Nearly Optimal Constructions of PIR and Batch Codes

Transcription

1 arxiv:700706v [csit] 5 Jun 07 Neary Optima Constructions of PIR and Batch Codes Hia Asi Technion - Israe Institute of Technoogy Haifa 3000, Israe shea@cstechnionaci Abstract In this work we study two famiies of codes with avaiabiity, namey private information retrieva PIR) codes and batch codes Whie the former requires that every information symbo has k mutuay disjoint recovering sets, the atter asks this property for every mutiset request of k information symbos The main probem under this paradigm is to minimize the number of redundancy symbos We denote this vaue by r P n, k), r B n, k), for PIR, batch codes, respectivey, where n is the number of information symbos Previous resuts showed that for any constant k, r P n, k) = Θ n) and r B n, k) = O n ogn)) In this work we study the asymptotic behavior of these codes for non-constant k and specificay for k = Θn ǫ ) We aso study the argest vaue of k such that the rate of the codes approaches, and show that for a ǫ <, r P n, n ǫ ) = on), whie for batch codes, this property hods for a ǫ < 05 I INTRODUCTION In this paper we study two famiies of codes with avaiabiity for distributed storage The first famiy of codes, caed private information retrieva PIR) Codes, requires that every information symbo has some k mutuay disjoint recovering sets These codes were studied recenty in [] due to their appicabiity for private information retrieva in a coded storage system They are aso very simiar to one-step majority-ogic decodabe codes that were studied a whie ago by Massey [7] and ater by Lin and others [5] and were prompted by appications of error-correction with ow-compexity The second famiy of codes, which is a generaization of the first one, was first proposed in the ast decade by Ishai et a under the framework of batch codes [3] These codes were originay motivated by different appications such as oad-baancing in storage and cryptographic protocos Here it is required that every mutiset request of k symbos can be recovered by k mutuay disjoint recovering sets Formay, we denote a k-pir code by[n, n, k] P to be a coding scheme which encodes n information bits to N bits such that each information bit has k mutuay disjoint recovering sets Simiary, a k-batch code wi be denoted by [N, n, k] B and the requirement of mutuay disjoint recovering sets is imposed for every mutiset request of size k The main figure of merit when studying PIR and batch codes is the vaue of N, given n and k Thus, we denote by Pn, k), Bn, k) the minimum vaue of N for which an [N, n, k] P,[N, n, k] B code exists, respectivey Since it is known that for a fixed k, im n B q n, k)/n = im n P q n, k)/n =, [3], we evauate these codes by their redundancy and define r B n, k) Bn, k) n, r P n, k) Pn, k) n One of the probems we study in the paper studies the argest vaue of k as a function of n) for which one can sti have r P n, k) = on) and r B n, k) = on), so the rate of the codes approaches We show that for PIR codes this hods for k = Θn ǫ ), for a ǫ <, whie for batch codes for a ǫ < / Since r P n, k), r B n, k) k, the resut for PIR codes is indeed optima Furthermore, in order to have a better understanding Eitan Yaakobi Technion - Israe Institute of Technoogy Haifa 3000, Israe yaakobi@cstechnionaci of the asymptotic behavior of the redundancy, we study the vaues r P n, k) and r B n, k) when k = Θn ǫ ) The resuts we achieve in the paper are based on two constructions The first one uses mutipicity codes which generaized Reed Muer codes and were first presented by Kopparty et a in [4] These codes were aso used for the construction of ocay decodabe codes [] The second construction we use is based on the subcube construction from [3] This basic construction