A Combination of Testability and Decodability by Tensor Products
aa r X i v : . [ c s . CC ] M a y A Combination of Testability and Decodability by Tensor Products
Michael Viderman ∗ Computer Science DepartmentTechnion — Israel Institute of TechnologyHaifa 32000, Israel [email protected]
April 28, 2018
Abstract
Ben-Sasson and Sudan (RSA 2006) showed that repeated tensor products of linear codes with a verylarge distance are locally testable. Due to the requirement of a very large distance the associated tensorproducts could be applied only over sufficiently large fields. Then Meir (SICOMP 2009) used this result(as a black box) to present a combinatorial construction of locally testable codes that match best knownparameters. As a consequence, this construction was obtained over sufficiently large fields.In this paper we improve the result of Ben-Sasson and Sudan and show that for any linear codes theassociated tensor products are locally testable. Consequently, the construction of Meir can be taken overany field, including the binary field.Moreover, a combination of our result with the result of Spielman (IEEE IT, 1996) implies a con-struction of linear codes (over any field) that combine the following properties: • have constant rate and constant relative distance; • have blocklength n and testable with n ǫ queries, for any constant ǫ > ; • linear time encodable and linear-time decodable from a constant fraction of errors.Furthermore, a combination of our result with the result of Guruswami et al. (STOC 2009) implies asimilar corollary regarding the list-decodable codes. Locally testable codes (LTCs) are error correcting codes that have a tester, which is a randomized algorithmwith oracle access to the received word x . The tester reads a sublinear amount of information from x andbased on this local view decides if x ∈ C or not. It should accept codewords with probability one, and rejectwords that are far (in Hamming distance) from the code with noticeable probability.Such codes are of interest in computer science due to their numerous connections to probabilisticallycheckable proofs (PCPs) and property testing (see the surveys [27, 13] for more information). By nowseveral different constructions of LTCs are known including codes based on low-degree polynomials overfinite fields [21, 1], constructions based on PCPs of proximity/assignment testers [2, 11] and sparse randomlinear codes [8, 18, 19]. In this paper we study a different family of LTC constructions, namely, tensor ∗ The research was partially supported by the European Community’s Seventh Framework Programme (FP7/2007-2013) undergrant agreement number 240258 and by grant number 2006104 by the US-Israel Binational Science Foundation. odes . Given two linear error correcting codes C ⊆ F n , R ⊆ F n over a finite field F , we define their tensor product to be the subspace R ⊗ C ⊆ F n × n consisting of n × n matrices M with entries in F having the property that every row of M is a codeword of R and every column of M is a codeword of C . If C = R we use C to denote C ⊗ C and for i > define C i = C ⊗ C i − . Note that the blocklength of C i is n i .Recently, tensor products were used to construct new families of LTCs [5, 22], new families of list-decodable codes [15], to give an alternative proof (see [23]) for IP=PSPACE theorem of [24, 25] etc.Ben-Sasson and Sudan [5] suggested to use tensor product codes as a means to construct LTCs com-binatorially. A natural hope would be to expect that given a code C ⊆ F n , whenever the task is to testwhether a word M ∈ F n is in C , the tester for C can choose a random row (or column) of M ; and if M was far from C , its typical row/column is far from C and hence can be tested on being in C . As wasshown in [28, 14, 9] this approach fails in general and is known to work only for the base codes having somenon-trivial properties [12, 6, 7].Nevertheless, Ben-Sasson and Sudan [5] showed that taking the repeated tensor products of any code C ⊆ F n with sufficiently large distance results in a locally testable code with sublinear query complexity.More formally, they showed [5, Theorem 2.6] that for every m ≥ if (cid:16) ∆( C ) − n (cid:17) m ≥ then C m is locallytestable using n queries. Note that the blocklength of C m is n m and query complexity is n . Hence, forexample, if m = 10 we obtain a code with blocklength N = n and query complexity N . = n , underassumption that (cid:16) ∆( C ) − n (cid:17) ≥ .Let us explain some issues that remained open. First of all, it was remained unclear if the assumptionabout a very large distance of the base codes is necessary. Moreover, the requirement on the distance ofthe base code is dependent on the number of tensor products ( m ) one should apply. Note that less querycomplexity (relatively to the blocklength) one should get more tensor product operations should be applied.Thus the requirement about the distance of the base code is increased when the number of queries oneshould get is decreased. We notice also that the larger distance implies the larger underlying field. It followsthat this result can not provide (via tensor products) arbitrary low sublinear query complexity ( N ǫ for everyconstant ǫ > ) over a fixed field.In this paper we ask the following question: is it possible to achieve a similar result to [5] but with norequirements about the base codes at all. A positive result to this question might seem surprising since itwould imply that any linear error-correcting code can be involved in the construction of LTCs via tensorproducts.We give a positive answer on this question and show that no assumptions about the base codes (orunderlying fields) are not needed. Our result does not make any assumptions about the base codes, andin particular we do not assume that the base codes involved in tensor products have very large distanceand hence it holds over any fields. This contrasts with previous works on the combinatorial constructions ofLTCs due to Ben-Sasson and Sudan [5] and Meir [22] which required very large base-code distance implyinglarge field size. The constructions of best known LTCs [4, 10, 22] were obtained over the large fields (whenfinally, the field size can be decreased through code concatenation). Our improvement of [5] implies thatthe construction of Meir [22] (which achieves LTCs of best known parameters) can be taken directly overany field (including the binary field). We think that this improvement has a non-negligible role since theLTCs construction of Meir is combinatorial and the combinatorial constructions of LTCs (or PCPs) shouldbe independent, as much as possible, of the algebraic terms such as “polynomials”, “field size”, “extensionfield” etc.Given the fact that error-correcting codes play an important role in a complexity theory, and in particular,2n different iterative protocols, it might be helpful to develop a general scheme for constructing the error-correcting codes that combine several different properties. E.g., it might be helpful to have a high-rate codeswhich combine such properties as local testing, efficient encoding and decoding from a constant fraction oferrors.We show that a combination of our results with the results of [26, 15] implies the construction of high-rate codes which are both testable with sublinear query complexity, linear-time encodable and efficientlydecodable (or list-decodable) from the constant fraction of errors. Organization of the paper.
