The Maximum Distance Separable (MDS) Codes Conjecture Jiyou Li - PowerPoint PPT Presentation

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes The Maximum Distance Separable (MDS) Codes Conjecture Jiyou Li lijiyou@sjtu.edu.cn Shanghai Jiao Tong University May 18th, 2013

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes Outline Introduction 1 2 Results Ball’s proof according to Ball’s slides 3 MDS codes for AG codes 4

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes A Simple Communication Model Message Source Receiver Source Encoder Channel Source Decoder

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes A Simple Communication Model: Example banana banana 00 Channel 00

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes A Simple Communication Model Message Source Receiver Source Encoder Channel Source Decoder ❇ ✓ ❇ ✓ ✬✩ ❇ ✓ ❇ ✓ Noisy! ✫✪

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes A Simple Communication Model apple banana 00 Channel 01 ❇ ✓ ❇ ✓ ✬✩ ❇ ✓ ❇ ✓ Noisy! ✫✪

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes An Error Correcting Communication Model Message Source Receiver Source Encoder Source Decoder Channel Encoder Channel Channel Decoder

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes An Example of Repetition Codes banana banana 00 00 00000 Channel 00001

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes A photo of Callisto

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes What is a code (Channel Encoder) Let F q be the finite field of q elements;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes What is a code (Channel Encoder) Let F q be the finite field of q elements; For integers 1 ≤ k ≤ n , an [ n , k ] q code C is a k -dimension subspace of F n q over F q ; C : F k q → F n q ;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes What is a code (Channel Encoder) Let F q be the finite field of q elements; For integers 1 ≤ k ≤ n , an [ n , k ] q code C is a k -dimension subspace of F n q over F q ; C : F k q → F n q ; The minimum distance d ( C ) of C is defined to be the smallest size of the support of a nonzero element in C ;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes What is a code (Channel Encoder) Let F q be the finite field of q elements; For integers 1 ≤ k ≤ n , an [ n , k ] q code C is a k -dimension subspace of F n q over F q ; C : F k q → F n q ; The minimum distance d ( C ) of C is defined to be the smallest size of the support of a nonzero element in C ; C is called an [ n , k , d ] q code if d ( C ) = d .

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes A code with minimum distance d ✬✩ ✬✩ ✬✩ ✬✩ ✬✩ ✉ ✉ ✉ ✉ r ✫✪ ✫✪ ✫✪ ✫✪ ✬✩ ✬✩ ✬✩ ✬✩ ✫✪ u ✉ ✉ ✉ ✉ d − 1 2 ✫✪ ✫✪ ✫✪ ✫✪ ✬✩ ✬✩ ✬✩ ✬✩ ✉ ✉ ✉ ✉ ✫✪ ✫✪ ✫✪ ✫✪ d

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes An Example of Repetition Code banana banana 00 00 000000 Channel 000001

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes Important parameters and MDS codes The information rate k n ;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes Important parameters and MDS codes The information rate k n ; The relative distance d n ;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes Important parameters and MDS codes The information rate k n ; The relative distance d n ; C is theoretically good if both k n and d n are large;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes Important parameters and MDS codes The information rate k n ; The relative distance d n ; C is theoretically good if both k n and d n are large; Singleton bound: k n + d n ≤ 1 + 1 n ;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes Important parameters and MDS codes The information rate k n ; The relative distance d n ; C is theoretically good if both k n and d n are large; Singleton bound: k n + d n ≤ 1 + 1 n ; If d = n − k + 1, then C is called a maximum distance separable ( MDS ) code.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes Important parameters and MDS codes The information rate k n ; The relative distance d n ; C is theoretically good if both k n and d n are large; Singleton bound: k n + d n ≤ 1 + 1 n ; If d = n − k + 1, then C is called a maximum distance separable ( MDS ) code. Examples: Reed-Solomon Codes

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes Generalized Reed-Solomon codes D = { x 1 , · · · , x n } ⊂ F q , | D | = n > 0. For 1 ≤ k ≤ n , denote by D n , k the subspace spanned by ( f ( x 1 ) , · · · , f ( x n )) ∈ F n q , where deg ( f ( x )) ≤ k − 1;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes Generalized Reed-Solomon codes D = { x 1 , · · · , x n } ⊂ F q , | D | = n > 0. For 1 ≤ k ≤ n , denote by D n , k the subspace spanned by ( f ( x 1 ) , · · · , f ( x n )) ∈ F n q , where deg ( f ( x )) ≤ k − 1; Since a polynomial of degree k − 1 has at most k − 1 roots, we have d = n − k + 1 and thus D n , k are ( MDS ) codes.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes The followings are all equivalent An MDS [ n , k , d ] linear code.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes The followings are all equivalent An MDS [ n , k , d ] linear code. A k × ( n − k ) matrix over F q such that every minor is nonzero.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes The followings are all equivalent An MDS [ n , k , d ] linear code. A k × ( n − k ) matrix over F q such that every minor is nonzero. A set of n vectors in F k q such that any k vectors in S are linearly independent.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes The followings are all equivalent An MDS [ n , k , d ] linear code. A k × ( n − k ) matrix over F q such that every minor is nonzero. A set of n vectors in F k q such that any k vectors in S are linearly independent. A set of n projective points in PG ( k − 1 , q ) such that there are at most k − 1 points in any hyperplane of PG ( k − 1 , q ) .

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes They are all equivalent   1 0 . . . 0 a 11 . . . a 1 , n − k . . . . . . 0 1 0 a 21 a 2 , n − k       0 0 . . . 1 a k 1 . . . a k , n − k k × n

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes A [ q + 1 , k , q − k + 2 ] q code  1 1 . . . 1 0  a 1 a 2 . . . a q 0    a 2 a 2 a 2  . . . 0   1 2 q  . . .  . . .   . . .   a k − 1 a k − 1 a k − 1 . . . 1 q 1 2

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes A [ q + 2 , 3 , q ] q MDS code When q is even,   1 1 . . . 1 0 0  . a 1 a 2 . . . a q 1 0  a 2 a 2 a 2 . . . 0 1 q 1 2

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes A [ q + 2 , 3 , q ] q MDS code When q is even,   1 1 . . . 1 0 0  . a 1 a 2 . . . a q 1 0  a 2 a 2 a 2 . . . 0 1 q 1 2 Question: Why not odd q ?

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes MDS conjecture Let M ( k , q ) be the maximum length n of an [ n , k , n − k + 1 ] q code;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes MDS conjecture Let M ( k , q ) be the maximum length n of an [ n , k , n − k + 1 ] q code; (Bush, 1952) If k ≥ q + 1, then M ( k , q ) = k + 1.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes MDS conjecture Let M ( k , q ) be the maximum length n of an [ n , k , n − k + 1 ] q code; (Bush, 1952) If k ≥ q + 1, then M ( k , q ) = k + 1. (Conjectured by Segre, 1955) If k ≤ q , then M ( k , q ) = q + 1,

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes MDS conjecture Let M ( k , q ) be the maximum length n of an [ n , k , n − k + 1 ] q code; (Bush, 1952) If k ≥ q + 1, then M ( k , q ) = k + 1. (Conjectured by Segre, 1955) If k ≤ q , then M ( k , q ) = q + 1, except the cases that when q is even and k = 3 or k = q − 1, in which cases M ( k , q ) = q + 2.