Broadcasting with side information

Broadcasting with side information Noga Alon ∗ Avinatan Hasidim † Ey al Lub etzky ‡ Uri Sta v § Amit W einstein ¶ Abstract A sender holds a word x consisting of n blo c ks x i , each of t bits, and wishes to broadcas t a co deword to m receivers, R 1 , ..., R m . Each rec eiver R i is in teres ted in one block, and has prior side informa tion consisting of so me subset of the other blocks. Let β t be the minim um nu mber of bits that has to b e transmitted when each block is of length t , and let β be the limit β = lim t →∞ β t /t . In words, β is the av era ge comm unication cost per bit in ea ch block (for long blo cks). Finding the co ding rate β , for such an informed broa dcast setting, gener a lizes se veral co ding theoretic parameter s r elated to Informed So ur ce Co ding on Demand, Index Co ding and Net work Co ding. In this work we show that usag e of larg e data blo cks may s trictly improv e up on the trivial enco ding which treats each bit in the block independently . T o this end, w e provide gener al bo unds on β t , a nd prove that for any constant C there is an e xplicit br o adcast setting in which β = 2 but β 1 > C . One of these examples answers a question of [ 15 ]. In addition, w e provide examples with the follo wing counterin tuitive direct-sum pheno mena. Consider a union of several mutually indep endent br oadcast settings. The optimal co de for the combined setting may yie ld a s ignificant saving in communication ov er co ncatenating optimal enco dings for the indiv idua l settings. This res ult also pr ovides new non-line a r co ding schemes which improv e up on the lar gest known gap b e t ween linear and no n-linear Net work Co ding, thus improving the results of [ 8 ]. The pro ofs are based o n a r elation b etw een this problem and results in the study of Witsen- hausen’s rate, OR gr aph pro ducts , colorings of Cayley gra phs, and the chromatic n umbers of Kneser gr aphs. ∗ Schools of Mathematics and Computer Science, Ra ymond and Beverly S ac kler F acult y of Exact Sciences, T el Aviv Un ivers ity , T el Aviv, 6997 8, Israe l and IA S, Princeton, NJ 08540, US A. Email: nogaa@tau.ac.il. Researc h supp orted in part by a USA- Israeli BSF gran t, by the Israel Science F oundation and by the Hermann Minko wski Minerv a Center for Geometry at T el Aviv Un ivers ity . † School of Computer Science, Hebrew Universit y Email: a v inatan@cs.huji. ac.il. ‡ Microsof t Researc h, One Microsoft W ay , Red mond, W A 98052-6399, USA. Email: eyal@micros oft.com. Researc h partially supp orted by a Charles Clore F oundation F ellowship. § School of Computer Science, Raymond and Bev erly Sac kler F aculty of Exact Sciences, T el Aviv Universit y , T el Aviv, 69978, Israel. Email: urista v@t au.ac.il. ¶ School of Computer Science, Raymond and Bev erly Sac kler F aculty of Exact Sciences, T el Aviv Universit y , T el Aviv, 69978, Israel. Email: amit w@tau.ac.il. 1 1 In tro duc tion Source co d ing d eals w ith a scenario in wh ic h a sender has some d ata str in g x he wishes to transmit through a broadcast c hann el to r e c eive rs . In this pap er w e consider a v arian t of source co ding whic h was first prop osed by Birk and Kol [ 6 ]. In this v ariant, called In formed Source Co ding On Demand (ISCOD), eac h receiv er h as some prior side information, co mp rising a p art o f the input w ord x . The send er is a w are of the p ortion of x kno wn to eac h r eceiv er. Moreo ver, eac h receiv er is in terested in just part of the data. W e f ormalize this source co ding setting as follo w s. Supp ose that a sender S wishes to broadcast a word x = x 1 x 2 . . . x n , w here x i ∈ { 0 , 1 } t for all i , to m receiv ers R 1 , . . . , R m . Each R j has some prior side in formation, consisting of s ome of the blo cks x i , and is intereste d in a single blo ck x f ( j ) . The send er wishes to transmit a co deword that will enable eac h and eve ry receiv er R j to reconstruct its m issing block x f ( j ) from its prior information. Let β t denote th e minim um p ossible length of suc h a b inary co de. Our ob jective in this pap er is to study the p ossible b eha