can be used to construct both PIR and batch codes Whie the idea in the works in [], [3] was to use mutidimensiona cubes in order to achieve arge vaues of k, here we take a different approach and position the information bits in a two dimensiona array and then form mutipe parity sets by taking different diagonas in the array The rest of the paper is organized as foows In Section II, we formay define the codes studied in this paper and review previous resuts In Section III, we review mutipicity codes Then, in Section IV we show how to use mutipicity codes to construct PIR codes, and in Section V we carry the same task for batch codes Then, in Section VI, we present our array construction and its resuts for PIR codes and batch codes Due to the ack of space some proofs in the paper are omitted II DEFINITIONS AND PRELIMINARIES Let F q denote the fied of size q, where q is a prime power A inear code of ength N and dimension n over F q wi be denoted by [N, n] q For binary codes we wi remove the notation of the fied The set [n] denotes the set of integers {,,, n} In this work we focus on two famiies of codes, namey private information retrieva PIR) codes that were defined recenty in [] and batch codes that were first studied by Ishai et a in [3] Formay, these codes are defined as foows Definition LetC be an[n, n] q inear code over the fiedf q ) The code C wi be caed a k-pir code, and wi be denoted by [N, n, k] q P, if for every information symbo x i, i [n], there exist k mutuay disjoint sets R i,0,, R i,k [N] such that for a j [k], x i is a function of the symbos in R i, j ) The codec wi be caed a k-batch code, and wi be denoted by [N, n, k] q B, if for every mutiset request of symbos{i 0, i,, i k }, there exist k mutuay disjoint sets R i0, R i,, R ik [N] such that for a j [k], x i j is a function of the symbos in R i j We sighty modified here the definition of batch codes In their conventiona definition, n symbos are encoded into some m tupes of strings, caed buckets, such that each batch ie request) of k information symbos can be decoded by reading at most some t symbos from each bucket In case each bucket can store a singe symbo, these codes are caed primitive batch codes, which is the setup we study here and for simpicity ca them batch codes In this work we study the binary and non-binary cases of PIR and batch codes

2 The main probem in studying PIR and batch codes is to minimize the ength N given the vaues of n and k We denote by P q n, k), B q n, k) the vaue of the smaest N such that there exists an [N, n, k] q P,[N, n, k] q B code, respectivey Since every batch code is aso a PIR code with the same parameters we get that B q n, k) P q n, k) For the binary case, we wi remove q from these and subsequent notations In [3], it was shown using the subcube construction that for any fixed k there exists an asymptoticay optima construction of [N, n, k] q B batch code, and hence im B qn, k)/n = im P q n, k)/n = n n Therefore, it is important to study how fast the rate of these codes converges to one, and so the redundancy of PIR and batch codes is studied We define r B n, k) q to be the vaue r B n, k) q Bn, k) q n and simiary, r P n, k) q Pn, k) q n In [], it was shown that for any fixed k 3 there exists an [N, n, k] PIR code where N = n+o n), so r P n, 3) = O n) and in [8] it was proved that r P n, 3) = Θ n), by providing a ower bound on the redundancy of 3-PIR codes These resuts assure aso that for any fixed k, r P n, k) = Θ n) and aso impied that for any fixed k, r B n, k) = Ω n) In [0], it was proved that for k = 3, 4, r B n, k) = Θ n), and for any fixed k 5, r B n, k) = O n ogn)) In this paper, we wi mosty study the vaues of r P n, k) and r B n, k), when k is a function of n, for exampe k = Θn ǫ ) One of the probems