NOTE. Some New Bounds for Cover-Free Families

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1 Jounal of Combinatoial Theoy, Seies A 90, (2000) doi: jcta , available online at on NOTE Some Ne Bounds fo Cove-Fee Families D. R. Stinson 1 and R. Wei Depatment of Combinatoics and Optimization, Univesity of Wateloo, Wateloo, Ontaio N2L 3G1, Canada and L. Zhu 2 Depatment of Mathematics, Suzhou Univesity, Suzhou , China Communicated by the Managing Editos Received Febuay 11, 1999 Let N((, ), T) denote the minimum numbe of points in a (, )-cove-fee family having T blocks. In this pape, e pove to ne loe bounds on N Academic Pess 1. INTRODUCTION Cove-fee families ee fist intoduced in 1964 by Kautz and Singleton [9] to investigate supeimposed binay codes. These stuctues have been discussed in seveal equivalent fomulations in subjects such as infomation theoy, combinatoics, and goup testing by numeous eseaches (see, fo example, [1, 2, 48, 12]). In 1988, Mitchell and Pipe [10] defined the concept of key distibution pattens, hich ae in fact a genealized type of cove-fee family. Some papes giving constuctions and bounds fo these objects include [3, 4, 11, 13]. Hee is the definition of a cove-fee family. Definition 1.1. Let X be an n-set and let F be a set of subsets (blocks) of X. (X, F) is called a (, )-cove-fee family (o (, )-CFF) povided 1 Reseach suppoted by NSERC Gants IRC and RGPIN Reseach suppoted by NSFC Gants Copyight 2000 by Academic Pess All ights of epoduction in any fom eseved. 224

2 NOTE 225 that, fo any blocks B 1,..., B # F and any othe blocks A 1,..., A # F, e have, i=1 B i 3. Note that the classical definition of cove-fee family is the case =1 of ou definiton. Let N((, ), T) denote the minimum numbe of points in any (, )- CFF having T blocks. The best knon uppe bounds on N((, ), T) (see [3, 4, 13]) use pobabilistic methods. In [13], the uppe bound as stated as hee N((, ), T) j=1 A j. (+) log T, &log p p=1& (+). + In this pape, e discuss loe bounds on N((, ), T). Loe bounds on N((1, ), T) have been studied by seveal eseaches. Spene's theoem states that N \(1, 1), \ n n 2 x++n fo all positive integes n2. The best loe bounds on N((1, ), T) fo geneal ae found in [2, 7, 12], hee diffeent poofs of the folloing theoem can be found. Theoem 1.1. Fo any 2, it holds that N((1, ), T)c 2 log T, log fo some constant c. The constant c in Theoem 1.1 is shon to be appoximately 12 in [2], appoximately 14 in [7] and appoximately 18 in[12]. Loe bounds on N((, ), T) fo abitay ee fist discussed in Dye et al [3]. Moe ecently, Engel [4] poved the folloing to impoved bounds.

3 226 NOTE Theoem 1.2. Theoem 1.3. N((, ), T)( +&1 )log(t&&+2). Fo any =>0, it holds that N((, ), T)(1&=) (+&2) +&2 log(t&&+2) (&1) &1 &1 (&1) fo all sufficiently lage T. We povide to ne bounds in this pape, hich ae stated as Theoem 3.2 and Theoem 4.4. The poofs e give ae puely combinatoial and quite simple, and ae based on a ne ecusive fomula that e pove in the next section. Ou loe bounds ae usually bette than the bounds given in [3, 4]. In paticula, ou bounds build on the bound of Theoem 1.2. If e fix (espectively, ) and let (espectively, ) vay, then ou esults ae alays stonge than the pevious bounds. Fo fixed, Theoem 3.2 impoves Theoem 1.2 by a facto of log, and Theoem 4.4 impoves Theoem 1.3 by a facto of 2 log. On the othe hand, hen =, Theoem 1.3 is bette than ou Theoem 4.4 by a facto of -. Logaithms used in this pape ae alays to the base to. 2. PRELIMINARY LEMMAS In this section, e pesent a fe easy peliminay lemmas that e ill use late. Lemma 2.1. Let A i be an abitay block in F=[A 1, A 2,..., A T ] and let B i A i. If F is (, )-cove-fee, hee 2, then (1) F 1 =[A j "B i :1jT, j{i] is (, &1)-cove-fee, and (2) F 1 =T&1. Poof. (1) Suppose that, t=1 (A et "B i )(A j1 "B i ) _ (A j2 "B i ) _ }}} _ (A j&1 "B i ), hee [e 1,..., e ] _ [j 1,..., j &1 ][1,..., T]"[i]. Then, t=1 hich is a contadiction. A et A j1 _ A j2 _ }}} _ A j&1 _ A i,

