Discee Mahemaics and Theoeical Compue Science DMTCS vol. 4:2, 202, 26 270 Secue Famepoof Codes Though Biclique Coves Hossein Hajiabolhassan,2 and Faokhlagha Moazami 3 Depamen of Mahemaical Sciences, Shahid Beheshi Univesiy, Tehan, Ian 2 School of Mahemaics, Insiue fo Reseach in Fundamenal Sciences (IPM), Tehan, Ian 3 Depamen of Mahemaics, Alzaha Univesiy, Tehan, Ian eceived 0 h Apil 202, evised 26 h Ocobe 202, acceped 5 h Novembe 202. Fo a binay code of lengh v,av-wod w poduces by a se of codewods {w,...,w } if fo all i =,...,v, we have w i 2{wi,...,wi }. We call a code -secue famepoof of size if = and fo any v-wod ha is poduced by wo ses C and C 2 of size a mos, hen he inesecion of hese ses is non-empy. A d-biclique cove of size v of a gaph G is a collecion of v complee bipaie subgaphs of G such ha each edge of G belongs o a leas d of hese complee bipaie subgaphs. In his pape, we show ha fo 2, an -secue famepoof code of size and lengh v exiss if and only if hee exiss a -biclique cove of size v fo he Knese gaph KG(, ) whose veices ae all -subses of a -elemen se and wo -subses ae adjacen if hei inesecion is empy. Then we invesigae some connecion beween he minimum size of d-biclique coves of Knese gaphs and cove-fee families, whee an (, w; d) cove-fee family is a family of subses of a finie se X such ha he inesecion of any membes of he family conains a leas d elemens ha ae no in he union of any ohe w membes. The minimum size of a se X fo which hee exiss an (, w; d) cove-fee family wih blocks is denoed by N((, w; d),). We pove ha fo >2and >s, we have bc d (KG(, )) bc m(kg(, s)), whee m = N(( s, s; d), 2s). Finally, we show ha fo any apple i<, apple j<w, and + w we have N((, w; d),) N(( i, w j; m),), whee m = N((i, j; d), w + i + j). Keywods: cove-fee family, secue famepoof code, biclique cove, Hadamad maix Inoducion Illegal copy is a majo poblem in digial daa. Famepoof codes ae one of many diffeen echniques o peven poducs agains illegal copy ha wee fis inoduced by Boneh and Shaw [2]. To poec digial daa, he disibuo maks each copy uniquely wih a codewod. These codewods have he popey ha an illegal copy can ace and back o he buye. Also, hese maks ae impossible o emove o change fo non-colluding buyes. Bu, wheneve some malicious buyes (ha ae called piaes) ae colluding, hey can compae hei copies and deec diffeen posiions. Colluding buyes have he abiliy o ease o change deeced posiions and consuc some illegal maks. In ode o fomulae hese condiions we Email: hhaji@sbu.ac.i. This eseach was in pa suppoed by a gan fom IPM (No. 900504). Email: f.moazami@alzaha.ac.i. 365 8050 c 202 Discee Mahemaics and Theoeical Compue Science (DMTCS), Nancy, Fance
262 Hossein Hajiabolhassan and Faokhlagha Moazami conside he following definiions and noaions. Le {0, } v and =. is called a (v, )-code and evey elemen of is said o be a codewod. We wie w i fo he ih componen of a wod w. Also, he incidence maix of is a v maix whose ows ae he codewods in. Suppose C = {w (u),w (u2),...,w (ud) } {0, } v. Fo i 2{, 2,...,v}, he ih componen is said undeecable fo he coaliion C if w (u) i = w (u2) i = = w (u d) i. Le U(C) be he se of undeecable componens fo C. The se F (C) ={x 2{0, } v : x U(C) = w (ui) U(C) fo all w (ui) 2 C} epesens all