ON THE MDS CONJECTURE Academy Contact Forum on Coding Theory and - PowerPoint PPT Presentation

ON THE MDS CONJECTURE Academy Contact Forum on Coding Theory and Cryptography VII Jan De Beule October 6, 2017

Codes Definition A (block) code is a set of words of equal length over an alphabet. Definition The Hamming distance between two codewords is the number of positions in which they differ. Definition The minimum distance of a code C is min { d ( x , y ) | x , y ∈ C } Jan De Beule On the MDS conjecture 1/ 58

Singleton bound Theorem (Singleton bound) Let C be a q-ary ( n , M , d ) -code.

Singleton bound Theorem (Singleton bound) Let C be a q-ary ( n , M , d ) -code. Then M ≤ q n − d + 1 Jan De Beule On the MDS conjecture 2/ 58

Linear codes Definition A (linear) [ n , k , d ] q -code is the set of vectors of a k -dimensional subspace of the n -dimensional vector space over F q and which has minimum distance d . Definition Let C be a linear [ n , k , d ] q code. Then C ⊥ = { x ∈ F n q | v · x = 0 ∀ v ∈ C } Jan De Beule On the MDS conjecture 3/ 58

Main theorem of linear codes Definition The parity check matrix of a linear code is the generator matrix of its dual code. Theorem Let C be a linear [ n , k ] code over F q . Then its minimum weight is d if and only if every set of d − 1 columns of the parity check matrix of C is linearly independent and there exists at least one set of d linearly dependent columns. Jan De Beule On the MDS conjecture 4/ 58

Singleton bound Theorem (Singleton bound) Let C be a q-ary ( n , M , d ) -code. Then k ≤ n − d + 1 . Jan De Beule On the MDS conjecture 5/ 58

MDS codes Definition A (linear) MDS code is a (linear) code achieving equality in the Singleton bound. Jan De Beule On the MDS conjecture 6/ 58

Examples of MDS codes Example � � A linear code over F q , generated by , 1 ≤ i ≤ k , g ij 1 ≤ j ≤ n , with g ij = λ j t i − 1 , j t j ∈ F q , λ j ∈ F q \ { 0 } is a Reed-Solomon code, of length n , dimension k over F q , and with minimum distance d = n − k + 1. Jan De Beule On the MDS conjecture 7/ 58

Examples of MDS codes . . .   1 1 1 1 . . . 0 t 1 t 2 t n − 1     . . . t 2 t 2 t 2 G = 0  n − 1  1 2   . . . . ...   . . . . . . . .   t k − 1 t k − 2 t k − 1 . . . 0 n − 1 1 2 RS ( n , k , n − k + 1 ) Jan De Beule On the MDS conjecture 7/ 58

Examples of MDS codes . . .   1 1 1 1 . . . 0 t 1 t 2 t q − 1    . . .  t 2 t 2 t 2 G = 0  q − 1  1 2   . . . . ...   . . . . . . . .   t k − 1 t k − 1 t k − 1 . . . 0 q − 1 1 2 RS ( q , k , q − k + 1 ) Jan De Beule On the MDS conjecture 7/ 58

Examples of MDS codes   . . . 1 1 1 1 0 . . .  t 1 t 2 t q − 1 0 0    . . . t 2 t 2 t 2   0 0 H = q − 1 1 2    . . . . .  ... . . . . .   . . . . .   t q − k t q − k t q − k . . . 0 1 q − 1 1 2 RS ( q + 1 , k , q − k + 2 ) Jan De Beule On the MDS conjecture 7/ 58

Examples of MDS codes  . . .  1 1 1 1 0 0 H = . . . t 1 t 2 t q − 1 0 1 0   . . . t 2 t 2 t 2 0 0 1 q − 1 1 2 RS ( q + 2 , q − 1 , 4 ) , only for q = 2 h Jan De Beule On the MDS conjecture 7/ 58

Arcs in vector spaces Definition A set S of vectors in F k q is an arc if every subset of S of size k is a basis for F k q . Jan De Beule On the MDS conjecture 8/ 58

Arcs in vector spaces Definition A set S of vectors in F k q is an arc if every subset of S of size k is a basis for F k q . Definition A set K of points in PG ( k − 1 , q ) is an arc if every subset of K of size k spans PG ( k − 1 , q ) Jan De Beule On the MDS conjecture 8/ 58

Arcs and linear MDS codes Theorem An arc of size n in F k q is equivalent with a linear [ n , n − k , k + 1 ] q code C. The dual code C ⊥ is a linear MDS code, with parameters [ n , k , n − k + 1 ] , and hence equivalent to an arc of size n in F n − k . q Jan De Beule On the MDS conjecture 9/ 58

Examples Example (frame) S = { ( λ 1 , 0 , . . . , 0 ) , ( 0 , λ 2 , . . . , 0 ) , . . . , ( 0 , 0 , . . . , λ k ) , ( 1 , 1 , . . . , 1 ) } , λ i ∈ F ∗ q Jan De Beule On the MDS conjecture 10/ 58

Examples Example (frame) S = { ( λ 1 , 0 , . . . , 0 ) , ( 0 , λ 2 , . . . , 0 ) , . . . , ( 0 , 0 , . . . , λ k ) , ( 1 , 1 , . . . , 1 ) } , λ i ∈ F ∗ q Example (normal rational curve) S = { ( 1 , t , t 2 , . . . , t k − 1 ) | t ∈ F q } ∪ { ( 0 , . . . , 0 , 1 ) } Jan De Beule On the MDS conjecture 10/ 58

