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 = ph 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 ≤ 2p − 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 (xk−1, xk−2y, . . . , xyk−2, yk−1), for n distinct projective points (x, y) = (x1, y1), . . . , (xn, yn). For n = q + 1, a matrix of normal rational type generates a doubly-extended Reed-Solomon code

1We assume throughout 2 ≤ k ≤ n − 2.

Plücker relations

The minors of a matrix satisfy Plücker relations. For M = x1 x2 x3 x4 y1 y2 y3 y4

• , let pi,j =
• xi

xj yi yj

• = xiyj − yixj.

Then (p1,4, p2,4, p3,4, 0) ∈ row M and 0 =

• p1,4

p2,4 p3,4 x1 x2 x3 y1 y2 y3

• = p1,4p2,3 − p2,4p1,3 + p3,4p1,2.

The relation excludes a 2 × 4 MDS matrix over F2 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 = ph. 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.

2And 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 = ph. Then n ≤ q + 1 for p ≤ k ≤ 2p − 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 = p1,4p2,3 − p2,4p1,3 + p3,4p1,2, interpolates the linear function pi,4 = y4xi − x4yi in the three points (xi, yi), i = 1, 2, 3. After division 0 = p1,4 p1,2p1,3 + p2,4 p2,1p2,3 + p3,4 p3,1p3,2 . The formula generalizes, for nonzero Plücker coordinates, to interpolation formulas for polynomials of higher degree.3

3Use 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 Ti,j = {c ∈ row M : ci = 0, cj = 1, ck = 0 for k = i, j}. Then, for distinct i, j, k ∈ [n],

• c∈Ti,j

ck

• c∈Tj,k

ci

• c∈Tk,i

cj = (−1)t+1 where t = |Ti,j| = |Tj,k| = |Tk,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 xq − x and let E(x) = gcd(F(x), G(x)). If deg F + deg G ≥ q + 2 then

• E(α)=0

1 F ′(α) 1 G′(α) = 0. The lemma is formulated for the affine line. A modified version holds for the projective line.

4Reference for all that follows

Proof for the lemma

Proof. Let Ff = Gg = xq − x. For α with F(α) = G(α) = 0, F ′(α)f(α) = G′(α)g(α) = −1, and thus 1 F ′(α) 1 G′(α) = f(α)g(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 NF(C) =

• x′∈F\C

det(x′, C). Lemma Let F, G ⊂ Pk−1 be arcs of rank k with |F| + |G| = q − 1 + 2k, and let F ∩ G = A ∪ E be a partition with |A| = k − 2. Then 0 =

• x∈E

NF(C)−1NG(C)−1. (C = A ∪ x)

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 NF(C)−1NG(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 = ph. 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 = ph. Then n ≤ q + 1 for p ≤ k ≤ 2p − 2. We need three further relations similar to 0 =

• x∈E

NF(C)−1NG(C)−1.

(a) The case k ≤ p

Proposition Let F, G be arcs of rank k with |F| + |G| = q − 1 + 2k, 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 0 = (r + 1)!

• X∈( E

r+1)

NF(C)−1NG(C)−1. (C = A ∪ X) ∼ [B1, Lemma 4.1].

(c) The case p ≤ k ≤ 2p − 2

Proposition Let F, G be arcs of rank k with |F| + |G| = q − 1 + 2k, 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 0 = (r+1)!

• X∈( E

r+1)

• τ⊂[m]

(−1)|τ|NF(C)−1NG(C)−1. (C = Aτ∪X) For a subset τ ⊂ [m] = {1, 2, . . . , m}, Aτ = A ∪ Yτ ∪ Z¯

τ, where

Yτ ⊂ 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

• f points on a normal rational curve if and only if for every

subarc G ⊂ F of size 2k − 2 and for every partition G = A ∪ E with |A| = k − 2, |E| = k, 0 =

• x∈E

NF(C)−1NG(C)−1. (C = E\x) ∼ [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 2p − 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 + 2k 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?