possible v-uples ha could be poduced by he coaliion C by compaing he d codewods. Definiion An -famepoof code is a subse {0, } v such ha fo evey C whee C apple, we have F (C) \ =C. Le FPC(v, b) denoes an -famepoof code {0, } v such ha = b. The codewods in he se ae called egiseed codewods. Theefoe, if we have an -famepoof code, hen he piae in he se C couldn poduce a egiseed illegal codewod ohe han hei maks; ha is no appopiae fo he piae copy. Fo moe deails abou famepoof codes; see [, 2,, 3, 4]. The following heoem was poved by Sinson, Tung, and Wei [0]. Theoem [0] Suppose is an FPC(v, b) wih b>2. Suppose D, whee D =2. Then hee exiss an unegiseed wod, say maj(d) 2{0, } v, such ha maj(d) 2 F (C) fo any C D wih C =. In view of he afoemenioned heoem, maj(d) is a codewod ha is poduced by he coaliion of evey -subse of he se D. Theefoe, in an FPC, hee exis some illegal maks such ha i is no possible o idenify a piae use. So hey consideed anohe condiion and defined secue famepoof codes in which disibuo is able o idenify a leas one piae of he guily coaliions. Definiion 2 Suppose ha is a (v, )-code. is said o be an -secue famepoof code if fo any C,C 2 wih C apple, C 2 apple, and C \ C 2 =?, we have F (C ) \ F (C 2 )=?. Also, is emed an SFPC(v, ), fo sho. In fac, when an illegal mak can be poduced by wo diffeen -subses, in an SFPC, hee exiss a leas one use in hei inesecion; whose can be consideed as a piae use. Sinson and Wei in [0] sudied he elaionship beween binay secue famepoof codes and combinaoial aspecs. In his pape, we esablish he elaionship beween his concep and biclique cove. By a biclique we mean a bipaie gaph wih veex se (X, Y ) such ha evey veex in X is adjacen o evey veex in Y. Noe ha evey empy gaph is a biclique. A d-biclique cove of a gaph G of size s is a collecion of s bicliques of G such ha each edge of G is in a leas d of he bicliques. The d-biclique coveing numbe of G, denoed by bc d (G), is defined o be he minimum numbe of s such ha hee exiss a d-biclique cove of size s fo he gaph G. Fo abbeviaion, le bc(g) sand fo bc (G). Definiion 3 Le X be an n-se and F = {B,...,B } be a family of subses of X. F is called an (, w; d)-cove-fee family if fo any wo subses I, J [] such ha I =, J = w, and I \ J =?, he following condiion holds ( \ B i ) \ ( [ B j ) d. i2i j2j
Secue Famepoof Codes Though Biclique Coves 263 We denoe i biefly by (, w; d) CFF(n, ). The minimum numbe of elemens of X fo which hee exiss an (, w; d) CFF wih membes is denoed by N((, w; d),). Fo convenience, we use he noaion N((, w),) insead of N((, w; ),). This paamee has been sudied exensively in he lieaue; see [4, 5, 7, 2]. The incidence maix of an (, w; d) CFF is a n binay maix A such ha a ij =wheneve j 2 B i and a ij =0ohewise. As usual, we denoe by [] he se {, 2,...,}, and denoe by [] he collecion of all -subses of[]. The gaph I (, w) is a bipaie gaph wih he veex se ( [] w, [] ) fo which a w-subse is adjacen o an -subse wheneve hei inesecion is empy. Theoem 2 [5] Fo any posiive ineges, w, d, and, whee N((, w; d),)=bc d (I (, w)). + w, we have Thoughou his pape, we only conside finie simple gaphs. Fo a gaph G, le V (G) and E(G) denoe [] is veex and edge ses, especively. The Knese gaph KG(, ) is he gaph wih veex se, and A is adjacen