we wi aso investigate is finding the argest ǫ for which r P n, k = Θn ǫ ) ) = on), and simiary for batch codes There are severa more constructions of PIR and batch codes, which we summarize beow ) r B n, n /3 ) n, [9] ) r B n, n ǫ ) n 7/8 for 7/3 ǫ /4, [9] 3) r B n, n ǫ ) n 4ǫ for /5 <ǫ 7/3, [9] 4) Bn, n) n 5, [] 5) r P n, n) = On og 3)/ ), [5] 6) r P n, n ǫ ) = On 05+ǫ ), [5] III MUTLIPLICITY CODES In this section we review the construction of mutipicity codes This famiy of codes was first presented by Kopparty et a in [4] as a generaization of Reed Muer codes by cacuating the derivatives of poynomias We foow the definitions of these codes as were presented in [4] and first start with the definition of the Hasse derivative For a fied F, et F[x,, x s ] = F[x] be the ring of poynomias in the variabes x,, x s with coefficients in F For a vector i = i,, i of non-negative integers, its weight wti) is s j= i j, and et x i denote the monomia s j= xi j j The tota degree of this monomia equas wti) For Px) F[x], et the degree of Px), degp), be the maximum tota degree over a monomias in Px) Definition For a poynomia Px) F[x] and a nonnegative vector i, the i-th Hasse derivative of Px), denoted by P i) x), is the coefficient of z i in the poynomia P x, z) = Px+z) F[x, z] Definition 3 Let m, d, s be nonnegative integers and et q be a prime power Let Σ = F q {i:wti)<m} = F s+m q For a poynomia Px,, x F q [x,, x s ], we define the order m evauation of P at w F s q, denoted by P <m) w), to be the vector P <m) w) = P i) w) ) i:wti)<m Σ The mutipicity code Cm, d, s, q) of order m evauations of degree d poynomias in s variabes is defined as foows The code is over Σ, has ength q s, and its coordinates are indexed by eements in F s q For each poynomia Px) F q [x,, x s ] with degp) d, there is a codeword in C given by: Enc m,d,s,q P) = P <m) w) ) w F s q Σ)qs That is, Cm, d, s, q)={ P <m) w) ) w F s q Σqs : P F q [x], degp) d} The foowing emma was proved in [4], Lemma 9 Lemma 4 The mutipicity code Cm, d, s, q) has reative distance at eastδ = mq d and rate d+s /s+m s )q s Lasty, we note that since the mutipicity code Cm, d, s, q) is a inear code it can aso be a systematic code and thus for the rest of the paper we assume these codes to be systematic; for more detais see Lemma 3 in [] For the rest of the paper and uness stated otherwise, we assume that m, d, s, q are positive integers IV PIR CODES FROM MULTIPLICITY CODES In [4], mutipicity codes were used to construct ocay decodabe codes in order to retrieve the vaue of a singe symbo with high probabiity, given that at most a fixed fraction of the codeword s symbo has errors [] Since we are not concerned with errors, we modify the recovering procedure so that each information symbo has a arge number of disjoint recovering sets For this end, we estabish severa properties on interpoation sets of poynomias which wi hep us ater to construct the recovering sets, and thus PIR and batch codes Lemma 5 Let Px) F q [x,, x s ] be an homogeneous poynomia such that degp) = d Let A,, A s be subsets of F q such that A i = d + Then the set A = A A s {} is an interpoation set of Px), where F q is the unitary eement of the fied The foowing definition wi be used in the construction of recovering sets for mutipicity codes Definition 6 Let F q be a fied, and S, S F s q where s is a positive integer We say that the sets S and S are disjoint under mutipication if for every x S andα F q \{0} it hods thatαx / S Lemma 7 Let Px) F q [x,, x s ] be an homogeneous poynomia such that degp) = d Then there exists q d+ s interpoation sets of Px), each of size d+) s, which are mutuay disjoint under mutipication Now we are in a good position to present the recovering procedure for mutipicity codes First, we show a genera