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

1Technical University of Munich, Germany 2Institute for Information Transmission Problems, Russia 3Skolkovo 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 = 2ℓ and Fq be a field of size q. Fix an integer m ≥ 1. Let X = (X1, . . . , Xm) and Fq[X] denote the ring of polynomials

• ver Fq.

Denote the set of lines in Fm

q by

Lm =

• (αT + β)|T∈Fq for α, β ∈ Fm

q

• .

For k < q, define the set of univariate polynomials of degree less than k Fq(k) = {f (T) ∈ Fq[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 Fq is the code LRSq(m, k) =

• (f (a))|a∈Fm

q : f (X) ∈ Fq[X] such that

∀L ∈ Lm : f |L ∈ Fq(k)

• .

𝑀𝑆𝑇𝑟(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 LRS4(2, 3) code. g(X1, X2) = X 2

1 X2

g|L = g(α1T + β1, α2T + β2) = (α1T + β1)2(α2T + β2) = (α2

1T 2 + β2 1)(α2T + β2)

= α2

1α2T 3 + α2 1β2T 2 + α2β2 1T + β2 1β2,

For α1 = α2 = 1, deg(g|L) = 3 = k and, thus, (g(a))|a∈F2

4 is not a codeword of LRS4(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 LRS4(2, 3) code. f (X1, X2) = X 2

1 X 2 2

f |L = f (α1T + β1, α2T + β2) = (α1T + β1)2(α2T + β2)2 = (α2

1T 2 + β2 1)(α2 2T 2 + β2 2)

= (α2

1β2 2 + α2 2β2 1)T 2 + (α2 1α2 2 + β2 1β2 2),

Thus, deg(f |L) ≤ 2 < k and, indeed, c = (f (a))|a∈F2

4 is a codeword of LRS4(2, 3). Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 7 / 25

Bad monomials Useful notations and definitions

Recall q = 2ℓ. Denote the binary representation of a ∈ Zq by (a(ℓ−1), . . . , a(0))2. Partial order relation ≤2 on Zq and Zm

q

For integers a, b ∈ Zq, we write a ≤2 b if a(i) ≤ b(i) for all i ∈ [0, ℓ). For vectors a, b ∈ Zm

q , we write a ≤2 b if aj ≤2 bj 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≥ → Zq as follows a (mod∗ q) =

• 0,

if a = 0, 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 Fq[T].

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 9 / 25

Bad monomials k∗-bad and good monomials

For d ∈ Zm

q , abbreviate the monomial Xd = m i=1X di i

∈ Fq[X]. Let deg(d) = m

i=1 di.

k∗-bad monomials We say that a monomial Xd with d ∈ Zm

q is k∗-bad if there exists i ∈ Zm q such that i ≤2 d and

deg(i) (mod∗ q) ∈ {k, k + 1, . . . , q − 1}. k-bad monomials We say that a monomial Xd with d ∈ Zm

q is k-bad if there exists i ∈ Zm 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

Bad monomials k∗-bad and good monomials

For d ∈ Zm

q , abbreviate the monomial Xd = m i=1X di i

∈ Fq[X]. Let deg(d) = m

i=1 di.

k∗-bad monomials We say that a monomial Xd with d ∈ Zm

q is k∗-bad if there exists i ∈ Zm q such that i ≤2 d and

deg(i) (mod∗ q) ∈ {k, k + 1, . . . , q − 1}. k-bad monomials We say that a monomial Xd with d ∈ Zm

q is k-bad if there exists i ∈ Zm 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

Bad monomials k∗-bad and good monomials

For d ∈ Zm

q , abbreviate the monomial Xd = m i=1X di i

∈ Fq[X]. Let deg(d) = m

i=1 di.

k∗-bad monomials We say that a monomial Xd with d ∈ Zm

q is k∗-bad if there exists i ∈ Zm q such that i ≤2 d and

deg(i) (mod∗ q) ∈ {k, k + 1, . . . , q − 1}. k-bad monomials We say that a monomial Xd with d ∈ Zm

q is k-bad if there exists i ∈ Zm 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

Bad monomials k∗-bad and good monomials

On connection between bad monomials and LRS codes

Lemma 1 (Guo-Kopparty-Sudan’2013) The LRSq(m, k) code includes the evaluation of polynomials from the linear span of k∗-good monomials over Fq[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