o B if A \ B =?. Ahomomophism fom G o H is a map : V (G)! V (H) such ha adjacen veices in G ae mapped ino adjacen veices in H, i.e., uv 2 E(G) implies (u) (v) 2 E(H). In addiion, if any edge in H is he image of some edge ing, hen is emed an ono-edge homomophism. In his pape, by A c we mean he complemen of he se A. In Secion 2, we show ha fo 2, an -secue famepoof code of size and lengh v exiss if and only if hee exiss a -biclique cove of size v fo he Knese gaph KG(, ). Also, we wish o invesigae some connecion beween he d-biclique coveing numbe of Knese gaphs and cove-fee families. Moeove, we pesen an uppe bound fo he biclique coveing numbe of Knese gaphs. In Secion 3, we will look moe closely a he biclique coveing numbe of Knese gaphs. Also, i is shown ha if hee exiss a Hadamad maix of ode 4d, hen N((, ; d), 8d 2) = 4d and bc 2d (K 8d )=4d. 2 Secue Famepoof Codes Fo a subse A i of [], he indicao veco of A i is he veco v Ai =(v,...,v ), whee v j =if j 2 A i and v j =0ohewise. Theoem 3 Le,, and v be posiive ineges, whee 2. An SFPC(v, ) exiss if and only if hee exiss a biclique cove of size v fo he Knese gaph KG(, ). Poof: Assume ha A is he incidence maix of an se A j as follows A j def = {i apple i apple, a ij =}. SFPC(v, ). Assign o he jh column of A, he Now, fo apple j apple v, consuc he bicliques G j wih veex se (X j,y j ), whee he veices ofx j ae all -subses of A j and he veices of Y j ae all -subses of A c j, i.e., [] \ A j. I is easily seen ha G j, fo apple j apple v, is a complee bipaie gaph of KG(, ). Le C C 2 be an abiay edge of KG(, ). So C,C 2 [], and C \ C 2 =?. Since A is he incidence maix of an SFPC(v, ), we have F (C ) \ F (C 2 )=?. This means ha hee exiss a bi posiion i such ha he ih bi of all code wods of C is c i, fo some c i 2{0, }, and also he ih bi of all codewods of C 2 is c i + (mod 2). So hee exiss a column of A such ha all enies coesponding o he ows of C ae equal o and all enies coesponding o he ows ofc 2 ae equal o 0, o vice vesa. Hence, C C 2 2 E(G i ). Convesely, assume
264 Hossein Hajiabolhassan and Faokhlagha Moazami ha we have a biclique cove of size v fo he gaph KG(, ). Ou objecive is o consuc an -SFPC. Label gaphs in his biclique cove wih G,...,G v, whee G i has as is veex se (X i,y i ). Le A i be he union of ses ha lie in X i. Conside he indicao vecos of A i s, fo apple i apple v, and consuc he maix A whose columns ae hese vecos. Assume ha C and C 2 ae wo disjoin subses of [] of size, i.e., C C 2 2 E(KG(, )). Le G i be he complee bipaie gaph ha coves he edge C C 2. Then in he ih column of he maix A all enies coesponding o he ows of C ae equal o and all enies coesponding o he ows of C 2 ae equal o 0, o vice vesa. Consequenly, F (C ) \ F (C 2 )=?. 2 A coveing of a gaph G is a subse K of V (G) such ha evey edge of G has a leas one end in K. The numbe of veices in a minimum coveing of G is called he coveing numbe of G and denoed by (G). In [0], Sinson, Tung, and Wei consuc an SFPC(2 2, 2 + ). Coollay [0] Fo any inege 0, hee exiss an SFPC(2 2, 2 + ). Poof: Easily, one can check ha he biclique coveing numbe of a gaph G wihou C 4 as a subgaph is equal o he coveing numbe of G. On he ohe hand, KG(2 +,) does no conain C 4 as a subgaph. So bc(kg(2 +,))) = (KG(2 +,)). Also, i is a well-known fac ha (KG(, )) =. An easy compuaion confims he asseion. 