In the following section we provide background regarding tensor codes andlocally testable codes. In Section 3 we state our main results. We prove our main theorem (Theorem 3.2) inSection 4. Finally, in Section 5 we prove our auxiliary statements.
Throughout this paper, F is a finite field, [ n ] denotes the set { , . . . , n } and F n denotes F [ n ] . All codesdiscussed in this paper will be a linear. Let C ⊆ F n be a linear code over F .For w ∈ F n let supp( w ) = { i | w i = 0 } , | w | = | supp( w ) | and wt( w ) = | w | n . We define the distance between two words x, y ∈ F n to be ∆ ( x, y ) = |{ i | x i = y i }| and the relative distance to be δ ( x, y ) = ∆( x,y ) n . The distance of a code is defined by ∆ ( C ) = min x = y ∈ C ∆ ( x, y ) and its the relative distance isdenoted δ ( C ) = ∆( C ) n . A [ n, k, d ] F -code is a k -dimensional subspace C ⊆ F n of distance d . The rate of thecode C is defined by rate( C ) = dim( C ) n . For x ∈ F n and C ⊆ F n , let δ ( x, C ) = δ C ( x ) = min y ∈ C { δ ( x, y ) } todenote the relative distance of x from the code C . We note that ∆ ( C ) = min c ∈ C \{ } { wt( c ) } . If δ ( x, C ) ≥ ǫ we say that x is ǫ -far from C and otherwise x is ǫ -close to C . We let dim( C ) denote the dimension of C .The vector inner product between u = ( u , u , . . . , u n ) ∈ F n and v = ( v , v , . . . , v n ) ∈ F n is definedto be h u, v i = P i ∈ [ n ] u i · v i . We let C ⊥ = { u ∈ F n | ∀ c ∈ C : h u, c i = 0 } be the dual code of C and C ⊥ t = n u ∈ C ⊥ | | u | = t o . In a similar way we define C ⊥≤ t = (cid:8) u ∈ C ⊥ | | u | ≤ t (cid:9) . For t ∈ F n and T ⊆ F n we say that t ⊥ T if h t, t ′ i = 0 for all t ′ ∈ T .For w ∈ F n and S = { j , j , . . . , j m } ⊆ [ n ] , where j < j < . . . < j m , we let w | S = ( w j , . . . , w j m ) be the restriction of w to the subset S . We let C | S = { c | S | c ∈ C } denote the restriction of the code C tothe subset S . The definitions appearing here are standard in the literature on tensor-based LTCs (e. g. [12, 5, 22, 7, 28]).For x ∈ F I and y ∈ F J we let x ⊗ y denote the tensor product of x and y (i. e., the matrix M withentries M ( i,j ) = x i · y j where ( i, j ) ∈ I × J ). Let R ⊆ F I and C ⊆ F J be linear codes. We define thetensor product code R ⊗ C to be the linear space spanned by words r ⊗ c ∈ F I × J for r ∈ R and c ∈ C .Some known facts regarding the tensor products (see e. g., [12]): • The code R ⊗ C consists of all I × J matrices over F whose rows belong to R and whose columnsbelong to C . • dim( R ⊗ C ) = dim( R ) · dim( C ) rate( R ⊗ C ) = rate( R ) · rate( C ) • δ ( R ⊗ C ) = δ ( R ) · δ ( C ) We let C = C and C t = C t − ⊗ C for t > . Note by this definition, C = C and C t = C t − ⊗ C t − for t > . We also notice that for a code C ⊆ F n and m ≥ it holds that rate( C m ) = (rate( C )) m , δ ( C m ) = ( δ ( C )) m and the blocklength of C m is n m .The main drawback of the tensor product operation is that this operation strongly decreases the rate andthe distance of the base codes. We refer the reader to [22] which showed how to use tensor products andavoid the decrease in the distance and the strong decrease in the rate. A standard q -query tester for a linear code C ⊆ F n is a randomized algorithm that on the input word w ∈ F n picks non-adaptively a subset I ⊆ [ n ] such that | I | ≤ q . Then T reads all symbols of w | I andaccepts if w | I ∈ C | I , and rejects otherwise (see [3, Theorem 2]). Hence a q -query tester can be associatedwith a distribution over subsets I ⊆ [ n ] such that | I | ≤ q .For purposes of composition we want to define a generalized tester (Definition 2.1) which does not makequeries, but selects and returns a “view” (a subset I ⊆ [ n ] ) which can be considered as a code by itself ( C | I ). Definition 2.1 (Tester of C and Test View) . A q -query tester D is a distribution D over subsets I ⊆ [ n ] such that | I | ≤ q . Let w ∈ F n (think of the task of testing whether w ∈ C ) and let I ⊆ [ n ] be a subset.We call w | I the view of a tester. If w | I ∈ C | I we say that this view is consistent with C , or when C is clearfrom the context we simply say w | I is consistent .When considering a tensor code C m ⊆ F n m , an associated