structure of the recovering sets, and then we argue that many disjoint sets can be constructed this way Theorem 8 Let m, d, s, q be such that d/m < q, and C = Cm, d, s, q) is the mutipicity code of ength q s over F s+m q Let A F s q be an interpoation set for homogeneous poynomias of degree at most m Then, for We say that Px) F q [x] is homogeneous if a the monomias of Px) have the same tota degree For Px) F q [x,, x s ] and R F s q, we say that R is an interpoation set of Px) if for every poynomia Qx) such that Px) = Qx) for every x R, it hods that Px) = Qx) for every x F s q

3 every y = y w ) w F s q C, and for any w 0 F s q, the set of coordinates indexed by the set R = {w 0 }+F q A {w 0 +λv : v A,λ F q \{0}} is a recovering set for the symbo y w0 Proof: The proof foows simiar ideas to the one from [4] Reca that every codeword y = y w ) w F s q C corresponds to a poynomia Px) F q [x], of degree at most d, where for a w F s q, y w = P <m) w) Every vector v in the interpoation set A is caed a direction and wi correspond to a ine containing w 0 in the direction v Reading the order m evauations of the poynomia Px) at these ines wi enabe us to recover the vaue of P <m) w 0 ) This procedure consists of two steps, described as foows Step : For every direction v A, define the foowing univariate poynomia p v λ) = Pw 0 + λv) def = d j=0 c v, jλ j F q [x] Since the vaues and the derivatives of Pw 0 + λv) for a λ F q \{0} are known, and degp v ) d, one can prove, as in [4], that p v λ) is unique, and thus can be recovered Step : From Step, one can get that p v λ) = P i) w 0 )v i λ wti) d = c v, j λ j, i j=0 and therefore for 0 j d, i:wti)= j P i) w 0 )v i = c v, j Considering ony the first m of these d+ equations, we get that u i = P i) w 0 ) is a soution for the equations system i:wti)= j u i v i = c v, j, 0 j < m d ) Now we prove that the equations system ) has a unique soution Indeed, if we denote Q j x) = i:wti)= j u i x i F q [x,, x s ] where 0 j < m, we get that the equations in ) are equivaent to Q j v) = c v, j for every v A But since for every j we know that Q j is an homogeneous poynomia of degree j, and A is an interpoation set for homogeneous poynomias of degree at most m, we get that the poynomia Q j x) is unique Therefore, we can recover the vaue of P <m) w 0 ) by soving the equations system ) The next theorem shows how to construct PIR codes from Mutipicity Codes d Theorem 9 For a m, d, s, q such that m < q, the code Cm, d, s, q) is a k-pir code [q s, n, k] P Q, where n = d+s s+m s ), k = q m s, and Q = q s+m Proof: According to Theorem 8, every interpoation set A for homogeneous poynomias of degree m defines a recovering set, which consists of the ines containing w 0 in the directions of v for a v A Therefore, in order to get disjoint recovering sets, a we need to do is to pick different ines According to Lemma 7, there are q m s interpoation sets for homogeneous poynomias of degree m which are mutuay disjoint under mutipication This means that each ine cannot appear in two sets, thus the recovering sets defined by these interpoation sets are disjoint The next theorem summarizes the resuts in this section Theorem 0 For every positive integer s, 0 < α <, and n sufficienty arge, there exists a k-pir code [N, n, k] P Q, overf Q with redundancy r = N n such that k = Θn α) ), Q = n Θnα), r = On α In particuar, for 0 ǫ <, it hods that r P n, k = Θn ǫ ) ) = O n δǫ)), where δǫ) = min s:s> {δ s ǫ)}, and δ s ǫ) = ǫ 5 05 δ δ 3 ǫ) δ 5 ǫ) δ 7 ǫ) δ 0 ǫ) ǫ od resuts ower bound ǫ Fig Asymptotic resuts for binary PIR codes s + s ǫ For a given vaue of ǫ, the vaue s that minimizesδǫ) is s = ǫ Now we use our ast resut in order to construct binary k-pir codes The main idea is to convert every symbo of the fied F Q to