Lower Bounds for Cover-Free Families

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1 Loe Bouds fo Cove-Fee Families Ali Z. Abdi Covet of Nazaeth High School Gade, Abas 7, Haifa Nade H. Bshouty Dept. of Compute Sciece Techio, Haifa, 3000 Apil, 05 Abstact Let F be a set of blocks of a t-set X. X, F is called, - cove-fee family, CFF povided that, the itesectio of ay blocks i F is ot cotaied i the uio of ay othe blocks i F. We give e asymptotic loe bouds fo the umbe of miimum poits t i a, -CFF he = F ɛ fo some costat ɛ /. Keyods: Cove-Fee Family, Loe Boud. Itoductio Let F be a set of blocks subsets of a t-set X. X, F is called, -covefee family, CFF povided that, fo ay blocks A, A,..., A F ad ay othe blocks B, B,..., B F e have A i B j. i= j= Sice usig De Moga a, CFF ca be tued ito, CFF, thoughout the pape e assume that. Cove-fee families ee fist itoduced i 964 by Kautz ad Sigleto [5]. Let N,, deote the miimum umbe of poits X i ay, - CFF havig F = blocks. The best ko loe boud fo N,, is [, 4, 7] N,, = Ω log

2 he ad Ω he >. The costat of the Ω is asymptotically /, /4 ad /8, espectively. Stiso et. al, [8], poved that N,, N,, + N,,. They the use it ith to pove to bouds. The fist boud is + + N,, Ω log + log he, [8, 6], ad N,, Ω + log + log 3 4 fo ay, [8]. To the best of ou koledge 4 is the best boud ko he. D yachkov et. al. beakthough esult, [3], implies that fo ad, + + N,, = Θ log + log 5 ad fo ad, N,, O + log + log. 6 I this pape e give a e loe boud fo, -CFF he >. We combie the to techiques used i [8, 6] ad [] to give the folloig asymptotic loe boud. Theoem. Fo ay k < / ad N,, ad fo + k k k k k k! + k + k +! l k = Ω + k k k+ k e k = Ω log + N,, = Θ. + +! l k+ log

3 Ou boud is Θ k e l k times geate tha the pevious boud i 4. I paticula, he k is costat, ou loe boud impoves the boud i 4 to N,, Ω + log k log + log. 7 A slightly bette boud ca be achieved he + k k k+ + k k k+ l /k+. Fo example, let = 4. The table i Figue compaes ou esults ith the pevious esults asymptotic values Pevious Loe Uppe Ou Loe Bouds 3, 4 Boud [3] Boud / / /3 /3 3/4 3/4 4/5 4 log 4 log 4 log > log 4/ log 3 log 4 log 5 Figue : Results fo = 4. Fist Loe Boud I this sectio e pove Lemma. Let /. If = Ω log + the N,, = Θ. 8 3

4 Otheise, N,, Ω + log. 9 + l Lemma follos fom the folloig Lemma. Let ɛ < be ay costat. Fo / e have + N,, mi ɛ + + l, ɛ 0 Poof. Let X, F be a optimal, -CFF. Let F = {F,..., F }, X = N = N,, ad assume ithout loss of geeality that X = [N] := {,..., N}. Defie v i {0, }, i =,..., N hee v i j = if ad oly if i F j. Let V = {v i i =,..., N}. Let V 0 be the set of v i of eight tv i i.e., j vi j equal to. Let m = + l ad coside the to sets V = {v i < tv i < m} ad V = {v i tv i m}. Obviously, V = V 0 V V is a patitio of V. Suppose V 0 ɛ ad max V, V ɛ + l. + + Coside W = {j,..., j j < < j } ad W W the set of all j,..., j hee o v i V 0, i =,..., N, satisfies v i j = = v i j =. Obviously, W = V 0 ɛ. Fix a elemet v V ad adomly ad uifomly choose j = j,..., j W. We have tv m P j W [v j = = v j = ] W ɛ. 4

5 Theefoe, the expectatio of the umbe of v V fo hich v j = = v j = is at most m V ɛ = m V ɛ + l ɛ ɛ + +. l + + Theefoe, thee is j = j,..., j W such that the umbe of v V that satisfies v j = = v j = is / +. Sice the eight of evey v V is geate tha, e ca choose e eties j,..., j {j,..., j } such that fo evey v V hee v j = = v j = thee is j l such that v j =. l No adomly ad uifomly choose := + distict k,..., k []. Let A be the evet that {k,..., k } {j,..., j } Ø. The pobability that A does ot happe is = The P[A v V v k = = v k = 0] m + V m + V + V e m ad V e m ɛ + ɛ = ɛ + + < l + l e + l e l l

6 Theefoe, P[A v V v k = = v k = 0] <. Theefoe, thee is {k,..., k } such that {k,..., k } {j,..., j } = Ø ad fo evey v V thee is k l {k,..., k } hee v kl =. No it is easy to see that thee is o v V hee v j = = v j =, v j = = v j = 0 ad v k = = v k = 0. This implies that F j i i= i= hich is a cotadictio. 3 The Secod Boud I this sectio e pove Theoem. F j i Lemma 3. Fo ay k / ad N,, F ki i= + k k k+ k k k! + k + k +! l k = Ω + +! l k Poof. We pove the lemma by iductio o. Fom Lemma the lemma holds fo = k. No assume the boud holds fo some ad evey that satisfies + k k k+. We o pove the boud fo + ad + k k k+ N, +, N,, + N, +, N + j,, j j= N,, + k k k! j + k + j= k +! l k j k k k! k + k +! l k j + k k k! k + k +! l k k k k! + k + k +! l k 6 j= 0 x + dx. 3

7 Hee, iequality comes fom [8]. Iequality follos fom the fact that N +, +, N,,. Iequality 3 follos fom the iductio hypothesis sice j = j + k k k+ j + k j k k+ = + j + k k k+. Refeeces [] N. Alo, V. Asodi. Leaig a Hidde Subgaph. SIAM J. Discete Math. 84. pp [] A. G. D yachkov ad V. V. Rykov. Bouds o the legth of disjuctive codes. Poblemy Peedachi Ifomatsii, 83, pp. 7 3, 98. [3] A. G. D yachkov, I. V. Voobev, N. A. Polyasky, V. Yu. Shchuki. Bouds o the ate of disjuctive codes. Poblems of Ifomatio Tasmissio. 50, pp [4] Z. Füedi. O -Cove-fee Families. J. Comb. Theoy, Se. A, 73. pp [5] W. H. Kautz ad R. C. Sigleto. Noadom biay supeimposed codes, IEEE Tas. Ifom. Theoy. 0, pp [6] X. Ma ad R. Wei. O Bouds of Cove-Fee Families. Desigs, Codes ad Cyptogaphy, 3, pp , 004. [7] M. Ruszikó. O the Uppe Boud of the Size of the -Cove-Fee Families. J. Comb. Theoy, Se. A. 66. pp [8] D. R. Stiso, R. Wei ad L. Zhu. Some Ne Bouds fo Cove-Fee Families. Joual of Combiatoial Theoy, Seies A. 90, pp

NOTE. Some New Bounds for Cover-Free Families

NOTE. Some New Bounds for Cover-Free Families Jounal of Combinatoial Theoy, Seies A 90, 224234 (2000) doi:10.1006jcta.1999.3036, available online at http:.idealibay.com on NOTE Some Ne Bounds fo Cove-Fee Families D. R. Stinson 1 and R. Wei Depatment