tester will be a distribution over subsets I ⊆ [ n ] m . Although the tester does not output accept or reject , the way a standard tester does, it can beconverted to output accept , reject as follows. Whenever the task is to test whether w ∈ C and a subset I ⊆ [ n ] is selected by the tester, the tester can output accept if w | I ∈ C | I and otherwise output reject . Definition 2.2 (LTCs and strong LTCs) . A code C ⊆ F n is a ( q, ǫ, δ ) -LTC if it has a q -query tester D suchthat for all w ∈ F n , if δ ( w, C ) ≥ δ we have Pr I ∼ D [ w | I / ∈ C | I ] ≥ ǫ .A code C ⊆ F n is a ( q, ǫ ) -strong LTC if it has a q -query tester D such that for all w ∈ F n , we have Pr I ∼ D [ w | I / ∈ C | I ] ≥ ǫ · δ ( w, C ) .We notice that a ( q, ǫ ) -strong LTC is a ( q, ǫδ, δ ) -LTC for every δ > . Note that given a code C ⊆ F n ,the subset I ⊆ [ n ] uniquely defines C | I . Moreover, the linearity of C implies that C | I is a linear subspace of F I . In the rest of this section we formally define the notion of robustness (Definition 2.4) as was introducedin [5]. To do that we start from the definition of local distance (Definition 2.3), which will be used inDefinition 2.4 and later in our proofs. Definition 2.3 (Local distance) . Let C be a code and w | I be the view on the coordinate set I obtained fromthe word w . The local distance of w from C with respect to I (also called the I -distance of w from C ) is ∆ ( w | I , C | I ) = min c ∈ C { ∆ ( w | I , c | I ) } and similarly the relative local distance of w from C with respect to I (relative I -distance of w from C ) is δ ( w | I , C | I ) = min c ∈ C { δ ( w | I , c | I ) } .4nformally, robustness implies that if a word is far from the code then, on average, a test’s view is farfrom any consistent view that can be accepted on the same coordinate set I . This notion was defined forLTCs following an analogous definition for PCPs [2, 10]. We are ready to provide a general definition ofrobustness. Definition 2.4 (Robustness) . Given a tester (i. e., a distribution) D for the code C ⊆ F n , we let ρ D ( w ) = E I ∼ D [ δ ( w | I , C | I )] be the expected relative local distance of input w .We say that the tester D has robustness ρ D ( C ) on the code C if for every w ∈ F n it holds that ρ D ( w ) ≥ ρ D ( C ) · δ C ( w ) .Let { C n } n be a family of codes where C n is of blocklength n and D n is a tester for C n . A family ofcodes { C n } n is robustly testable with respect to testers { D n } n if there exists a constant α > such that forall n we have ρ D n ( C n ) ≥ α . The tester we consider in this paper is the plane tester (suggested in [5]).
Definition 3.1 (Plane Tester) . Let m ≥ . Let M ∈ F n m be an input word and think of testing whether M ∈ C n m . The plane tester D picks (non-adaptively) a random b ∈ [3] and random i ∈ [ n ] , and returns ( b, i ) -plane (the corresponding local view is M | ( b,i ) ). Note that if M is a candidate word to be in C m then M | ( b,i ) is a candidate word to be in C m − .Now we state our main technical theorem which says that the tensor product of any base code (withconstant relative distance) is robustly testable. This extends the result of [5] which showed that this claimholds for base codes with a very large distance. Theorem 3.2 (Main Theorem) . Let C ⊆ F n be a linear code and m ≥ . Let D be the plane tester for C m .Then ρ D ( C m ) ≥ ( δ ( C )) m m . The proof of Theorem 3.2 is postponed to Section 4. Theorem 3.2 extends the main result of Ben-Sasson and Sudan [5] since it implies that the m -wise tensor product of linear codes is robust for any linearbase codes with constant relative distance. In particular, the tensor product can be applied over any field,including the binary field. So, as explained in the introduction, the combinatorial construction of LTCs in[22] can be taken over any field (regardless of the field size).Ben-Sasson and Sudan [5] explained that plane testers can be composed and the robustness of the planetesters implies the strong local testability. For the sake of completeness we state this claim formally inCorollary 3.3, and provide a proof-sketch in Section 5 (see [5, 6] for more information about compositionof the testers). Corollary 3.3.