2 In he nex heoem, we show he elaionship beween he d-biclique cove of Knese gaphs and covefee families. Theoem 4 Fo any posiive ineges, d, and, whee 2, i holds ha bc 2d (KG(, )) apple N((, ; d),) apple 2bc d (KG(, )). Poof: Fis, assume ha we have an opimal (, ; d) CFF(n, ), i.e., n = N((, ; d),) wih incidence maix A. Assign o he jh column of A, he se A j as follows A j def = {i apple i apple, a ij =}. Conside he biclique G j wih veex se (X j,y j ), whee he veices of X j ae all -subses of A j and he veices of Y j ae all -subses of A c j. Also, wo veices ae adjacen if he subses coesponding o hese veices ae disjoin. I is no difficul o see ha G j s, fo apple j apple N((, ; d),), fom a 2d-biclique cove of KG(, ). So bc 2d (KG(, )) apple N((, ; d),). Convesely, assume ha we have a d-biclique cove of KG(, ). Label gaphs in his biclique cove wih G,...,G l, whee G i has as is veex se (X i,y i ). Le A i be he union of ses ha lie in X i and B i be he union of ses ha lie in Y i. Obviously, A i and B i ae disjoin. Conside he indicao vecos of A i s and B i s, fo i =,...,l. Consuc he maix A whose columns ae hese vecos. Then A is he incidence maix of an (, ; d) CFF(2l, ). So N((, ; d),) apple 2bc d (KG(, )). 2 Coollay 2 Fo any posiive ineges, and, whee 2, i holds ha bc 2 (KG(, )) apple N((, ),) apple 2bc(KG(, )). A simila esul has been obained by Sinson e al. [0]. Theoem 5 [0] Le and be posiive ineges, whee 2, hen. If hee exiss an SFPC(v, ), hen hee exiss an (, ) CFF(2v, ).
Secue Famepoof Codes Though Biclique Coves 265 2. Any (, )-cove fee family is an -secue famepoof code. Sinson e al. [0] have obained he following uppe bounds fo SFPC. Theoem 6 [0] Suppose ha and ae posiive ineges and p = 2 2. If hen hee exiss an SFPC(v, ). v ln( 2( )( )) ln p, The following uppe bound is shown in Deng e al. [3], by he Lovász Local Lemma. Theoem 7 [3] Thee exiss an whee SFPC(v, ) if v ln(e(( )( ) ( 2 )( 3 ))) ln p, p = 2. 2 Le p = 2. Using he appoximaion e x x, if we se x = 2 2, hen we can see ha 2 ln p 22. In he nex heoem, by he alenaion mehod, we pesen a bound ha is a sligh impovemen of Theoem 6. Also, i is a sligh impovemen of Theoem 7 povided ha is no lage elaive o. Theoem 8 Le and be posiive ineges. If >3 and hen hee exiss an v 2 2 ( + ln( d 2 e SFPC(v, ). Poof: We show ha if v b2 2 ( + ln( d 2 e Q b 2 c )) i=0 Q b 2 c )) i=0 ( 2i+ ), ( 2i+ )c, hen hee exiss a biclique [] cove of size v fo he Knese gaph KG(, ). Le A be d 2 e. Fo evey membe of A, say A i, we can consuc he biclique G i wih veex se (X i,y i ), whee he veices of X i ae all -subses of A i and he veices of Y i ae all -subses of A c i. We define B o be he collecion of hese bicliques. Le p 2 [0, ] be abiay, lae, we specify an opimized value fo p. Le us pick, andomly and independenly, each biclique of B wih pobabiliy p and F be he andom se of all bicliques picked and le Y F be he se of all edges AB of he gaph KG(, ) which ae no coveed by he se F. The expeced value of F is clealy d 2 e p. Fo evey edge AB, p(ab 2 Y F)=( p) l whee l =2 2 d 2 e. So he expeced value of he F + Y F is a mos d 2 e p + 2 ( p) 2( 2 d 2 e ). If we se F 0 = F S Y F, hen clealy all edges of he gaph KG(, ) ae coveed by F 0. So we wan o esimae p such ha F 0 is minimum. Fo convenien, we bound p apple e p o obain E( F + Y F ) apple d 2 e p + 2 e 2( 2 d 2 e )p. The igh hand side is minimized a p = ln, which = d 2 e b 2 c and =2 2 d 2 e whee p 2 [0, ] if is sufficienly lage espec o (e.g., >3). So we have an SFPC(v, ) ha