Early results Theorem (Bose, 1947) Let S be an arc of F 3 q , q not even. Then | S | ≤ q + 1 . Jan De Beule On the MDS conjecture 11/ 58

Early results Theorem (Bose, 1947) Let S be an arc of F 3 q , q not even. Then | S | ≤ q + 1 . Theorem (Bush, 1952) Let S be an arc of F k q , k ≥ q. Then | S | ≤ k + 1 . In case of equality, S is equivalent with a frame. Jan De Beule On the MDS conjecture 11/ 58

Theorem (Segre, 1955) Let S be an arc of F 3 q of size q + 1 , and let q be odd. Then S is equivalent with a normal rational curve. This theorem confirms a conjecture of Järnefelt and Kustaanheimo (1952). Jan De Beule On the MDS conjecture 12/ 58

Theorem (Segre, 1955) Let S be an arc of F 3 q of size q + 1 , and let q be odd. Then S is equivalent with a normal rational curve. This theorem confirms a conjecture of Järnefelt and Kustaanheimo (1952). Theorem (Segre, 1955) Let K be an arc of PG ( 2 , q ) of size q + 1 , and let q be odd. Then K is equivalent with a conic. Jan De Beule On the MDS conjecture 12/ 58

One more example Lemma Let q be even. Consider an arc K of size q + 1 , then there exists a nucleus , i.e. a point n such that every line on n meets K in exactly one point. Jan De Beule On the MDS conjecture 13/ 58

One more example Lemma Let q be even. Consider an arc K of size q + 1 , then there exists a nucleus , i.e. a point n such that every line on n meets K in exactly one point. Definition Let q be even. An arc of PG ( 2 , q ) of size q + 1 is called an oval .

One more example Lemma Let q be even. Consider an arc K of size q + 1 , then there exists a nucleus , i.e. a point n such that every line on n meets K in exactly one point. Definition Let q be even. An arc of PG ( 2 , q ) of size q + 1 is called an oval . An arc of size q + 2 is called a hyperoval . Jan De Beule On the MDS conjecture 13/ 58

Segre’s lemma of tangents Jan De Beule On the MDS conjecture 14/ 58

Segre’s lemma of tangents Let q be odd, and Let S be an arc of size q + 1. Lemma (Segre, 1955) abc = − 1 Jan De Beule On the MDS conjecture 14/ 58

Segre’s lemma of tangents (1967) Let q be odd, and Let S be an arc of size at most q + 1. Let t := q + 2 − | S | . Jan De Beule On the MDS conjecture 15/ 58

Segre’s lemma of tangents (1967) Let q be odd, and Let S be an arc of size at most q + 1. Let t := q + 2 − | S | . Lemma (Segre, 1967) t � a i b i c i = − 1 i = 1 Corollary An arc of size q + 1 in PG ( 2 , q ) , q odd, is necessarily the set of points of a conic. Jan De Beule On the MDS conjecture 15/ 58

Questions of Segre (i) What is the upper bound on the size for an arc in F k q ? (ii) For which values of k , q , q > k , is each ( q + 1 ) -arc in F k q a normal rational curve? (iii) For a given k , q , q > k , which arcs of F k q are extendable to a ( q + 1 ) -arc? Jan De Beule On the MDS conjecture 16/ 58

MDS conjecture Conjecture (MDS conjecture) Let k ≤ q. An arc of F k q has size at most q + 1 , unless q is even and k = 3 or k = q − 1 , in which case it has size at most q + 2 . Jan De Beule On the MDS conjecture 17/ 58

MDS conjecture Conjecture (MDS conjecture) Let k ≤ q. An arc of F k q has size at most q + 1 , unless q is even and k = 3 or k = q − 1 , in which case it has size at most q + 2 . Conjecture A linear MDS code of dimension k over F q has length at most q + 1 , unless k = 3 or k = q − 1 , in which case it has length at most q + 2 . Jan De Beule On the MDS conjecture 17/ 58

summary of old and more recent results ◮ The MDS conjecture is known to be true for all q ≤ 27, for all k ≤ 5 and k ≥ q − 3 and for k = 6 , 7 , q − 4 , q − 5, see overview paper of J. Hirschfeld and L. Storme, pointing to results of Segre, J.A. Thas, Casse, Glynn, Bruen, Blokhuis, Voloch, Storme, Hirschfeld and Korchmáros. ◮ many examples of hyperovals , see e.g. Cherowitzo’s hyperoval page, pointing to examples of Segre, Glynn, Payne, Cherowitzo, Penttila, Pinneri, Royle and O’Keefe. ◮ Recent results on hyperovals over small fields by Peter Vandendriessche. Jan De Beule On the MDS conjecture 18/ 58

Some more examples ◮ An example of a ( q + 1 ) -arc in F 5 9 , different from a normal rational curve, (Glynn): S = { ( 1 , t , t 2 + η t 6 , t 3 , t 4 ) | t ∈ F 9 , η 4 = − 1 } ∪ { ( 0 , 0 , 0 , 0 , 1 ) } ◮ An example of a ( q + 1 ) -arc in F 4 q , q = 2 h , gcd ( r , h ) = 1, different from a normal rational curve, (Hirschfeld): S = { ( 1 , t , t 2 r , t 2 r + 1 ) | t ∈ F q } ∪ { ( 0 , 0 , 0 , 1 ) } ◮ Segre oval: q = 2 h , gcd ( i , h ) = 1 S = { ( 1 , t , t 2 i ) | t ∈ F q } ∪ { ( 0 , 0 , 1 ) } Jan De Beule On the MDS conjecture 19/ 58