Let C ⊆ F n be a linear code and m ≥ is a constant. Then C m is a ( n , α m ) -strong LTC,where α m > is a constant that depends only on m and δ ( C ) . Note that the blocklength of C m is n m . Corollary 3.3 implies that any linear code can be used to define a locally testable code with sublinearquery complexity. Claim 3.4 shows that if a linear code C is linear-time encodable then so is C i for anyconstant i . Later we will use this claim together with Corollary 3.3 to show Corollary 3.5.5 laim 3.4. Let m ≥ be a constant. If C ⊆ F n is a linear-time encodable linear code then C m islinear-time encodable. The proof of Claim 3.4 is postponed to Section 5. Now, we combine Corollary 3.3 and Claim 3.4 toshow a simple construction of strong LTCs with arbitrary small sublinear query complexity and arbitraryhigh rate from any linear code with sufficiently high rate.
Corollary 3.5.
Let C ⊆ F n be a linear code and let m ≥ be a constant. Then C m ⊆ F n m is a ( n , α m ) -strong LTC, where α m > is a constant that depends only on m and δ ( C ) .In particular, for every ǫ > , m = ⌈ ǫ ⌉ , N = n m and C ⊆ F n such that rate( C ) ≥ (1 − ǫ ) /m wehave C m ⊆ F N is a ( N ǫ , α ) -strong LTC and rate( C m ) ≥ − ǫ , where α > is a constant that dependsonly on ǫ . Moreover, if C is a linear-time encodable then C m is a linear-time encodable. Usually, in the areas of locally testable and locally decodable codes the main interest was given to theconstant query complexity. Recently, Kopparty et al. [20] showed the construction of high-rate locally de-codable codes with sublinear query complexity (see [20] for the motivation behind this range of parameters).Since then, the interest to the other range of parameters, and in particular, to sublinear query complexity wasincreased.We would like to stress that Corollary 3.5 is quite powerful for this range of parameters (sublinear querycomplexity and high rate). First of all, there are different constructions of linear-time encodable codeswith constant rate and constant relative distance [16, 17, 26], and them all can be involved to define high-rate LTCs with sublinear query complexity that are linear-time encodable. The other advantage of suchconstructions is that the repeated tensor product of the base code is known to inherit some properties ofthe base codes besides local testability. E.g., Gopalan et al. [15] showed that the tensor product operationpreserves list-decodability properties. Furthermore, we know about nice constructions of error-correctingcodes that can be efficiently encoded and decoded (list-decoded) from a constant fraction of errors (see e.g.,[16, 17]).In Section 3.1 we show how testing with sublinear query complexity can be combined with a linear timeencoding and decoding. Then, in Section 3.2 we show that Corollary 3.5 can be combined with the result of[15] to define asymptotically good codes that can be encodable in linear time, testable with sublinear querycomplexity and list-decodable in polynomial time.
Proposition 3.6 shows that the tensor product operation preserves the “unique-neighbor” decoding property.In particular, if C is a linear code that is linear time unique-neighbor decodable from a constant fractionof errors then so is C . Hence this observation, together with a result of, e.g. [26], can result in theconstruction of asymptotically good locally testable codes with sublinear query complexity that can belinear-time encoded and decoded to the closest neighbor after a constant fraction of errors. Proposition 3.6.
Assume C ⊆ F n is a linear code that is linear-time decodable from α · n errors. Then C = C ⊗ C is a linear code that is linear-time decodable from α · n errors.Proof. Let
Dec C be a linear-time decoder for the code C that can correct any α · n errors. Note that inparticular, Dec C correct any α · n erasures in the linear time. We define the linear-time decoder Dec C forthe code C that will correct any α · n errors.To do this, let M ∈ F n × n be an input word. The decoder Dec C will decode every row of M using Dec C and every column of M using Dec C . Note that every entry of M is contained in (exactly) one row6nd one column. Call the entry ( i, j ) of M an inconsistent if row decoding gives to M | ( i,j ) a different valuefrom column decoding, and otherwise the entry is called consistent.We call the row (column) of M bad if it contains at least αn inconsistent entries. Let Bad r be anumber of bad rows and Bad c be a number of bad columns. It holds that Bad r · αn ≤ α n and hence Bad r ≤ αn/ . Similarly, Bad c ≤ αn/ .The decoder Dec C removes all bad rows and bad columns that have at least αn/ inconsistent entriesand obtains a large submatrix of size at least (1 − α/ n × (1 − α/ n . It is easy to see that all consistententries in the above large submatrix were decoded correctly.In the last step, the decoder Dec C decodes, using Dec C , every row of the large submatrix of M (ofsize at least (1 − α/ n × (1 − α/ n ) from at most αn/ erasures and obtains a submatrix of sizeat least (1 − α/ n × n . Now, it decodes every column of the submatrix to the full matrix. It can be easilyverified that the decoder Dec C obtains a correct codeword of C and runs in linear time.While the results of [26] were improved, for our purpose (Corollary 3.8) this result is sufficient. Theorem 3.7 ([26]) . There exists an (explicit) family of linear error correcting codes C ⊆ F n such that rate( C ) = Ω(1) , δ ( C ) = Ω(1) , C is a linear-time encodable and linear-time decodable from the constantfraction of errors. A combination of Theorem 3.7, Proposition 3.6 and Claim 3.4 results in the following corollary.