266 Hossein Hajiabolhassan and Faokhlagha Moazami v apple ( d 2 e ) 2( 2 d 2 e Fuhemoe, i is saighfowad o check ha 2 e ) ( + ln( d b 2 c )). ( d 2 ) Q e apple 2( 2 22 ( d ) 2 2i+ ). e Hence, he bound follows. 2 i=0 3 Biclique Cove of Knese Gaphs By he menioned esuls in he pevious secion, i may be of inees o find some bounds fo he biclique coveing numbe of Knese gaphs. Theoem 9 Fo any posiive ineges d,, s, and, whee >2 and >s, we have whee m = N(( s, s; d), 2s). bc d (KG(, )) bc m (KG(, s)), Poof: Le {G,G 2,...,G l } be an opimal d-biclique cove of KG(, ). Also, assume ha G i has as is veex se (X i,y i ). Le A i and B i be he union of ses ha lie in X i and Y i, especively. Fo any apple i apple l, conside he biclique G 0 i, as a subgaph of KG(, s), wih veex se (X0 i,y0 i ), whee X0 i is he se of all s-subses of A i and Yi 0 is he se of all s-subses of B i. One can check hag 0 i s cove all edges of KG(, s). Moeove, any edge UV 2 E(KG(, s)) is conained in a leas m-bicliques, whee m = N(( s, s; d), 2s). To see his, conside he bipaie gaph I {U,V } (as an induced subgaph of KG(, )) wih veex se (X U,Y V ), whee X U = {W U W [],W \ V = ;, W = } Y V = {W V W [],W \ U = ;, W = }. I is a simple mae o check ha I {U,V } and I 2s ( s, s) ae isomophic. Also, if G j coves any edge of I {U,V }, hen UV is conained in G 0 j. Consequenly, by Theoem 2 he asseion follows. 2 In view of he poof of Theoem 9, similaly, one can exend any biclique of I (, w) o a biclique of I ( i, w j). Consequenly, we have he following heoem. Theoem 0 Le d,, w, and be posiive ineges, whee + w. Fo any apple i<and apple j<w, we have N((, w; d),) N(( i, w j; m),), whee m = N((i, j; d), w + i + j). The facional biclique coveing numbe bc (G) is defined as follows bc bc d (G) bc d (G) (G) =inf = lim. d d d! d I is known ha if a gaph is edge-ansiive, hen one can compue is facional biclique coveing numbe, see [9] fo moe abou facional gaph heoy. Theoem [9] Fo evey non-empy edge-ansiive gaph G, bc (G) = E(G) B(G), whee B(G) is he maximum numbe of edges among he bicliques of G.
Secue Famepoof Codes Though Biclique Coves 267 The nex esul is a consequence of Theoem 0. Theoem 2 [2] Le d,, w, i, j, and be non-negaive ineges, whee apple j<w. Then N((, w; d),) N((i, j; d), w + i + j) min w japplemapple +i + w, apple i<, and m w+i+j m w+j. Poof: In view of definiion of facional biclique cove and since bc s (G) apple s.bc (G) fo evey posiive ineges and s, one can conclude ha fo evey gaph G, we have.bc (G) apple bc (G). By his fac and using Theoem 0, we have N((, w; d),) N((i, j; d), w + i + j)bc (I ( i, w j)). () The gaph I (, w) is an edge-ansiive gaph. Theefoe, using Theoem and an easy calculaion bc (I (, w)) = min 0 + 00 = w 0 00 w Theefoe, accoding o he inequaliy i follows ha = min wapplemapple m w m w N((, w; d),) N((i, j; d), w + i + j) min w japplemapple +i. m w+i+j m w+j We know ha he image of a biclique unde a gaph homomophism is a biclique. This