Lifted Reed-Solomon Codes with Application to Batch Codes Lukas - PowerPoint PPT Presentation

Lifted Reed-Solomon Codes with Application to Batch Codes Lukas Holzbaur 1 Rina Polyanskaya 2 Nikita Polyanskii 1,3 Ilya Vorobyev 3 1 Technical University of Munich, Germany 2 Institute for Information Transmission Problems, Russia 3 Skolkovo Institute of Science and Technology, Russia Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 1 / 25

Outline 1. Introduction to lifted Reed-Solomon codes. 2. Bad monomials notion and its connection with lifted Reed-Solomon codes. 3. How to count bad monomials. 4. Code rate and distance of lifted Reed-Solomon codes. 5. Batch codes based on lifted Reed-Solomon codes. Construction and asymptotic behaviour. Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 2 / 25

Outline 1. Introduction to lifted Reed-Solomon codes. 2. Bad monomials notion and its connection with lifted Reed-Solomon codes. 3. How to count bad monomials. 4. Code rate and distance of lifted Reed-Solomon codes. 5. Batch codes based on lifted Reed-Solomon codes. Construction and asymptotic behaviour. Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 2 / 25

Outline 1. Introduction to lifted Reed-Solomon codes. 2. Bad monomials notion and its connection with lifted Reed-Solomon codes. 3. How to count bad monomials. 4. Code rate and distance of lifted Reed-Solomon codes. 5. Batch codes based on lifted Reed-Solomon codes. Construction and asymptotic behaviour. Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 2 / 25

Outline 1. Introduction to lifted Reed-Solomon codes. 2. Bad monomials notion and its connection with lifted Reed-Solomon codes. 3. How to count bad monomials. 4. Code rate and distance of lifted Reed-Solomon codes. 5. Batch codes based on lifted Reed-Solomon codes. Construction and asymptotic behaviour. Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 2 / 25

Outline 1. Introduction to lifted Reed-Solomon codes. 2. Bad monomials notion and its connection with lifted Reed-Solomon codes. 3. How to count bad monomials. 4. Code rate and distance of lifted Reed-Solomon codes. 5. Batch codes based on lifted Reed-Solomon codes. Construction and asymptotic behaviour. Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 2 / 25

Lifted Reed-Solomon Codes Lifting and basic notations Let q = 2 ℓ and F q be a field of size q . Fix an integer m ≥ 1. Let X = ( X 1 , . . . , X m ) and F q [ X ] denote the ring of polynomials over F q . Denote the set of lines in F m q by � ( α T + β ) | T ∈ F q for α , β ∈ F m � L m = . q For k < q , define the set of univariate polynomials of degree less than k F q ( k ) = { f ( T ) ∈ F q [ T ] : deg( f ) < k } . Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 3 / 25

Lifted Reed-Solomon Codes Lifted RS Code Definition Lifted Reed-Solomon code (Guo-Kopparty-Sudan’2013) The m -dimensional lift of the Reed-Solomon code over F q is the code � � q : f ( X ) ∈ F q [ X ] such that LRS q ( m , k ) = ( f ( a )) | a ∈ F m . ∀ L ∈ L m : f | L ∈ F q ( k ) 𝑀𝑆𝑇 𝑟 (2, 𝑙) code Red , yellow and green are codewords of the RS code of length 𝑟 and dimension 𝑙 Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 4 / 25

Lifted Reed-Solomon Codes Lifted RS Code Definition Lifted Reed-Solomon codes have found numerous applications for constructing Locally correctible codes Locally testable codes Codes with the disjoint-repair-group-property Private information retrieval codes We also show that they are good as batch codes. Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 5 / 25

Lifted Reed-Solomon Codes Lifted RS Code Definition Lifted Reed-Solomon codes have found numerous applications for constructing Locally correctible codes Locally testable codes Codes with the disjoint-repair-group-property Private information retrieval codes We also show that they are good as batch codes. Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 5 / 25

Lifted Reed-Solomon Codes Examples Example 1 Let m = 2, k = 3 and q = 4. Consider possible codewords of the LRS 4 (2 , 3) code. g ( X 1 , X 2 ) = X 2 1 X 2 g | L = g ( α 1 T + β 1 , α 2 T + β 2 ) = ( α 1 T + β 1 ) 2 ( α 2 T + β 2 ) 1 T 2 + β 2 = ( α 2 1 )( α 2 T + β 2 ) 1 α 2 T 3 + α 2 1 β 2 T 2 + α 2 β 2 = α 2 1 T + β 2 1 β 2 , For α 1 = α 2 = 1, deg ( g | L ) = 3 = k and, thus, ( g ( a )) | a ∈ F 2 4 is not a codeword of LRS 4 (2 , 3). Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 6 / 25

Lifted Reed-Solomon Codes Examples Example 2 Let m = 2, k = 3 and q = 4. Consider possible codewords of the LRS 4 (2 , 3) code. f ( X 1 , X 2 ) = X 2 1 X 2 2 f | L = f ( α 1 T + β 1 , α 2 T + β 2 ) = ( α 1 T + β 1 ) 2 ( α 2 T + β 2 ) 2 1 T 2 + β 2 2 T 2 + β 2 = ( α 2 1 )( α 2 2 ) 1 ) T 2 + ( α 2 = ( α 2 1 β 2 2 + α 2 2 β 2 1 α 2 2 + β 2 1 β 2 2 ) , Thus, deg ( f | L ) ≤ 2 < k and, indeed, c = ( f ( a )) | a ∈ F 2 4 is a codeword of LRS 4 (2 , 3). Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 7 / 25