Corollary 3.8.
For every constant ǫ > there exists an (explicit) family of linear error correcting codes C ⊆ F N (obtained by tensor products on the codes from Theorem 3.7) that • have constant rate and constant relative distance, • linear time encodable and linear time decodable from the constant fraction of errors, • are ( N ǫ , α ) -strong LTCs, where α = α ( ǫ ) > is a constant. In this section we recall some constructions of the list-decodable codes. We start by defining the list-decodable codes.
Definition 3.9 (List-decodable codes) . A code C is a ( α, L ) -list decodable if for every word w ∈ F n , δ ( w, C ) ≤ α we have |{ c ∈ C | δ ( c, w ) ≤ α }| ≤ L . The code is said to be ( α, L ) -list decodable in time T if there exists algorithm which on the input w ∈ F n such that δ ( w, C ) ≤ α outputs all codewords c ∈ C such that δ ( c, w ) ≤ α (at most L codewords).Guruswami et al. [15] showed that the list-decodability is preserved in the tensor product operation.More formally, they showed the following theorem stated in [15, Theorem 5.7]. Theorem 3.10 ([15]) . Let F be a finite field and q = | F | . Given two linear codes C , C ⊆ F n , for every ǫ > , the number of codewords of C ⊗ C within distance η ∗ = min( δ η , δ η ) − ǫ of any receivedword is bounded by l ( C ⊗ C , η ∗ ) ≤ q δ ǫ ln l η ǫ ln l η ǫ .Further, if C and C can be efficiently list decoded up to error rates η , η and C is a linear code,then C ⊗ C can be list decoded efficiently up to error rate η ∗ . Specifically, if T denotes the time com-plexity of list decoding C and C , then the running time of the list decoding algorithm for C ⊗ C is O (4 q δ ǫ ln l η ǫ ln l η ǫ · T n n ) . Theorem 3.11 ([15]) . Let C be a linear code with distance δ , list decodable up to an error rate η . For every δ > , the m -wise tensor product code C m can be list decoded up to an error rate δ m − η − ǫ with a listsize exp(( O ( ln l ( η ) /ǫǫ )) m ) . Moreover, if m ≥ is constant and C is polynomial-time list decodable then therunning time of the list decoding algorithm for C m is polynomial (depending on m ). The next fact is known due to the several constructions of list-decodable codes.
Fact 3.12.
There exist linear error-correcting codes of constant rate and constant relative distance that canbe encoded in linear time and list-decoded in polynomial time.We use the combination of Theorem 3.11, Fact 3.12, Claim 3.4 and Corollary 3.5 to conclude the fol-lowing corollary.
Corollary 3.13.
Let F be any field. For every constant ǫ > there exists a code C ′ ⊆ F N such that C ′ = C m , where C ⊆ F n is a linear code, rate( C ) = Ω(1) , δ ( C ) = Ω(1) and C is ( ρ, L ) -list decodablein polynomial time. • C ′ is a ( N ǫ , α ) -strong LTC, where α = α ( ǫ ) > is a constant, • C ′ is linear time encodable and list-decodable in polynomial time from the constant fraction of errors, • rate( C ′ ) ≥ Ω(1) and δ ( C ′ ) = Ω(1) . Throughout this paper we assume that C ⊆ F n is a linear code. We shall consider an m -wise tensor product,i.e., C m ⊆ F n m . Note that the blocklength of C m is n m . Throughout this paper we assume that m ≥ and for the case of m = 2 we refer a reader to [6, 7, 12, 28, 14, 9, 9]). We start this section by defining theconcepts of points, lines and planes (some of the terms were defined following [5]). A point in such a code can be associated with an m -tuple ( i , i , ..., i m ) such that i j ∈ [ n ] . Next we definean axis parallel line, or shortly, a line which can be associated with a subset of points. For b ∈ [ m ] and i ∈ [ n ] we say that l is a ( b, ( i , i , ..., i b − , i b +1 , ..., i m )) -line if l = { ( i , i , ..., i b − , i, i b +1 , ..., i m ) | for all j ∈ [ m ] \ { b } we have i j = i j } . Note that ( b, ( i , i , ..., i b − , i b +1 , ..., i m )) -line is parallel to the b -th axis. A line l contains a point p if p ∈ l .Note that a ( b, ( i , i , ..., i b − , i b +1 , ..., i m )) -line contains a point p = ( j , i , ..., j m ) if for all k ∈ [ m ] \ { b } we have i k = j k . Two (different) lines intersects on the point p if both lines contain the point p .We say that pl is a ( b, i ) -plane if pl = { ( i , i , ..., i m ) | i b = i and for all j ∈ [ m ] \ { b } we have i j ∈ [ n ] } . A ( b, i ) -plane contains a point p = ( i , i , ..., i m ) if i b = i , i.e., b -th coordinate of the point is i . A ( b, i ) -plane contains a line l if it contains all points of the line. We say