ogq) binary symbos We say that fn) = Ωn a ) is for a τ > 0, fn) = Ωn a τ ) Simiary we define fn) = On a+ ) if for a τ > 0, fn) = On a+τ ) Theorem For every positive integer s, 0 <α <, and n sufficienty arge, there exists a binary k-pir code[n, n, k] P, with redundancy r = N n such that k = Θ n ogn) ) α +α ), r = O n α s+α) ogn)) ) s+α) In particuar, for 0 ǫ <, r P n, k = Ωn ǫ ) ) = O ) n δǫ)+, where δǫ) = mins:s> {δ s ǫ)} and δ s ǫ) = ǫ s ǫ) ss ), and r P n, k = Θn ǫ ) ) = on) The anaysis so far deat with constructing k-pir when k = Θn ǫ ) and 0 ǫ < Now we show how to use these resuts to construct k-pir codes forǫ The idea is to concatenate a sufficient copies of k -PIR codes, when k = Ωn ) such that each bit wi have k recovering sets Theorem For aǫ and n sufficienty arge, there exists a binary k-pir code [N, n, k] P, such that k = Θnǫ ) and N = On ǫ+ ) The ength achieved by the PIR construction in Theorem is neary optima Reca that the ength of k-pir codes is Ωk) since every non-trivia recovering set must contain at east one redundancy bit Fig summarizes the resuts of binary PIR codes we achieved in this section together with the previous resuts We pot the curves δ s ǫ) for s = 3, 5, 9, 0 from Theorem as we as the resuts for ǫ from Theorem The ower bound on the redundancy is given by min{k, n} V BATCH CODES FROM MULTIPLICITY CODES It turns out that mutipicity codes can be aso an exceent too to construct batch codes Unike the PIR case, recovering different entries in the codeword wi cause intersection in the corresponding ines, and thus intersecting recovering sets In order to overcome this obstace, we reduce the degree d of the poynomias such that a fewer number of points is needed α

4 from every ine This wi aow different ines to avoid points which are used by other ines That way, every recovering set can drop out points which are used by other sets, resuting in disjoint recovering sets Lemma 3 For a m, s, q, d, k such that d mq km s ) and k q m s, the code Cm, d, s, q) is a k-batch code [q s, n, k] B d+s Q, where n = s+m and Q = qs+m Proof: The caim regarding the code dimension and fied size can be proven simiary to PIR codes Now we prove that every mutiset request of size k can be recovered As we saw in the recovering procedure for PIR codes, every recovering set contains m s different ines Since different ines can intersect on at most one point, and there are k recovering sets, it suffices to prove that Step in the recovering procedure can be competed even when km s points on the ine are not used But since the minimum distance of Cm, d, s =, q) equas q d m > kms +, it can be shown in a very simiar way to PIR codes, that the poynomia p v λ) in Step can be uniquey recovered, and thus aso Step can be competed Unike the PIR case, it turns out that ony the vaue s = is usefu for batch codes, thus getting the foowing theorem Theorem 4 For every 0 < α < 05 and n sufficienty arge, there exists a k-batch code[n, n, k] B Q overf Q with redundancy r = N n such that k = Θn 05 α ), r = On α ), Q = n Θn α) In particuar, for 0 < ǫ < 05, it hods that r B n, k = Θn ǫ ) ) = O n δǫ)), whereδǫ) = ǫ As in the PIR case, the ast resut can be extended for binary batch codes Theorem 5 For every 0 < α < 05 and n sufficienty arge, there exists a binary k-batch code [N, n, k] B with redundancy r = N n such that k = Θ n/ ogn)) 05 α), r = O n α 3 ogn)) α 3 ) In particuar, for 0 < ǫ < 05, it hods that r B n, k = Ωn ǫ ) ) = O n δǫ)+ ), where δǫ) = ǫ 3, and r B n, k = Θn ǫ ) ) = on) Forǫ 05 there exists a binary k-batch code [N, n, k] B of dimension n such that k = Θnǫ ) and N = On 05+ǫ+ ) VI ARRAY CONSTRUCTION FOR PIR AND BATCH CODES Our point of departure for this section is the subcube construction from [3] which was aso used in [] to construct