � N F ( C ) − 1 N G ( C ) − 1 . 0 = ( C = E \ x ) x ∈ E ∼ [RS], [RL]

Conclusion The only MDS codes that we understand well are those of normal rational type. This includes all MDS codes of dimension 2. Leveraging this knowledge in a clever way ([S], [B1], [B2]) we can make statements about arbitrary MDS codes of dimension k up to 2 p − 2. We wrote the main lemmas of [B1], [B2] in a different format and in [D] give short proofs . For arcs F and G with large intersection, any choice of U ⊂ G such that | F | + | G \ U | = q − 1 + 2 k yields a relation. (approach used in [C], [B3] to obtain further results) The relations for a pair of arcs F and G hold more generally for any combination of arcs. Can this be used to push results further?