leads us o he following lemma. Lemma Le G and H be wo gaphs and : G! H be an ono-edge homomophism. Also, assume ha d and ae posiive ineges and fo any edge e 2 E(H), bc d ( (e)). Then bc d (G) bc (H). Poof: Le {K,K 2,...,K l } be an opimal d-biclique cove of G. One can check ha fo any 0 apple i apple l, (K i ) is a biclique and he family { (K ), (K 2 ),..., (K l )} is a -biclique cove of H. 2 Poposiion Fo any posiive ineges and, whee >2, we have bc d (KG(, )) bc 3d (KG( 2, )). Poof: Fis, we pesen an ono-edge homomophism fom KG(, ) o KG( 2, ). To see his, fo evey veex A of KG(, ), define (A) :=A 0 as follows. If A does no conain boh and, hen define A 0 := A \{maxa}. Ohewise, se A 0 := {x}[a\{, }, whee x is he maximum elemen of [ 2] absen fom A. I is simple o check ha he subgaph induced by he invese image of any edge of KG( 2, ) conains an induced cycle of size six o an induced maching of size hee. Hence, in view of Lemma, if {K,...,K l } is a d-biclique cove of KG(, ), hen { (K ),..., (K l )} is a 3dbiclique cove of KG( 2, ). 2 The afoemenioned esuls moivae us o conside he following quesion. Quesion Le d,, and be posiive ineges, whee >2. Wha is he exac value of bc d (KG(, ))?. 2
268 Hossein Hajiabolhassan and Faokhlagha Moazami Deemining he exac value of he paamee N((, w; d),), even fo special, w, d and, is an ineesing and challenging poblem ha is sudied in he lieaue; see [5, 6, 7, 8]. An n n maix H wih enies + and is called a Hadamad maix of ode n wheneve HH = ni. I is no difficul o see ha any wo columns of H ae also ohogonal. If we pemue ows o columns o if we muliply some ows o columns by, hen his popey does no change. Two such Hadamad maices ae called equivalen. Fo a given Hadamad maix, we can find an equivalen one fo which he fis ow and he fis column consis eniely of +ï 2s. Such a Hadamad maix is called nomalized. We will denoe by K m,m he complee bipaie gaph wih a pefec maching emoved. Obviously, K m,m is isomophic o I m (, ). Theoem 3 Le d be a posiive inege such ha hee exiss a Hadamad maix of ode 4d, hen. bc 2d (K 8d )=4d, 2. N((, ; d), 8d 2) = bc d (K 8d 2,8d 2 )=4d. Poof: Le H =[h ij ] be a Hadamad maix of ode 4d. Suppose ha K 8d has {u,...,u 4d,v,...,v 4d } as is veex se. Fo he jh column of H, wo ses X j and Y j ae defined as follows X j := {u i h ij =+}[{v i h ij = } & Y j := {u i h ij = }[{v i h ij =+}. By consucing a bipaie gaph G j wih veex se (X j,y j ), indeed, we assign a biclique o each column. I is well-known ha fo any wo ows of a Hadamad maix, he numbe of columns fo which coesponding enies in hese ows ae diffeen in sign, ae equal o 2d. So, fo i 6= j he edges u i u j, v i v j, and u i v j of he gaph K 8d ae coveed by 2d bicliques. Also, fo he edge u i v i, hee exis 4d bicliques ha cove i. Accoding o he above agumen evey edge is coveed a leas 2d imes, so bc 2d (K 8d ) apple 4d. On he ohe hand, fo evey gaph G, we have E(G) B(G) edges among he bicliques of G. Theefoe, apple bc d(g) d 4d 2 apple bc 2d(K 8d )., whee B(G) is he maximum numbe of Since bc 2d (K 8d ) is an inege, we have 4d apple bc 2d (K 8d ) which complees he poof. Fo he poof of he second pa, assume ha H is a nomalized Hadamad maix of ode 4d. Delee he fis ow of H and denoe i by H 0 =[h 0 ij ]. Also, assume ha K 8d 