Bad monomials k∗-bad and good monomials

On connection between bad monomials and LRS codes

Lemma 1 (Guo-Kopparty-Sudan’2013) The LRSq(m, k) code includes the evaluation of polynomials from the linear span of k∗-good monomials over Fq[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

Bad monomials k∗-bad and good monomials

On connection between bad monomials and LRS codes

Fix an integer r ≤ m. Let k = q − r = 2ℓ − r. Let Sj(ℓ) be a subset of the k-bad tuples Sj(ℓ) = {d ∈ Zm

q : ∃i ≤2 d with deg(i) = k + jq}

Let sj(ℓ) = |Sj(ℓ)|. The number of k-bad monomials is then bounded by s0(ℓ) from one side and by m−1

i=0 si(ℓ)

from the other side.

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 12 / 25

Bad monomials k∗-bad and good monomials

On connection between bad monomials and LRS codes

Fix an integer r ≤ m. Let k = q − r = 2ℓ − r. Let Sj(ℓ) be a subset of the k-bad tuples Sj(ℓ) = {d ∈ Zm

q : ∃i ≤2 d with deg(i) = k + jq}

Let sj(ℓ) = |Sj(ℓ)|. The number of k-bad monomials is then bounded by s0(ℓ) from one side and by m−1

i=0 si(ℓ)

from the other side.

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 12 / 25

How to count bad monomials Notations

Let b

≥a

• be the number of ways to choose an (unordered) subset of at least a elements from a

fixed set of b elements. For a < 0 or a > b, we assume that b

a

• = 0.

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 13 / 25

How to count bad monomials Proposition

Proposition 1 The system of recurrence relations

       s0(ℓ + 1) s1(ℓ + 1) . . . sj(ℓ + 1) . . . sm−1(ℓ + 1)        = Am        s0(ℓ) s1(ℓ) . . . sj(ℓ) . . . sm−1(ℓ)       

holds true, where the square m × m matrix Am is given by

Am =            

m

≥1

• m
• ...

m

≥3

• m

2

• m

1

• m
• ...

. . . . . . . . . . . . ... . . .

• m

≥2j+1

• m

2j

• m

2j−1 m 2j−2

• ...
• m

2j−m+2

• .

. . . . . . . . . . . ... . . .

• m

≥2m−1 m 2m−2 m 2m−3 m 2m−4

• ...

m

m

• 

           .

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 14 / 25

How to count bad monomials Proposition

Proposition 1 implies that         s0(ℓ + 1) s1(ℓ + 1) . . . sj(ℓ + 1) . . . sm−1(ℓ + 1)         = Aℓ

m

        s0(1) s1(1) . . . sj(1) . . . sm−1(1)         Let Λ be the set of eigenvalues of Am. We define λm to be the largest element from Λ. It follows directly from the structure of Am that 2m−1 ≤ λm ≤ 2m.

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 15 / 25

How to count bad monomials Proposition

Corollary 1 For an integer r ≤ m, the number of (q − r)-bad monomials is Θ(λℓ

m) = Θ(qlog λm) as q → ∞.

Let r ≤ q (the restriction r ≤ m is no longer necessary, i.e., r could be very large). Corollary 2 For an integer r < q, the number of (q − r)∗-bad monomials is Θ(rm−log λmqlog λm) as q → ∞.

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 16 / 25

Code rate and distance of lifted Reed-Solomon codes Code rate and distance of lifted RS codes

Theorem 1 (Code rate and distance of lifted RS code) The length n, the rate R and the minimal distance dmin of the LRSq(m, q − r) code are n = qm, R = 1 − Θ

• (q/r)log λm−m

as q → ∞, dmin ≥ r q n. This theorem improves the estimate of the code rate R = 1 − O((q/r)pm) presented in (Guo-Kopparty-Sudan’2013) for m ≥ 3 and is consistent with the result of (Polyanskii-Vorobyev’2019) for m = 3.