that two (different) planes are intersected if8oth planes contain at least one common point. Note that two (different) planes: ( b , i ) -plane and ( b , i ) -plane are intersected iff b = b , moreover, they are intersected on all points p = ( i , . . . , i m ) such that i = i b and i = i b , i.e., are intersected on n m − points.Assume that pl is a ( b , i ) -plane and pl is a ( b , i ) -plane such that b < b (in particular b = b ).Let pl ∩ pl = { ( i , . . . , i m ) | i b = i , i b = i } be an intersection of two planes and C m | pl ∩ pl be a code C m restricted to the points in pl ∩ pl . Note that δ ( C m | pl ∩ pl ) = δ ( C m − ) = δ ( C ) m − .Given a word M ∈ F n m , b ∈ [ m ] and i ∈ [ n ] we let M ( b,i ) be a restriction of M to the ( b, i ) -plane, i.e.,to all points of the plane. We say that M ( b,i ) is a ( b, i ) -plane of M . Similarly, for the point p = ( i , . . . , i m ) let M | p be a restriction of M to the point p and for the line l we let M | l be a restriction of M to the line l .We say that M | l is a line l of M . Let M ∈ F n m be an input word. We prove that ρ D ( M ) ≥ ( δ ( C )) m − m δ ( M, C m ) .For every plane pl of M let r ( pl ) be the closest codeword of C m − to M | pl (if there are more than onesuch codewords fix anyone arbitrarily). Intuitively, the plane pl of M “thinks” that the symbols of M | pl should be changed to r ( pl ) . In this sense every plane of M has its own “opinion”. Then we have ρ D ( M ) = E pl ∼D [ δ ( M | pl , r ( pl ))] . (4.1)We say that the ( b , i ) -plane and the ( b , i ) -plane disagree on the point p = ( i , . . . , i m ) if ( b , i ) -plane and ( b , i ) -plane are intersected, both contain the point p and r ( pl ) | p = r ( pl ) | p . We say that twoplanes disagree on the line l if both planes are intersected, both contain the line l and r ( pl ) | l = r ( pl ) | l .Note that if ( b , i ) -plane pl and ( b , i ) -plane pl are intersected and disagree on at least one pointthen letting reg = pl ∩ pl we have r ( pl ) | reg = r ( pl ) | reg and moreover, δ ( r ( pl ) | reg , r ( pl ) | reg ) ≥ ( δ ( C )) m − . This is true since r ( pl ) | reg , r ( pl ) | reg ∈ C m − are non-equal codewords of C m − and δ ( C m − ) = ( δ ( C )) m − .Let E ∈ F n m be a binary matrix such that E | p = 1 if at least two planes disagree on the point p , andotherwise E | p = 0 . For the point p we say that the point is almost fixed if E | p = 0 but p is contained insome plane pl such that r ( pl ) | p = M | p . Intuitively, a point p is almost fixed if all planes containing thispoint agree on this point but “think” that its value in M ( M | p ) should be changed (to r ( pl ) | p ).We let T oF ix = { p = ( i , i , . . . , i m ) | p is almost fixed } and let N umT oF ix = | T oF ix | . Proposition 4.1.
It holds that ρ D ( M ) ≥ wt( E ) m + NumT oF ixn m .Proof. Equation 4.1 says that ρ D ( M ) is a relative distance of a typical plane of M (which is a word in F n m − ) from C m − . Note that for every point p = ( i , . . . , i m ) : if E | p = 0 then p / ∈ N eedT oF ix . Thatmeans for every point p at most one condition is satisfied: E | p = 0 or p ∈ N eedT oF ix .Note also that for every point p ∈ N eedT oF ix , for all planes pl of M we have ( M | pl ) | p = r ( pl ) | p .Now, every point p is contained in m different planes. Hence if E | p = 0 then for at least one plane pl (of m planes containing the point p ) it holds that r ( pl ) | p = M | p .Hence a relative distance between a typical plane ( pl ) of M and r ( pl ) is at least wt( E ) m + NumT oF ixn m .Next we define an important concept of “heavy planes (lines)” in the inconsistency matrix E . Intuitively,a heavy plane (line) of the matrix E is a plane (line) where many inconsistencies occur, i.e., many non-zerosymbols. 9 efinition 4.2 (Heavy lines and planes) . A line l of E is called heavy if | E | l | ≥ δ ( C ) · n . A plane ( b, i ) of E is called heavy if | E | ( b,i ) | ≥ ( δ ( C ) · n ) m − .Lemma 4.3 is our main observation in the proof of Theorem 3.2. It says that any non-zero element of E is located in some heavy plane of E . Lemma 4.3 (Main Lemma) . Let p = ( i , i , . . . , i m ) be a point such that E p = 0 . Then p is contained insome heavy plane of E . The proof of Lemma 4.3 is postponed to Section 4.2.1. Corollary 4.4 shows that it is sufficient to removeat most wt( E ) · ( δ ( C )) m − / · n planes from E to get a zero submatrix. Corollary 4.4.
