On the Maximum Number of Codewords of X-Codes of Constant Weight Three

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1 On the Maxiu Nuber of Codewords of X-Codes of Constant Weight Three arxiv: v1 [cs.it] 2 Mar 2019 Yu Tsunoda Graduate School of Science and Engineering Chiba University 1- Yayoi-Cho Inage-Ku, Chiba Eail: yu.tsunoda@chiba-u.jp Abstract X-codes for a special class of linear aps which were originally introduced for data copression in LSI testing and are also known to give special parity-check atrices for linear codes suitable for error-erasure channels. In the context of circuit testing, an (,n,d,x X-code copresses n-bit output data R fro the circuit under test into bits, while allowing for detecting the existence of an up to d-bit-wise anoaly in R even if up to x bits of the original uncopressed R are unknowable to the tester. Using probabilistic cobinatorics, we give a nontrivial lower bound for any d 2 on the axiu nuber n of codewords such that an (,n,d,2 X-code of constant weight exists. This is the first result that shows the existence of an infinite sequence of X-codes whose copaction ratio tends to infinity for any fixed d under severe weight restrictions. We also give a deterinistic polynoial-tie algorith that produces X- codes that achieve our bound. I. INTRODUCTION ery-large-scale integration (LSI testing is vital part of digital circuit production and ais to iniize the nuber of defective circuits due to iperfect anufacturing processes. In typical digital circuit testing, the tester applies test patterns to the circuit under test and checks whether it outputs correct responses. In short, LSI testing ais to detect a discrepancy between expected and observed responses of the circuit under test. While this type of siple coparison-based testing ay see trivial to perfor, it is not always an easy job. Indeed, the ever growing coplexity of odern LSI circuits is aking the volue of required input and output patterns extreely large, resulting in prohibitively long test tie and unacceptably large tester-eory requireents [1]. For this reason, various cost reduction techniques, such as scan-based logic built-in self-test (BIST [2], have been developed to ake odern digital circuits testable []. One iportant developent in test cost reduction is response data copression, where a well-designed copactor hashes the expected and observed responses fro the circuit under test in such a way that, if the original observed data contain unexpected bits signaling a defect, the copressed versions of expected and observed responses also exhibit discrepancies []. With this ethod, the aount of data we should copare becoes saller than if we naively copare every actual output against the expected behavior bit by bit. Yuichiro Fujiwara Division of Matheatics and Inforatics Chiba University 1- Yayoi-Cho Inage-Ku, Chiba Eail: yuichiro.fujiwara@chiba-u.jp This idea of copressing responses is very siilar to nonadaptive group testing such as pooling designs for DNA library screening [5]. However, there is a key difference between group testing in bioinforatics and test copression for LSI circuits, which is the existence of unknowable bits, called Xs, in the case of LSI testing. Ideally, the tester would like to perfectly predict the behavior of a non-defective circuit for any input pattern. However, this is not the case in general with a coplex odern circuit due to various factors such as uninitialized eory eleents, bus contention, floating triple-states, and iperfect siulations []. Thus, the data to be copressed during circuit testing ay contain Xs, that is, logic values that are unknowable to the tester beforehand, coplicating the otherwise classic codingtheoretic proble of hashing a pair of data sets while avoiding collisions if they differ. X-copact [], [7] is a siple ethod to hash responses while aintaining test quality even in the presence of Xs. A response data copactor for X-copact is called an X-code and restricted to a linear ap []. In the language of LSI testing, an (,n,d,x X-code copresses n-bit output fro the circuit under test into bits while allowing for detecting the existence of up to d-bit-wise discrepancies between the observed and correct responses even if up to x bits of the correct behavior are unknowable. Hence, all else being equal, given d and x, X-codes of higher copaction ratio n are desirable. It is notable that X-codes are also known to be useful for error-erasure separation in coding theory [9], where achieving larger n for given, d, and x is again desirable. Because X-codes are linear aps, we ay regard an (,n,d,x X-code as a well-designed n atrix H over the finite field F 2 of order 2 that copresses n-diensional vectors a F n 2 into the corresponding -diensional vectors Ha T F 2. When seen this way, desirable X-codes are those with ore coluns and fewer rows that achieve a higher copaction ratio. The ability to detect discrepancies and high copaction ratio are not the only required properties, however. For practical reasons such as power requireents, copactor delay, and wirability, it is also desirable for the nuber of 1s in each colun to be as sall as possible [], [10]. However, a colun of weight less that or equal to x in an (,n,d,x X-code