Bad monomials Useful notations and definitions Recall q = 2 ℓ . Denote the binary representation of a ∈ Z q by ( a ( ℓ − 1) , . . . , a (0) ) 2 . Partial order relation ≤ 2 on Z q and Z m q For integers a , b ∈ Z q , we write a ≤ 2 b if a ( i ) ≤ b ( i ) for all i ∈ [0 , ℓ ). For vectors a , b ∈ Z m q , we write a ≤ 2 b if a j ≤ 2 b j for all j ∈ [ m ]. Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 8 / 25

Bad monomials Useful notations and definitions By Z ≥ denote the set of non-negative integers. Operation ( mod ∗ q ) Define an operation ( mod ∗ q ) : Z ≥ → Z q as follows � 0 , if a = 0 , a ( mod ∗ q ) = b ∈ [ q − 1] , if a � = 0 , a = b ( mod q − 1) . Obviously, if a (mod ∗ q ) = b , then T a = T b ( mod T q − T ) in F q [ T ]. Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 9 / 25

k ∗ -bad and good monomials Bad monomials q , abbreviate the monomial X d = � m i =1 X d i For d ∈ Z m ∈ F q [ X ]. i Let deg( d ) = � m i =1 d i . k ∗ -bad monomials We say that a monomial X d with d ∈ Z m q is k ∗ -bad if there exists i ∈ Z m q such that i ≤ 2 d and deg( i ) ( mod ∗ q ) ∈ { k , k + 1 , . . . , q − 1 } . k -bad monomials We say that a monomial X d with d ∈ Z m q is k-bad if there exists i ∈ Z m q such that i ≤ 2 d and deg( i ) = k (mod q ). A monomial is said to be k ∗ -good (or k-good ) if it is not k ∗ -bad (or k -bad). Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 10 / 25

k ∗ -bad and good monomials Bad monomials q , abbreviate the monomial X d = � m i =1 X d i For d ∈ Z m ∈ F q [ X ]. i Let deg( d ) = � m i =1 d i . k ∗ -bad monomials We say that a monomial X d with d ∈ Z m q is k ∗ -bad if there exists i ∈ Z m q such that i ≤ 2 d and deg( i ) ( mod ∗ q ) ∈ { k , k + 1 , . . . , q − 1 } . k -bad monomials We say that a monomial X d with d ∈ Z m q is k-bad if there exists i ∈ Z m q such that i ≤ 2 d and deg( i ) = k (mod q ). A monomial is said to be k ∗ -good (or k-good ) if it is not k ∗ -bad (or k -bad). Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 10 / 25

k ∗ -bad and good monomials Bad monomials q , abbreviate the monomial X d = � m i =1 X d i For d ∈ Z m ∈ F q [ X ]. i Let deg( d ) = � m i =1 d i . k ∗ -bad monomials We say that a monomial X d with d ∈ Z m q is k ∗ -bad if there exists i ∈ Z m q such that i ≤ 2 d and deg( i ) ( mod ∗ q ) ∈ { k , k + 1 , . . . , q − 1 } . k -bad monomials We say that a monomial X d with d ∈ Z m q is k-bad if there exists i ∈ Z m q such that i ≤ 2 d and deg( i ) = k (mod q ). A monomial is said to be k ∗ -good (or k-good ) if it is not k ∗ -bad (or k -bad). Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 10 / 25

k ∗ -bad and good monomials Bad monomials On connection between bad monomials and LRS codes Lemma 1 (Guo-Kopparty-Sudan’2013) The LRS q ( m , k ) code includes the evaluation of polynomials from the linear span of k ∗ -good monomials over F q [ X ]. Thus, the code rate of the lifted Reed-Solomon code is equal to the fraction of good monomials. Lemma 2 (Informal) For q − m ≤ k < q , the number of k ∗ -bad monomials can be well approximated by the number of k -bad monomials. Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 11 / 25

k ∗ -bad and good monomials Bad monomials On connection between bad monomials and LRS codes Lemma 1 (Guo-Kopparty-Sudan’2013) The LRS q ( m , k ) code includes the evaluation of polynomials from the linear span of k ∗ -good monomials over F q [ X ]. Thus, the code rate of the lifted Reed-Solomon code is equal to the fraction of good monomials. Lemma 2 (Informal) For q − m ≤ k < q , the number of k ∗ -bad monomials can be well approximated by the number of k -bad monomials. Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 11 / 25

k ∗ -bad and good monomials Bad monomials On connection between bad monomials and LRS codes Fix an integer r ≤ m . Let k = q − r = 2 ℓ − r . Let S j ( ℓ ) be a subset of the k -bad tuples S j ( ℓ ) = { d ∈ Z m q : ∃ i ≤ 2 d with deg( i ) = k + jq } Let s j ( ℓ ) = | S j ( ℓ ) | . The number of k -bad monomials is then bounded by s 0 ( ℓ ) from one side and by � m − 1 i =0 s i ( ℓ ) from the other side. Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 12 / 25