There exists S , . . . , S m ⊆ [ n ] such that n −| S | + n −| S | + . . . + n −| S m | ≤ | E | ( δ ( C ) n ) m − · m and letting S = S × S × . . . × S m we have E | S = 0 .Proof. Let
HeavyP lanes = { ( b, i ) | ( b, i ) is a heavy plane } to be a subset of pairs associated with heavyplanes. For b ∈ [ m ] let S b = { i ∈ [ n ] | ( b, i ) ∈ HeavyP lanes } and S b = [ n ] \ S b .We claim that | HeavyP lanes | ≤ | E | ( δ ( C ) · n ) m − · m . This is true since every heavy plane contains at least ( δ ( C ) · n ) m − non-zero elements of E and the total number of non-zero elements of E is | E | . Furthermore,every non-zero element of E is contained in at most m (heavy) planes. Thus n − | S | + n − | S | + . . . + n − | S m | = P b ∈ [ m ] | S b | ≤ | E | ( δ ( C ) n ) m − · m .Now, note that Lemma 4.3 implies that for every point p = ( i , i , . . . , i m ) such that E | p = 0 iscontained in some heavy plane, i.e., in some plane from HeavyP lanes . Hence if all heavy planes areremoved from E we obtain a zero submatrix. So, it follows that E | S = 0 .Proposition 4.5 says that if after removing small fraction of planes from M we obtain a submatrix thatis close to the legal submatrix then M is close to C m . Proposition 4.5.
Let S , S , ..., S m ⊆ [ n ] be such that n − | S | + n − | S | + ... + n − | S m | ≤ τ n < δ ( C ) · n and let S = S × S × . . . × S m . Let C ′ = C | S ⊗ C | S ⊗ . . . ⊗ C | S m . Recall that M | S is a submatrixof M obtained by removing at most τ -fraction of planes. Assume that ∆ ( M | S , C ′ ) ≤ α · n m . Then δ ( M, C m ) ≤ τ + α . The proof of Proposition 4.5 appears in Section 4.2.2. Let us prove Theorem 3.2.
Proof of Theorem 3.2.
By Proposition 4.1 we have ρ D ( M ) ≥ wt( E ) m + NumT oF ixn m . If wt( E ) ≥ ( δ ( C )) m m thenwe are done. Otherwise, assume that wt( E ) < ( δ ( C )) m m .Corollary 4.4 implies that it is sufficient to remove at most | E | ( δ ( C ) n ) m − · m < δ ( C ) · n planes from E toget a zero submatrix. Proposition 4.5 implies that δ ( M, C m ) ≤ E )( δ ( C )) m − · m + NumT oF ixn m .Let β = m ( δ ( C )) m − . Then ρ D ( M ) · β ≥ ( wt( E ) m + NumT oF ixn m ) · β ≥ δ ( M, C m ) and ρ D ( M ) ≥ ( δ ( C )) m − m · δ ( M, C m ) . In this section we prove Lemma 4.3. 10 roof od Main Lemma 4.3.
By definition of E we know that there are (at least) two planes that disagree onthe point p . Assume without loss of generality (symmetry) that the planes p = (1 , i ) and p = (2 , i ) disagree on the point p . We will prove that either p is a heavy plane or p is a heavy plane.Consider the intersection of p and p , i.e., reg = pl ∩ pl = { ( i , i , j , j , . . . , j m ) | j k ∈ [ n ] } . Notethat p ∈ reg . Let ln be a line, which is parallel to the axis and contains a point p (recall that m ≥ ). Thenthe planes p and p disagree on this line (since they disagree on the point p contained in the line ln ), i.e., r ( p ) | ln = r ( p ) | ln . But r ( p ) | ln , r ( p ) | ln ∈ C by definition. This implies that ∆ ( r ( p ) | ln , r ( p ) | ln ) ≥ δ ( C ) · n , i.e., for at least δ ( C ) · n points p ∈ ln it holds that r ( p ) | p = r ( p ) | p .Let BadP oints = { p ∈ ln | p and p disagree on p } . Note that | BadP oints | ≥ δ ( C ) · n . Note that | BadP oints | ≥ δ ( C ) · n . Let BadP lanes = { (3 , i ) − plane | i ∈ [ n ] , ∃ p ∈ BadP oints s.t. p ∈ (3 , i ) − plane } .Note that | BadP lains | ≥ δ ( C ) · n .We claim that for every plane ∈ BadP lanes we have that either plane disagrees with p on some point p ∈ BadP oints or with p on some point p ∈ BadP oints . Hence at least one of p , p disagrees with atleast · | BadP lanes | ≥ · δ ( C ) n planes from BadP lanes . Without loss of generality assume that p disagrees with at least · δ ( C ) · n planes from BadP lanes .Let
BadP lanes p = { plane ∈ BadP lanes | plane disagrees with p } . Note that all planes from BadP lanes are non-intersecting and thus all planes from
BadP lanes p are non-intersecting. Every plane pl ∈ BadP lanes p disagrees with the plane p on some point and hence disagree on at least ( δ ( C ) n ) m − points in their intersection region ( pl ∩ p ) since r ( pl ) | pl ∩ p = r ( p ) | pl ∩ p ∈ C m − .Let total = { p = ( i , j , . . . , j m ) | ∃ plane ∈ BadP lanes p s.t. p ∈ p ∩ plane, r ( plane ) | p = r ( p ) | p } .We have | total | ≥ ( δ ( C ) n ) m − · δ ( C ) · n = ( δ ( C ) · n ) m − since every intersection region (as above) contains atleast ( δ ( C ) n ) m − inconsistency points and there are at least · δ ( C ) · n such regions. We stress that we donot count any inconsistency point more than once, since the planes in BadP lanes p are non-intersecting.Hence the plane p disagree with other planes in at least ( δ ( C ) · n ) m − points (on the plane). Thus E | p has at least ( δ ( C ) · n ) m − non-zero symbols. We conclude that p is a heavy plane of E and the point p iscontained in the plane p . In this section we prove Proposition 4.5.