PIR codes The idea of this construction is to position the information bits in a two-dimensiona array, and add a simpe parity bit for each row and each coumn Our approach here is to extend this construction by considering aso diagonas with different sopes As there are many different sopes, this can greaty increase the number of recovering sets However, we wi have to guarantee that using the diagonas wi sti resut with disjoint recovering sets By a sight abuse of notation, in this section we et the set [n] denote the set of integers {0,,, n } We use the notation x m to denote the vaue of x mod m) Definition 6 Let A be an r p array, with indices [r] [p] For s [p] we define the foowing set of sets P s r, p) = {D s,0, D s,,, D s,p }, where for t [p], D s,t ={0, t),, t+ s p ),,r, t+r )s p )} The idea behind Definition 6 is to fix a sope s [p] and then define p diagona sets which are determined by the starting point on the first row and the sope We use these sets in order to construct array codes, where every diagona determines a parity bit for the bits on this diagona Construction Array Construction) Let r, p, n be positive integers such that n = rp, and S [p] a subset of size k We define the encoder E r,p,s, as a mapping E r,p,s : {0, } n {0, } k p as foows We denote S = {s 0, s,, s k } where 0 s 0 < s < < s k p The input vector x {0, } n is represented as an r p array, that is x = x i, j ) [r] [p] and is encoded to the foowing kp redundancy bitsρ,t, for [k], and t [p], ρ,t = x i, j D s,t Let E r,p,s x) = ρ 0,0,,ρ 0,p,,ρ k,0,,ρ k,p ), and the codecr, p, S) is defined to be Cr, p, S) = {x, E r,p,s x)) : x {0, } n } We first ist severa usefu properties Lemma 7 For a r, p, and s [p] the set P s r, p) is a partition of[r] [p] Lemma 8 For a r p and S [p] If p is prime, then for a s = s S and t, t [p], D s,t D s,t We ony state here the resut of this construction for PIR codes, as we focus here mainy on batch codes Theorem 9 Let n = p, where p is a prime number, and k n The code Cr = p, p, S = [k]) is a k-pir code with redundancy k n In particuar, for a k n, r P n, k) = Ok n) For batch codes, this construction can resut with good batch codes as we as batch codes with restricted size for the recovering sets [3] Formay, a k-pir code, k-batch code, in which the size of each recovering set is at most r wi be caed an r, k)-pir code, r, k)-batch code, respectivey The idea here is to choose the set S in a way that for every bit, each of its recovering sets intersects with at most one recovering set of any other bit This property for constructing batch codes from PIR codes was proved in [9] and is stated beow Lemma 0 LetC be anr, k)-pir code Assume that for every distinct indices i, j [n], it hods that each recovering set of the ith bit intersects with at most one recovering set of the jth bit Then, the code C is an r, k)-batch code The main chaenge is to find sets S that wi generate recovering sets which satisfy the condition in Lemma 0 For that, we use the foowing definition Definition Let r be a positive integer, and S be a set of nonnegative integers We say that the set S does not contain an r- weighted arithmetic progression moduo p if there do not exist s, s, s 3 S and 0 < x, y < r, where x+ y < r, such that xs + ys = x+ y)s 3 mod p Given this definition, we prove the foowing theorem Theorem Let r p and S [p], S = k If p is prime, and S does not contain an r-weighted arithmetic progression moduo p, then the code C = Cr, p, S) is anr, k)-batch code of dimension rp