2 akes no essential contribution to the achievable copaction ratio [], [11]. Hence, our focus will be on X-codes with the largest possible nuber M x+1 (,d,x of coluns for given nuber of rows and other two test quality paraeters d and x under the condition that the colun weights are all restricted to x+1, naely optial X-codes of constant weight x+1. The siplest case is when x = 1, where the copactor is only required to tolerate a single X. In this case, an (,n,d,1 X-code of constant weight 2 can be shown to be equivalent to a graph of girth d+2 in graph theory [10]. More details on X-codes of constant weight 2 and their connection to graph theory can be found in [10] and references therein. While tolerance of a single X is sufficient in soe cases, ultiple Xs can occur in practice. Unfortunately, our knowledge on X-codes of constant weight x+1 for x 2 is quite liited. As we will briefly review in the next section, even for the next siplest case of x = 2, the precise asyptotic behavior of M (,d,2 is only known for d = 1, which is M (,1,2 = Θ( 2 [11]. For larger d, as far as the authors are aware, the only non-trivial result is an upper bound, which states that M (,d,2 = o( 2 for d, proved by using a tool fro extreal graph theory [12]. While this bound suggests that the asyptotic behavior of M (,d,2 is not so siple, whether M x+1 (,d,x can be superlinear for any fixed d and x has reained an open proble. Here, we ake a substantial step towards understanding the asyptotic behavior of M (,d,2 by proving the existence of an infinite sequence of (,n,d,2 X-codes of constant weight whose copaction ratio tends to infinity for any d. Theore 1.1: For any positive integer, Ω( for d = 2, M (,d,2 = Ω( 5 for d =, Ω( 5 for d. We first give a short and nonconstructive proof based on probabilistic cobinatorics and then provide a deterinistic polynoial-tie algorith that produces X-codes that achieve the above lower bound. In Section II, we briefly review the basic properties of X- codes and known results. Section III gives our new bound on M (,d,2 as well as a deterinistic algorith for constructing X-codes attaining this bound that runs in tie polynoial in for fixed d. Section I concludes this paper with soe rearks. II. PRELIMINARIES Here, we give a foral atheatical definition of X-codes. Basic facts and known results are also briefly reviewed. Let, and n be positive integers. The superiposed su of two binary coluns a = (a 1,a 2,...,a T, b = (b 1,b 2,...,b T F 2 is defined to be a b = (a 1 b 1,a 2 b 2,...,a b T, where a i b i = 0 if a i = b i = 0 and 1 otherwise. The addition of a + b between two coluns a,b F 2 is assued to be the coordinatewise su over F 2 as usual. A binary colun a is said to be contained in another binary colun b if a b = b. For positive integer d and non-negative integer x, an n binary atrix H is an (,n,d,x X-code if the superiposed su of any x coluns does not contain the addition of any other up to d coluns. The coluns of an X-code are the codewords, while the nuber of rows is the length. By definition, an (,n,d,x X-code with d 2 is an (,n,d 1,x X-code. For d 2 and x 1, an (,n,d,x X-code is also an (,n,d+1,x 1 X-code []. It is notable that the definition of an (,n,1,x X-code coincides with that of an x-disjunct atrix [1] of size n for group testing. Disjunct atrices are also known as cover-free failies [1] and superiposed codes [15]. An (,n,d,0 X-code fors a parity-check atrix for a linear code of length n, diension at least n, and iniu distance at least d+1. For x 1, an (,n,d,x X-code can be seen as a special parity-check atrix that can treat errors and erasures separately over an error-erasure channel [9]. Now, to see how X-codes work in LSI testing, let us consider the following (,,1,1 X-code H. H = Assue that a and b are -diensional vectors over {0, 1, X} that represent the observed and expected responses fro the circuit under test, respectively. Because X represents an unknowable logic value, coputation involving X is defined by a + X = X + a = X, 0 X = X 0 = 0, and 1 X = X 1 = X. LSI testing with the X-code H copares -diensional vectors Ha T and Hb T instead of a and b. The property of H as a (,,1,1 X-code guarantees that a discrepancy of no ore than one bit between the actual output a and the correct output b can be detected by coparing the shrunk responses a and b even if up to 1 bit of the correct behavior is unknowable. For exaple, when a = (0,1,1,0,0,0 and b = (X,1,1,1,0,0, the shrunk responses are Ha T = (0,1,0,1 T and Hb T = (X, X,1,1 T. As can be seen easily, there exists a discrepancy between the last bits of Ha T and Hb T, so that we can find the circuit under test is defective. As a linear function for copaction, all else being equal, it is desirable for an (,n,d,x X-code to have as large n as possible for given,d, and x. Recall that M x+1 (,d,x is the axiu nuber n for which there exists an (,n,d,x X-code of constant weight x+1. For d = 1 and x = 2, it is known that M (,1,2 ( 1 with equality if and only if 1, (od [11]. Because M (,d,2 M (,1,2 by definition, the above upper bound holds for all d 2 as well. While the above inequality says that M (,1,2 = Θ( 2, it is also known that M (,d,2 = o( 2 for any d [12]. Therefore, we have the following theore.