4 NOTE 227 (2) Fist e note that A j "B i =% < fo any j=% i. Next, e sho fo any j 1, j 2 {i that A j1 "B i {A j2 "B i. This is seen easily since A j1 "B i =A j2 "B i implies that A j1 A j2 _ A i, hich is a contadiction, since 2. The esult follos. K The folloing lemma can be poved in a simila ay. Lemma 2.2. Let A i be an abitay block in F=[A 1, A 2,..., A T ]. If F is (, )-cove-fee, hee 2. Then (1) F 2 =[A j & A i :1jT, j{i] is (&1, )-cove-fee, (2) F 2 =T&1. The folloing lemma is simple. Lemma 2.3. N((, ), T)=N((, ), T). Poof. (X, F) isa(, )-CFF if and only if (X, F )isan(, )-CFF, hee F =[F : F # F ]. K No e pove the folloing ecusive fomula that is the basis of ou ne bounds. Lemma 2.4. N((, ), T)N((, &1), T&1)+N((&1, ), T&1). Poof. Suppose (X, F) isa(, )-CFF ith X =n=n((, ), T) and F =T. Choose a block A i # F and let n 1 = A i. Then (X"A i, F 1 )isa (, &1)-CFF ith n&n 1 points, and (X & A i, F 2 ) is a (&1, )-CFF ith n 1 points (hee F 1 and F 2 ae defined in Lemmas 2.1 and 2.2, espectively). It is clea that n&n 1 N((, &1), T&1) and n 1 N((&1, ), T&1). Thus N((, ), T)=n=n&n 1 +n 1 N((, &1), T&1)+N((&1, ), T&1). K 3. THE FIRST BOUND To discuss the fist bound, e define \+ + g(,, T)= log T. log(+)

5 228 NOTE The function g has the folloing popety, hich can be easily poved using the fact that the function log xlog(x&1) is deceasing fo x>1. Lemma 3.1. Fo, 2 and T+, it holds that g(,, T)g(, &1, T&1)+g(&1,, T&1). Fom the above lemma, e obtain the folloing bound. Theoem 3.2. Fo, 1 and T+>2, e have N((, ), T)2c \+ + log T, log(+) hee c is the same constant as in Theoem 1.1. Poof. Fist conside the case =1. Fom Theoem 1.1, e have N((, ), T)c 2 log T. log We ill sho that hich is equivalent to shoing that c 2 log log T2c +1 log T, log(+1) 2 log(+1) (+1) log 2. Since the left side of the above inequality is an inceasing function of, e have since log(+1) 4 log 3>2, log 3

6 NOTE 229 The case =1 is simila, in vie of Lemma 2.3. Fo the geneal case hee, 2, e pove the bound by induction on + as N((, ), T)N((&1, ), T&1)+N((, &1), T&1) 2cg(&1,, T&1)+2cg(, &1, T&1) 2cg((, ), T). Hee, the fist inequality comes fom Lemma 2.4, the second one comes fom an induction assumption, and the thid one comes fom Lemma 3.1. K 4. THE SECOND BOUND Ou second bound is consideably lage than ou fist bound. Hoeve, e can only pove that this bound holds asymptotically. Define \+ h(, )= + (+). log \+ + We ill discuss some popeties of the function h(, ), but fist e need an easy peliminay lemma. Lemma 4.1. Fo positive integes and, e have (+) + = 1 + : i=0 i + +&i\ i + > \ + +. The folloing lemma establishes an impotant popety of the function h(, ). Lemma 4.2. Fo integes, 2, it holds that h(, )<h(&1, )+h(, &1).

7 230 NOTE Poof. Fo, 2, e have h(&1, )+h(, &1)&h(, ) \+&1 &1 + (+&1) \+&1 + (+&1) = + log \+&1 log &1 + \+&1 + \+&1 &1 + (+&1) \+&1 + (+&1) = + log \+&1 log &1 + \+&1 + \+&1 + (+) \+&1 &1 + (+) & & log \+ log + \+ + = \+&1 &1 + (+) + log &log \ + + log \+&1 &1 + log \ log &(+) log \+ + & \+&1 + log \ log \ +&1 \+ & + (+) log \+ + No suppose (note that the function h is symmetic in and, so e can make this assumption ithout loss of geneality). Then Fo, e have \ + + by Lemma 4.1, so e have \ +&1 &1 + \ +& (+)+ > \+ + (+) log + &log \ + + >0.