5 Proof: Assume that S = {s 0, s,,, s k } One can verify using Lemma 7 and 8 that for every [r] [p] the foowing sets R = {ρ,t } {x i, j : i, j ) D s,t \{}}, for [k] are k mutuay disjoint recovering sets for x i, j, where t [p] is chosen such that D s,t We denote DR ) = D s,t Thus C is r, k)-pir, and it remains to prove that C satisfies the condition of Lemma 0 Assume in the contrary that there exist two bits,i, j ) [r] [p] such that has a recovering set R that intersects with two recovering sets R i, j ), R i, j ) of i, j ) Assume b R R i, j ) and b R R i, j ) where b, b are codeword entries It can be verified that b, b don t correspond to parity bits Therefore, we denote b = x i, j, b = x i, j, for i, j ),i, j ) [r] [p] Denote DR ) = D s,t, DR i, j ) ) = D s,t, DR i, j ) ) = D s 3,t 3 for s, s, s 3 S and t, t, t 3 [p] Thus we get that i, j ),i, j ) D s,t,i, j ),i, j ) D s,t, and i, j ),i, j ) D s 3,t 3 From Lemma 7 and 8 we deduce that s = s = s 3 and i = i = i Assume wog i < i < i It foows that: j = t + i s p, j = t + i s p, j = t 3 + i s 3 p, j = t + i s p j = t + i s p j = t 3 + i s 3 p This impies that i i )s + i i )s 3 p = i i )s p, which is a contradiction since S does not contain an r-weighted arithmetic progression moduo p In order to compete the construction of batch codes, we are eft with the probem of finding arge sets S which satisfy the condition in Theorem That is, given r and p, our goa is to find the argest such a set S A simpe greedy agorithm can give the foowing resut Theorem 3 Let r, p be positive integers, such that p is prime Then there exists a set S with no r-weighted arithmetic progression moduo p of size at east k, where k is the argest integer such that p > k r The foowing theorem foows from these observations Theorem 4 For every r, k, et n = rp, where p is the smaest prime number such that k r < p Then, there exists an r r, k)-batch code of dimension n and rate r+k In particuar, the redundancy of the code equas kp According to Theorem 4 we are now at a point to construct k-batch codes with good redundancy Coroary 5 For any n and k such that k = o n), there exists a k-batch code of dimension n and redundancyon 3 k 5 3) In particuar, for 0 <ǫ < /, r B n, n ǫ ) = On /3+5ǫ/3 ) Proof: For n and k, et us choose r = n 3/k 3, and p is the smaest prime number such that k r < p Then, according to Theorem 4, there exists an r, k)-batch code of dimension pr > n and redundancy kp That is, the redundancy satisfies kp = Θk 3 r ) = Θn 3 k 5 3) The second statement in the coroary is estabished for k = n ǫ in the ast equation Let us denote r B k = n ǫ ) = On δ ) In Fig we pot the resuts on the asymptotic behavior of the redundancy of batch δ Batch codes Theorem 5) Batch codes Coroary 5) ǫ 05 ower bound od resuts ǫ Fig Asymptotic resuts for binary batch codes codes These pots are received from Coroary 5 in this section and Theorem 5 from Section V Note that the array construction improves the redundancy ony for ǫ < Lasty, we report on two more resuts that can be derived using the Array Construction Note that the second resut improves upon the one from [0], which states that r B n, 5) = O n ogn) Theorem 6 For every 0 < α <, k = On α ), and fixed r 3, there exists anr, k)-batch code with rate r r+k Theorem 7 Let n = p where p is a prime number The code C, that extends Cr = p, p, S = [5]) by adding a goba parity bit, is a 5-batch code with redundancy 5p+ = Θ n), and therefore r B n, 5) = Θ n) REFERENCES [] S Buzago, E Yaakobi, Y Cassuto, and P H Siege, Consecutive switch codes, Proc IEEE Int Symp Inf Theory, pp , Barceona, Spain, Juy 06 [] A Fazei, A Vardy, and E Yaakobi, PIR with ow storage overhead: Coding instead of repication, arxiv:505064, May 05 [3] Y Ishai, E Kushievitz, R Ostrovsky, and A Sahai, Batch codes and their appications, Proc of the 36-sixth Annua ACM Symposium on Theory of Computing, pp 6 7, Chicago, ACM Press, 004 [4] S Kopparty, S Saraf, and S Yekhanin, High-rate codes with subineartime decoding, in 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