3 Theore 2.1 ( [11], [12]: It holds that Θ( 2 for d = 1, M (,d,2 = O( 2 for d = 2,, o( 2 for d. Although an attept has been ade to construct atrices siilar to (,n,d,2 X-codes of constant weight for large d in [12], to the best of the authors knowledge, no nontrivial lower bounds on M (,d,2 are known for d 2. III. MAIN RESULTS This section is divided into two subsections. In Section III-A, we derive a general lower bound on M (,d,2 by using the probabilistic ethod in cobinatorics [1]. Section III-B derandoizes the probabilistic proof to deonstrate that an (,n,d,2 X-code of constant weight which attains the derived lower bound can be constructed deterinistically in tie polynoial in. A. General lower bound on M (,d,2 Here, we prove the following general asyptotic bound. Theore.1: For sufficiently large, it holds that α for d = 2, M (,d,2 β 5 for d =, γ 5 for d, where α = 1 ( , 179 β = , and (15 (711 1 γ = 5 ( Note that Theore 1.1 iediately follows fro this bound. To prove the above theore, we will first show the following two leas. Lea.2: For sufficiently large, there exists an (,c,2,2 X-code of constant weight with c = 1 ( Lea.: For any d and any p [0,1], there exists an (,N(,d,p,d,2 X-code of constant weight, where ( (( (( ( (( N(,d,p = p p + p ( (( + i i 1 p i+2. i+2 i= To show these leas, we eploy a well-known class of cobinatorial designs. A set syste of order v is an ordered pair (,B such that is a finite set of points with = v and B is a faily of subsets of, called blocks. The pointby-block incidence atrix of a set syste (,B is the binary B atrix H = (h i,j such that rows and coluns are indexed by points and blocks, respectively, and h i,j = 1 if the ith point is contained in the jth block and h i,j = 0 otherwise. For a subset B of B, the odd-point union U of B is defined to be U = {a {B B a B} is odd}. It is straightforward to see that an (,n,d,x X-code whose rows and coluns are indexed by and B, respectively, is equivalent to a set syste (,B of order with B = n such that no union of x blocks contains the odd-point union of any other d or fewer blocks as a subset. We denote by ( the set of -subsets, called triples, of. X-codes of constant weight can be characterized by soe subsets of (. A configuration in a set syste (,B is a subset C of B. When C = i, a configuration C is an i- configuration. A configuration is (d, x-forbidden if it appears in no (,n,d,x X-codes of constant weight. We denote by C,d,x the set of (d,x-forbidden configurations on the point set. For instance, for {a,b,c,d}, a -configuration C = {{a,b,c},{b,c,d},{a,b,d}} is (1,2-forbidden, that is, C C,1,2, because the union{a,b,c,d} = {a,b,c} {b,c,d} contains {a,b,d} as a subset, aking it ipossible to appear in an (,n,1,2 X-code on the point set. The probabilistic proof of Leas.2 and. relies on the fact that it is enough to show the existence of a set syste that avoids all (d,2-forbidden configurations. The set C,d,2 of (d, 2-forbidden configurations can be partitioned into the following sets of configurations: C,d,2 = i d+2 C,d,2 (i, where C,d,2 (i is the set of i-configurations that are (d,2- forbidden. Note that C,d,2 (1 and C,d,2 (2 are epty sets since the set syste (, ( consists of distinct triples. Proof of Lea.2: Let = {1,2,...,}. Take a set B of triples by picking eleents of ( uniforly at rando with probability p. Let X = {C C,2,2 C B} be the rando variable that counts the nuber of (2, 2-forbidden configurations in B. Note that because discarding a triple T in B reoves the configuration containing T, deleting at ost one triple fro each (2, 2-forbidden configuration in B gives an (,n,2,2 X-code with n B X. Therefore, there exists an (,n,2,2 X-code with n E( B X. Let D be the set of -configurations in (, ( that consists of eleents of. Since every configuration in ( consists of distinct triples, the largest nuber of eleents of in a configuration in C,2,2 ( is, that is, C,2,2 ( D. Siilarly, C,2,2 ( D, where D is the set of - configurations in (, ( that consists of eleents of. Therefore, by linearity of expectation, we have E( B X = E( B E(X ( = p C,2,2 (i p i i= ( p D i p i i=