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 17 / 25

Code rate and distance of lifted Reed-Solomon codes Code rate and distance of lifted RS codes

Theorem 1 (Code rate and distance of lifted RS code) The length n, the rate R and the minimal distance dmin of the LRSq(m, q − r) code are n = qm, R = 1 − Θ

• (q/r)log λm−m

as q → ∞, dmin ≥ r q n. This theorem improves the estimate of the code rate R = 1 − O((q/r)pm) presented in (Guo-Kopparty-Sudan’2013) for m ≥ 3 and is consistent with the result of (Polyanskii-Vorobyev’2019) for m = 3.

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 17 / 25

Code rate and distance of lifted Reed-Solomon codes Code rate and distance of lifted RS codes

Figure: The largest eigenvalue λm of Am, the resulting convergence rate m − log(λm) derived in this

work, and the convergence rate pm of (Guo-Kopparty-Sudan’2013) for different values of m.

m λm m − log(λm) pm 2 3.0000 4.1504 × 10−1 4.1504 × 10−1 3 7.2361 1.4479 × 10−1 1.1360 × 10−2 4 15.5436 4.1747 × 10−2 2.8233 × 10−3 5 31.7877 9.6043 × 10−3 4.6986 × 10−4 6 63.9217 1.7653 × 10−3 1.1742 × 10−4 7 127.9763 2.6714 × 10−4 2.9353 × 10−5 8 255.9939 3.4467 × 10−5 2.8664 × 10−8 9 511.9986 3.8959 × 10−6 2.6872 × 10−9

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 18 / 25

Batch codes based on lifted Reed-Solomon codes Introduction to batch codes

Batch codes (Ishai et al’2004) Let C be a code of length N and dimension K over Fq, which encodes a string x to a string c. C is called an s-batch code, if for every multiset request {xi1, . . . , xis} with ij ∈ [K], there exist s mutually disjoint sets R1, . . . , Rs ⊂ [N] and functions φ1, . . . , φs such that for all c ∈ C and for all j ∈ [s], xij = φj(c|Rj).

Storage Requested symbols

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 19 / 25

Batch codes based on lifted Reed-Solomon codes Introduction to batch codes

Applications of batch codes

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 20 / 25

Batch codes based on lifted Reed-Solomon codes Batch codes based on lifted Reed-Solomon codes

Theorem 2 (Batch codes from lifted Reed-Solomon codes) Fix integers q, m and r < q. The LRSq(m, q − r) code is an s-batch code for s = qm−2r.

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 21 / 25

Batch codes based on lifted Reed-Solomon codes Batch codes based on lifted Reed-Solomon codes

Sketch of the proof of Theorem 2

Suppose LRSq(m, q − r) is given and symbols {xi1, . . . , xis} are requested. We will recover this request symbol-by-symbol. Because of the linearity of the lifted RS code, we can assume that {xi1, . . . , xis} are codeword symbols, i.e. xij = f (aj) for some aj ∈ Fm

q .

Take a line L1 going through the point a1. To recover symbol xi1 = f (a1), we read all symbols f (b) with b ∈ L1 \ {a1}. Since f |L1 has degree less than q − 1, we can reconstruct the univariate polynomial f |L1 and evaluate f (a1) = xi1. By induction and averaging arguments, there exists a line Lj passing through aj and containing at most j/qm−2 points from L1, . . . , Lj−1. Since j ≤ s = qm−2r and f |Lj has degree less than q − r, it is possible to interpolate f |Lj by reading unused points on Lj and to evaluate f (aj) = xij

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 22 / 25

Batch codes based on lifted Reed-Solomon codes Batch codes based on lifted Reed-Solomon codes

Sketch of the proof of Theorem 2

Suppose LRSq(m, q − r) is given and symbols {xi1, . . . , xis} are requested. We will recover this request symbol-by-symbol. Because of the linearity of the lifted RS code, we can assume that {xi1, . . . , xis} are codeword symbols, i.e. xij = f (aj) for some aj ∈ Fm

q .

Take a line L1 going through the point a1. To recover symbol xi1 = f (a1), we read all symbols f (b) with b ∈ L1 \ {a1}. Since f |L1 has degree less than q − 1, we can reconstruct the univariate polynomial f |L1 and evaluate f (a1) = xi1. By induction and averaging arguments, there exists a line Lj passing through aj and containing at most j/qm−2 points from L1, . . . , Lj−1. Since j ≤ s = qm−2r and f |Lj has degree less than q − r, it is possible to interpolate f |Lj by reading unused points on Lj and to evaluate f (aj) = xij

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 22 / 25

Batch codes based on lifted Reed-Solomon codes Batch codes based on lifted Reed-Solomon codes

Sketch of the proof of Theorem 2

Suppose LRSq(m, q − r) is given and symbols {xi1, . . . , xis} are requested. We will recover this request symbol-by-symbol. Because of the linearity of the lifted RS code, we can assume that {xi1, . . . , xis} are codeword symbols, i.e. xij = f (aj) for some aj ∈ Fm

q .