Proof of Proposition 4.5.
Note that for every i ∈ [ n ] we have | S i | > n − δ ( C ) · n . The following simpleclaim was proven in [5, Proposition 3.1]. For the sake of completeness we provide its proof.Every codeword c ′ of C ′ can be extended to a unique codeword c of C m . To see this note that theprojection of C to C | S i is bijective. It is surjective because it is a projection, and it is injective because | S i | >n − ∆ ( C ) . So, the projection of C to C ′ is bijection, because both codes are of dimension (dim( C )) m .Thus, every word in C ′ has a unique preimage in C .We turn to prove Proposition 4.5. We know that M can be modified in at most α -fraction of points p ∈ S to get M | S ∈ C ′ . Then, by the claim above, M can be modified (outside the submatrix M | S ) to a codewordof C m by changing at most τ -fraction of symbols (since all symbols outside the submatrix M | S are at most τ -fraction of all symbols). We conclude that δ ( M, C m ) ≤ τ + α . In this section we first proof Corollary 3.3. 11 roof Sketch:
For i ≥ let D i be the plane tester for the code C i . Note that the tester D m returns a localview that is a candidate to be in the code C m − . Hence D m − can be invoked on the local view of D m , etc.So, the testers D m , D m − , . . . , D can be composed to result in a n -query tester D comp for the code C m .The robustness of the composed tester will be ρ D comp ( C m ) ≥ ρ D m ( C m ) · ρ D m − ( C m − ) · . . . · ρ D ( C ) .To see this let w ∈ F n m be a word such that δ ( w, C m ) = δ . Then the local view of the tester D m isexpected to be ρ D m ( C m ) · δ far from C m − . When D m − will be invoked, its local view will be ρ D m ( C m ) · ρ D m − ( C m − ) · δ far from C m − , etc. Finally, the local view of D will be ( ρ D m ( C m ) · ρ D m − ( C m − ) · . . . · ρ D ( C )) · δ far from C .Theorem 3.2 says that ρ D m ( C m ) ≥ ( δ ( C )) m m . Hence for constant m ≥ it holds that ρ D comp ( C m ) > is a constant that depends only on δ ( C ) and m .Now, let α m = ρ D comp ( C m ) and note that the query complexity of D comp is n . Let M ∈ F n m suchthat δ ( M, C m ) = δ and think of testing whether M in C m . We argue that Pr I ∼D comp [ M | I ∈ C m | I ] ≥ α m .This is true since otherwise, Pr I ∼D comp [ M | I / ∈ C m | I ] < α m , and then the robustness ρ D comp ( C m ) < α m · (even assuming that whenever M | I / ∈ C m | I we have δ ( M | I , C m | I ) = 1 ). Contradiction.This proves that C m is a ( n , α m ) -strong LTC.Now we prove Claim 3.4. Proof of Claim 3.4.
Let k = dim( C ) . Let E C be an encoder for the code C , which receives a message x ∈ F k and outputs a codeword E C ( x ) ∈ C such that C = (cid:8) E C ( x ) | x ∈ F k (cid:9) . Assume that E C hasrunning time T = O ( k ) . Note that this implies that n ≤ T = O ( k ) since the blocklength can not exceedthe running time of the encoder.For every i ≥ we define E C i to be the encoder for C i , i.e., C i = n E C i ( x ) | x ∈ F k i o . We will arguethat the running time of E C i is i · n i − · T . Since n ≤ T = O ( k ) we will conclude that for any constant i ≥ the running time of E C i is linear (in k i ).We prove the claim by induction on i . The encoder E C = E C was defined and its running time is T = 1 · n − · T . Assume that we defined the encoder E C i − for the code C i − and its running time is ( i − · n ( i − − · T .Let us define the encoder E C i for the code C i . Note that the code C i has message length k i and itsblocklength is n i . Hence the message x ∈ F k i can be viewed as a matrix k × k i − and the encoder E C i willfirst encode (by the encoder E C i − ) every row of the matrix, obtaining the matrix k × n i − . Then E C i willencode every column of the obtained matrix to get a codeword of C i . The running time of the encoder C i is k · (( i − · n i − T ) + n i − T ≤ (( i − · n i − T ) + n i − T = i · n i − · T . Acknowledgements
We thank Eli Ben-Sasson and Or Meir for helpful discussions. The author thanks also Eli Ben-Sasson formany invaluable discussions regarding the “robustness” concept.
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