4 = ( p (( (( p + ( (( p. By settingp = 2( , the right-hand side of the above inequality is 1 ( f( with f( = o(, as desired. It is notable that precisely counting the nuber of configuration in C,2,2 (i instead of D i can only give asyptotically the sae bound. Lea. can be obtained by essentially the sae probabilistic arguent as in the proof of Lea.2. Proof of Lea.: Consider the rando variable Y = {C C,d,2 C B} that counts the nuber of (d,2- forbidden configuration in B and follow the sae arguent as in the proof of Theore.1 to show the existence of an (,n,d,2 X-code of constant weight with n E( B Y. Note that for any (i+2-configuration C in C,d,2 (i+2 with i d, it holds that i 1 T +, 2 T C where equality holds when i is even and C consists of i triples A 1,A 2,B 1,B 2,...B i such that A 1 A 2 =, (A 1 A 2 B j = φ for any 1 j i, and each eleent in the union 1 j i B j is contained in exactly 2 triples of {B 1,B 2,...,B i }, or when i is odd and C consists of i triples A 1,A 2,B 1,B 2,...B i such that A 1 A 2 = and there exists exactly one triple B in {B 1,...,B i } such that (A 1 A 2 B = 1, and each eleent in the union 1 j i B j except for an eleent in (A 1 A 2 B is contained in exactly 2 triples of {B 1,B 2,...,B i 2 }. Therefore, by linearity of expectation, we have E( B Y ( = p ( p C,d,2 (i+2 p i+2 i=1 (( (( + ( i= p + ( + i 1 (( p (( + i 1 i+2 p i+2 as desired. Now, we prove Theore.1. Proof of Theore.1: By Lea.2, it holds that M (,2,2 α, where α = 1 ( For d =, by setting p = 2 ( to axiize N(,d,p, we have N(,d,p = β 5 + g( with β = and g( = o(. 5 By Lea., it holds (15 (711 1 that M (,,2 β 5. Siilarly, for d, by setting p = 2 ( , we have N(,d,p = γ 5 +h( ( with γ = and h( = o( 5. By Lea., for d it holds that M (,d,2 γ 5, as desired., B. Construction algorith To extract a deterinistic algorith fro our probabilistic proof in the previous section, we follow the approach of [12], which uses the ethod of conditional expectations [1]. Let T i, 1 i ( (, be the triples in in arbitrary order. While the proof of Leas.2 and. randoly picks each triple in (, here we deterinistically decide whether to pickt i one by one frot 1 throught (. Note that the picked triples ay contain soe forbidden configurations in C,d,2. To reove the forbidden configurations, the final deletion process is done the sae way as in the probabilistic proof by discarding at ost one triple fro each realized forbidden configuration. Now we describe our derandoized algorith in detail. To record our decision on whether we pick a triple at each step, define t i = 1 if T i is included and 0 otherwise. For a given binary sequence t 1...t i of length i and forbidden configuration C C,d,2, define r = {T j C t j = 1,1 j i} and s(c = { p C r if {T j C t j = 0} = 0, 0 otherwise. Given the first i decisions t 1...t i on the triples, assue for the oent that we pick each of reaining ( i triples independently and uniforly at rando with probability 2( for d = 2, p = 2 ( for d =, 2 ( for d. This is equivalent to hypothetically regarding each t j for i+ 1 j ( as an independent rando variable such that t j = 1 with probability p and 0 with probability 1 p. Let B = {T j t j = 1,1 j ( }, and define X = {C C,d,2 C B}. B and X represent the sets of picked triples and fored forbidden configurations after the hypothetical rando sapling, respectively. If we started with fixed i decisions t 1,...,t i and perfored the rando sapling for the reaining triples, the nuber of triples after the deletion process would be at least the conditional expectation E( B X t i,...t i = {T j t j = 1,1 j i} (( + i p s(c. C C,d,2 For b = 0,1, define E((t i,t i+1=b = E( B X t i,...t i,t i+1 = b, which is the conditional expectation when the rando variable t i+1 is realized as b. When i = 0, we define Since E((t 0,t 1 = b = E( B X t 1 = b. E( B X t i,...t i = pe((t i,t i+1=1 +(1 pe((t i,t i+1=0,