2,8d 2 has (X, Y ) as is veex se whee X = {u,...,u 4d,v,...,v 4d }, Y = {u 0,...,u 0 4d,v0,...,v4d 0 }, and u iu 0 i,v ivi 0 62 E(K 8d 2,8d 2 ). Assign o he jh column of H 0, wo ses X j and Y j as follows X j := {u i h 0 ij =+}[{v i h 0 ij = } & Y j := {u 0 i h0 ij = }[{v0 i h0 ij =+}. By he same agumen in he fis pa of he poof and using he well-known fac ha in H 0 evey wo disinc ows i, j and fo any a, b 2 {, +} hee ae exacly dcolumns ha he coesponding enies ae a and b in he ows i and j, especively, one can see ha evey edge is coveed a leas d imes. So 2d bc d (K 8d 2,8d 2 ) apple 4d. On he ohe hand, 4d 4d apple bc d(k 8d 2,8d 2 ), and 2d 4d <. Theefoe, 4d apple bc d (K 8d 2,8d 2 ) which esablishes he second pa. 2
Secue Famepoof Codes Though Biclique Coves 269 Acknowledgemens This pape has been evised and esubmied fo eview while Hossein Hajiabolhassn was on leave a Technical Univesiy of Denmak (he academic yea 202-203). Hossein Hajiabolhassan is gaeful o pofesso Casen Thomassen fo his hospialiy and suppo. Also, he auhos wish o hank he anonymous efeees fo hei useful commens. Refeences [] Simon R. Blackbun. Famepoof codes. SIAM J. Discee Mah., 6(3):499 50 (eleconic), 2003. [2] Dan Boneh and James Shaw. Collusion-secue fingepining fo digial daa. IEEE Tans. Infom. Theoy, 44(5):897 905, 998. [3] D. Deng, D. R. Sinson, and R. Wei. The Lovász local lemma and is applicaions o some combinaoial aays. Des. Codes Cypog., 32(-3):2 34, 2004. [4] A. D yachkov, P. Vilenkin, and S. Yekhanin. Uppe bounds on he ae of supeimposed (s, l)- codes based on engel s inequaliy. In Poceedings of he Inenaional Conf. on Algebaic and Combinaoial Coding Theoy (ACCT), pages 95 99, 2002. [5] H. Hajiabolhassan and F. Moazami. Some new bounds on cove-fee families hough biclique coves. Discee Mah., 32(24):3626 3635, 202. [6] Hyun Kwang Kim, Vladimi Lebedev, and Dong Yeol Oh. Some new esuls on supeimposed codes. J. Combin. Des., 3(4):276 285, 2005. [7] Sh. Kh. Kim and V. S. Lebedev. On he opimaliy of ivial (w, )-cove-fee codes. Poblemy Peedachi Infomasii, 40(3):3 20, 2004. [8] P. C. Li, G. H. J. van Rees, and R. Wei. Consucions of 2-cove-fee families and elaed sepaaing hash families. J. Combin. Des., 4(6):423 440, 2006. [9] E. R. Scheineman and D. H. Ullman. Facional gaph heoy. Wiley-Inescience Seies in Discee Mahemaics and Opimizaion. John Wiley & Sons Inc., New Yok, 997. A aional appoach o he heoy of gaphs, Wih a foewod by Claude Bege, A Wiley-Inescience Publicaion. [0] D. R. Sinson, Tan van Tung, and R. Wei. Secue famepoof codes, key disibuion paens, goup esing algoihms and elaed sucues. J. Sais. Plann. Infeence, 86(2):595 67, 2000. Special issue in hono of Pofesso Ralph Sanon. [] D. R. Sinson and R. Wei. Combinaoial popeies and consucions of aceabiliy schemes and famepoof codes. SIAM J. Discee Mah., ():4 53 (eleconic), 998. [2] D. R. Sinson and R. Wei. Genealized cove-fee families. Discee Mah., 279(-3):463 477, 2004. In honou of Zhu Lie. [3] Gábo Tados. Opimal pobabilisic fingepin codes. In Poceedings of he Thiy-Fifh Annual ACM Symposium on Theoy of Compuing, pages 6 25 (eleconic), New Yok, 2003. ACM. [4] Gábo Tados. Opimal pobabilisic fingepin codes. J. ACM, 55(2):A. 0, 24, 2008.
270 Hossein Hajiabolhassan and Faokhlagha Moazami