Take a line L1 going through the point a1. To recover symbol xi1 = f (a1), we read all symbols f (b) with b ∈ L1 \ {a1}. Since f |L1 has degree less than q − 1, we can reconstruct the univariate polynomial f |L1 and evaluate f (a1) = xi1. By induction and averaging arguments, there exists a line Lj passing through aj and containing at most j/qm−2 points from L1, . . . , Lj−1. Since j ≤ s = qm−2r and f |Lj has degree less than q − r, it is possible to interpolate f |Lj by reading unused points on Lj and to evaluate f (aj) = xij

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 22 / 25

Batch codes based on lifted Reed-Solomon codes Batch codes based on lifted Reed-Solomon codes

Sketch of the proof of Theorem 2

Suppose LRSq(m, q − r) is given and symbols {xi1, . . . , xis} are requested. We will recover this request symbol-by-symbol. Because of the linearity of the lifted RS code, we can assume that {xi1, . . . , xis} are codeword symbols, i.e. xij = f (aj) for some aj ∈ Fm

q .

Take a line L1 going through the point a1. To recover symbol xi1 = f (a1), we read all symbols f (b) with b ∈ L1 \ {a1}. Since f |L1 has degree less than q − 1, we can reconstruct the univariate polynomial f |L1 and evaluate f (a1) = xi1. By induction and averaging arguments, there exists a line Lj passing through aj and containing at most j/qm−2 points from L1, . . . , Lj−1. Since j ≤ s = qm−2r and f |Lj has degree less than q − r, it is possible to interpolate f |Lj by reading unused points on Lj and to evaluate f (aj) = xij

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 22 / 25

Batch codes based on lifted Reed-Solomon codes Batch codes based on lifted Reed-Solomon codes

Theorem 3 (Asymptotics of parameters) Given a positive integer m, for any real ε with m−2

m

≤ ε < m−1

m

and a power of two q, there exists an K ε-batch code of length N = qm and dimension K over Fq such that the redundancy, N − K, satisfies N − K = O

• K (m−log λm)ε+((m−1) log λm/m−m+2)

. Up to our best knowledge, this gives the best known redundancy of s-batch codes when s = K ε and 0.27 ≤ ε ≤ 0.65

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 23 / 25

Batch codes based on lifted Reed-Solomon codes Batch codes based on lifted Reed-Solomon codes

Theorem 3 (Asymptotics of parameters) Given a positive integer m, for any real ε with m−2

m

≤ ε < m−1

m

and a power of two q, there exists an K ε-batch code of length N = qm and dimension K over Fq such that the redundancy, N − K, satisfies N − K = O

• K (m−log λm)ε+((m−1) log λm/m−m+2)

. Up to our best knowledge, this gives the best known redundancy of s-batch codes when s = K ε and 0.27 ≤ ε ≤ 0.65

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 23 / 25

Conclusion

Conclusion

In this paper, the code rate of lifted Reed-Solomon codes has been investigated. Our results are two-fold.

• 1. We have improved the estimate on the rate of the m-dimensional lifts of the RS codes

when the field size is large. In particular, we have shown that for r = O(1), the LRSq(m, q − r) code has rate 1 − Θ(qlog λm−m) as q → ∞.

• 2. Additionally, we have shown that a LRSq(m, q − r) code is also a s-batch code with

s = rqm−2. This improves the known upper bounds on the redundancy of batch codes in some parameter regimes.

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 24 / 25

Conclusion

Thank you for your attention!

Contact us if you have any questions: Emails: lukas.holzbaur@tum.de, rev-rina@yandex.ru, nikita.polyansky@gmail.com, vorobyev.i.v@yandex.ru

Rina Polyanskaya (IITP RAS) ISIT 2020 June 21 - 26, 2020 25 / 25