5 it holds that E( B X t i,...t i ax{e((t i,t i+1=1,e((t i,t i+1=0 }. Therefore, by starting fro no decisions on the triples and picking T i at the ith step if and only if E((t i,t i+1=1 > E((t i,t i+1=0, we end up with at least E( B X triples after the deletion process, which is precisely the guaranteed nuber of codewords of an X-code by Theore.1. Algorith 1 describes the above deterinistic procedure. Algorith 1 Derandoized algorith for Theore.1 Input: Point set of cardinality Output: (, B,d,2 X-code (,B of constant weight 1: B φ 2: Fix the order of {T 1,T 2,...T ( } = ( arbitrarily : C,d,2 set of all forbidden configurations in ( : for i = 1 to ( do 5: if (E((t i 1,t i = 1 > E((t i 1,t i = 0 then : B B {T i } 7: t i 1 : else 9: t i 0 10: end if 11: end for 12: while C C,d,2 s.t. C B do 1: B B \{T}, where T is an arbitrary triple in C 1: end while 15: return (, B In the reainder of this section, we show that Algorith 1 runs in tie polynoial in. Our analysis here is quite rough but enough to show that it is efficient in a technical sense. First, note that listing all forbidden configurations in C,d,2 only takes tie polynoial in because C,d,2 = d 1 + O(. The steps for picking triples require coputing two conditional expectations ( ties each. Since coputing a conditional expectation takes at osto( C,d,2 tie, d 1 9+ the steps for picking triples can be done in O(. Checking whether a given triple is contained in B takes O(log B tie by using the binary search. Therefore, the coplexity of the final deletion process is bounded fro above by C,d,2 log (. Hence, the total run tie will not exceed d 1 9+ O(, as required. I. CONCLUSION We have derived a lower bound on the axiu nuber n for which an (,n,d,2 X-code of constant weight exists. This is the first nontrivial lower bound on M (,d,2 for general d and deonstrates that constant-weight X-codes can substantially reduce the aount of response data under the presence of a ulti-bit discrepancy, ultiple unknowable bits, and a severe constraint on fan-out (see [11] and references therein for the background on the fan-out issue. We have also proved that such X-codes can be constructed deterinistically in tie polynoial in. This was done by first proving their existence through a probabilistic arguent and then derandoizing it by the ethod of conditional expectations. It is notable that this approach was also shown effective in [12] for a ore specific situation where ultiple Xs are rather rare but do occur. It would be interesting to see how widely this approach can be applied to siilar probles. Finally, it should be noted that while we have ade nontrivial progress towards understanding the asyptotic behavior of M (,d,2, there still reains a substantial gap between the sharpest upper and lower bounds on M (,d,2. In fact, for general d and x, the proble of deteriningm x+1 (,d,x is nearly copletely open. We hope that future work addresses these challenging areas. ACKNOWLEDGMENT This work was supported by JSPS KAKENHI Grant Nuber JP1J20 (Y.T. and KAKENHI Grant Nuber JP17K12 (Y.F.. REFERENCES [1] E. J. McCluskey, D. Burek, B. Koeneann, S. Mitra, J. H. Patel, J. Rajski, and J. A. Waicukauski, Test data copression, IEEE Design Test Coput., vol. 20, pp. 7 7, Mar./Apr [2] E. J. McCluskey, Logic Design Principles with Ephasis on Testable Sei-Custo Circuits. Englewood Cliffs, NJ: Prentice-Hall, 19. [] P. Girard, N. Nicolici, and X. Wen, Eds., Power-Aware Testing and Tes Strategies for Low Power Devices. New York: Springer, [] R. Kapur, S. Mitra, and T. W. 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