8 NOTE 231 Thus h(&1, )+h(, &1)&h(, ) \+&1 &1 & \+&1 + + (+) log log \+&1 \ +&1 + = log \+ & + log \+ log \+ + log \ +&1 log \+ ++. Thus, e need to sho that (+) log + + &log \ log \ log \ +&1 &(+) log+ + + \ +(+) log + hich is equivalent to shoing that Simplifying, e get o (+) log + & + +(+) log + log \+ + >0, \ + + (+)\ + (+) (+) > + \+. + \ + ((+)) + hich follos fom Lemma 4.1. \ + ) + ((+)) (+) > \+, + \ + + \ + > + \+ +, The folloing lemma is easy to veify. K

9 232 NOTE Lemma 4.3. Fo all 2, it holds that No e pove ou second bound. 0.7 (+1)2 log(+1) < 2 log. Theoem 4.4. that Fo any integes, 1, thee exists an intege T, such \+ N((, ), T)0.7c + (+) log T log \+ + fo all T>T,, hee c is the same constant as in Theoem 1.2. Poof. Fist fo =1, e have N((1, ), T)c 2 log T>0.7c (+1)2 log log(+1) log T by Lemma 4.3. The case =1 is equivalent, in vie of Lemma 2.3. Thus e can assume that, 2, and hence +4. Fo s4, let : s =min By Lemma 4.2, e have : s >1. Since )+h(, &1) : +s,, 2 = {h(&1, h(, ). lim T log T log(t&1) =1, thee exists a sequence of integes T s, s=4, 5,..., such that : s log T log(t&1) fo all T>T s and T s T s&1 +1, fo all s4. No e pove the conclusion by induction on s=+. LetT>T s ;then T&1>T s&1. By induction e have

10 NOTE 233 N((, ), T)N((&1, ), T&1)+N((, &1), T&1) 0.7c }(h(&1, )+h(, &1)) log(t&1) h(&1, )+h(, &1) =0.7c h(+) log(t&1) h(+) 0.7c } : + } h(+) log(t&1) 0.7c } h(+) logt, as equied. K Ou feeling is that the above bound is tue not only fo T sufficiently lage, but in fact fo all T. It also appeas unlikely that ou bound could be impoved significantly by an appoach based on Lemma 2.4 togethe ith Theoem 1.2. This is based on the folloing expeimental evidence. Define p(1, )=p(, 1)= 2 log and let p(, )=p(&1, )+p(, &1) fo, 2. We computed p(, ) fo +400, and found that h(, )<p(, )<1.762 h(, ) fo all such values of and. ACKNOWLEDGMENT The authos thank the efeee fo valuable comments and fo binging Engel's pape [4] to ou attention. REFERENCES 1. K. A. Bush, W. T. Fedee, H. Pesotan and D. Raghavaao, Ne combinatoial designs and thei application to goup testing, J. Statist. Plann. and Infeence 10 (1984), A. G. Dyachkov and V. V. Rykov, Bounds on the length of disjunctive codes, Poblemy Peedachi Infomatsii 18 (1982), 713. [In Russian] 3. M. Dye, T. Fenne, A. Fieze, and A. Thomason, On key stoage in secue netoks, J. Cyptology 8 (1995), K. Engel, Inteval packing and coveing in the boolean lattice, Combin. Pobab. Comput. 5 (1996), P. Edo s, P. Fankl, and Z. Fu edi, Families of finite sets in hich no set is coveed by the union of to othes, J. Combin. Theoy Se. A 33 (1982), P. Edo s, P. Fankl, and Z. Fu edi, Families of finite sets in hich no set is coveed by the union of othes, Iseal J. Math. 51 (1985), Z. Fu edi, On -cove-fee families, J. Combin. Theoy Se. A 73 (1996),

11 234 NOTE 8. F. K. Hang and V. T. So s, Non-adaptive hypegeometic goup testing, Studia Sci. Math. Hunga. 22 (1987), W. H. Kautz and R. C. Singleton, Nonandom binay supeimposed codes, IEEE Tans. Infom. Theoy 10 (1964), C. J. Mitchell and F. C. Pipe, Key stoage in secue netoks, Discete Appl. Math. 21 (1988), K. A. S. Quinn, Bounds fo key distibution pattens, J. Cyptology, in pess. 12. M. Ruszinko, On the uppe bound of the size of the -cove-fee families, J. Combin. Theoy Se. A 66 (1994), D. R. Stinson, Tan van Tung, and R. Wei, Secue famepoof codes, key distibution pattens, goup testing algoithms and elated stuctues, J. Statist. Plann. Infeence, in pess.

Research Article On Alzer and Qiu s Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function

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