viors of β t , fo cusin g on the more natur al scenario of transmitting large data blo c ks (namely a large t ). The motiv ation for informed sour ce co ding is in applications su c h as Video on Demand. In suc h applications, a net wo rk, or a satellite , h as to br oadcast information to a set of clien ts. During the first transmission, ea c h receiv er misses a part of the d ata. Hence , eac h clien t is now interested in a d ifferen t (small) p art of the data, and has a prior side inform ation, consisting of the p art of the data he receiv ed [ 21 ]. Note that our assumption that eac h rece iver is in terested only in a single blo c k is not necessary . T o see this, on e can sim ulate a r eceiv er intereste d in r blo cks b y r receiv ers, eac h interested in one of these b lo c ks, and all ha ving th e same side in formation. The pr oblem ab o ve generalizes the problem of I ndex Co din g, whic h was first p resen ted by Birk and Kol [ 6 ], and later stud ied b y Bar-Y ossef, Birk, J a yram and Kol [ 4 ] and by Lub eztky and Sta v [ 15 ]. Ind ex Co ding is equiv alen t to a sp ecial case of our problem, in whic h m = n , f ( j ) = j for all j ∈ [ m ] = { 1 , . . . , m } and the size of the d ata blo c ks is t = 1. O ur w ork can also b e considered in the conte xt of Netw ork Cod ing, a term wh ich was coined by Ahlsw ede, Cai, Li, and Y eung [ 3 ]. In a Net w ork C o ding problem it is ask ed whether a give n comm unication net wo rk (with limited capacitie s o n eac h link) can m eet its requir emen t, passing a certain amoun t of information fr om a set of source vertice s to a s et of targets. It will b e easier to describ e our sour ce cod ing problems in terms of a certain hyp ergraph. W e define a dir e cte d hyp er g r aph H = ( V , E ) on the set of vertic es V = [ n ]. Eac h vertex i of H corresp onds to an input blo ck x i . The set E of m edges corresp ond s to th e receiv ers R 1 , . . . , R m . F or the receiv er R j , E con tains a dir ected edge e j = ( f ( j ) , N ( j )), wh ere N ( j ) ⊂ [ n ] denotes the set of blo c ks which are kno wn to receiv er R j . Clearly the str ucture of H captures the definition of the b roadcast s etting. W e thus denote b y β t ( H ) the min imal num b er of b its required to br oadcast the information to all the receiv ers when the blo ck length is t . Let H b e such a directed hyp ergraph. F or an y pair of in tegers t 1 and t 2 , wh en the blo ck length is t 1 + t 2 , it is p ossible to enco de the first t 1 bits, then separately e nco d e the remaining t 2 bits. By concatenating th ese t wo co des we get β t 1 + t 2 ( H ) ≤ β t 1 ( H ) + β t 2 ( H ), i.e. β t ( H ) is s ub-additiv e. F ek ete’s Lemm a thus imp lies that the limit lim t →∞ β t ( H ) /t exists and e quals inf t β t ( H ) /t . W e 2 define β ( H ) to b e this limit: β ( H ) := lim t →∞ β t ( H ) t = inf t β t ( H ) t . In w ords, β is the av erage asymptotic n umb er of enco ding bits needed p er bit in eac h inpu t b lo c k. T o study this p roblem, we will also consider the follo wing relate d one. L et k · H denote the disjoin t union of k copies of H . Define β ∗ t ( H ) := β 1 ( t · H ). In words, β ∗ t represent s the minimal n umb er of bits required if the net w ork top ology is r eplicated t indep endent times 1 . A similar sub-additivit y argumen t j ustifies the definition of the limit β ∗ ( H ) := lim t →∞ β ∗ t ( H ) t = inf t β ∗ t ( H ) t . By viewing eac h rec eive r in the broadcast net w ork as t receiv ers, eac h intereste d in a single bit, w e can compare this scenario w ith the setting of in dep end en t copies. Clearly , the receiv ers in the first scenario ha v e additional information and hence β t ( H ) ≤ β ∗ t ( H ) for an y t . T aking limits we get β ( H ) ≤ β ∗ ( H ). There are sev eral lo we r b ounds for β ( H ). One such s imple b oun d, wh ic h we denote by ~ α ( H ), is the maximal size of a set S of v ertices satisfying the follo wing: F or eve ry v ∈ S there exists some e = ( v , J ) ∈ E so that J ∩ S = ∅ . A simple counting argumen t sho ws that ~ α ( H ) ≤ β ( H ), giving 2 ~ α ( H ) ≤ β ( H ) ≤ β ∗ ( H ) ≤ β 1 ( H ) . (1) 1.1 Our Results Let H = ([ n ] , E ) b e a directe d h yp ergraph for a broadcast net w ork, and set t = 1. It will b e con v enient to address the more pr ecise notion of the numb er of c o dewor ds in a broadcast code whic h s atisfies H . W e say that C , a b r oadcast co de for H , is optimal , if it con tains the minimum p ossible num b er of co dewords (in which case, β 1 ( H ) = ⌈ log 2 |C |⌉ ). W e sa y that t wo input-strings x, y ∈ { 0 , 1 } n are confu sable if there exists a receiv er e = ( i, J ) ∈ E such that x i 6 = y i but x j = y j for all j ∈ J . This implies that the input-strin gs x, y can not b e enco ded with the same cod ew ord. Denoting b y γ th e maximal card in alit y of a set of input-strings w hic h is pairwise unconfusable. The first tec hnical result of this pap er relates β ∗ and γ . Theorem 1.1. L et H and γ b e define d as ab ove. The fol lowing holds for any inte ger k :  2 n γ  k ≤ |C | ≤ l  2 n γ  k k n log 2 m , (2) wher e C is an optimal c o de for k · H . In p articular, β ∗ ( H ) = lim k →∞ β 1 ( k · H ) k = n − log 2 γ . 1 Such a scenario can o ccur when the top ology is standard (e.g. resulting from using a common application or operation system). Therefo re it is identical across netw orks, albeit with d ifferent data. 2 The b oun d ~ α given here generalizes the boun d given in [ 4 ] to directed hyp ergraphs, as w ell as to β (rather than just to β 1 ). Another b ound in the Index Cod ing mo del is the MAIS (max im um acy clic induced subgraph) b oun d giv en in [ 4 ], that can also b e generalized t o our mo del. 3 A su rprising corollary of the ab o ve theorem is that β ∗ ma y b e strictly smaller than β 1 . Indeed, as β ∗ deals with the case of disjoint in stances, it is not int uitively clear that th is should b e the case: one would th ink that there can b e no ro om for imp ro ving up on β 1 ( H ) when replicat ing H into t disjoin t copies, giv en the total indep endence b et we en these copies (no kno w ledge on bloc ks fr om other co pies, indep endently c hosen inputs). Note that eve n in the somewhat r elated Inf orm ation Theoretic notion of the Shann on capacit y of graphs (corresp onding to c hannel co d ing rather than source co ding), though it is kno wn that the capacit y of a disjoint u nion may exceed the sum of the individual capacities (see [ 1 ]), it is easy to v erify that disjoint unions of the same g r aph can nev er ac hiev e this. The follo wing th eorem demonstrates the p ossible gap b etw een β 1 ( t · H ) /t and β 1 ( H ) ev en in a v ery limited setting, whic h coincides with Index Co ding. T his solv es the op en p roblem present ed by [ 15 ] already for the smallest p ossible n = 5. Theorem 1.2. Define a br o adc ast network H = ( Z 5 , E ) b ase d on the o dd cycle C 5 : F or e ach i ∈ Z 5 , ther e is a dir e cte d e dge ( i, { i − 1 , i + 1 } ) , wher e the arithmetic is mo dulo 5. Then β 1 ( H ) = 3 , wher e as β ∗ ( H ) = 5 − log 2 5 ≈ 2 . 68 . It is wo rth noting that in the example a b o ve , the optimal co d e for H cont ains 8 co dew ords , whereas in the limit, eac h disjoin t copy of H costs 6 . 4 co dew ords , hence this surprising direct-sum phenomenon carries b ey ond an y inte ger rounding issues. In addition, in Section 3.3 (Theorem 3.15 ) w e generalize the ab ov e example to an y broadcast netw ork which is based on a complemen t of an o dd cycle. The f ollo win g theorem extends the ab ov e results on the ga p b etw een β ∗ and β 1 ev en further, b y pr o viding an example where β ∗ is b ound ed wh ereas β 1 can b e arbitrarily large: Theorem 1.3. Ther e exists an explicit infinite family of br o adc ast networks for which β ∗ ( H ) < 3 is b ounde d and yet β 1 ( H ) is unb ounde d. Finally , recalling ( 1 ), one w ould exp ect that in man y cases β should b e strictly smaller than β ∗ , as the receiv ers p ossess m ore side information. Ho w ever it is not clear h o w m uch can b e gained by this additional information. W e construct an example w here not only is there a difference b et w een the t wo, but β is constan t w hile β ∗ is unb ounded. Theorem 1.4. Ther e exists an explicit infinite family of br o adc ast ne tworks for which β ( H ) = 2 is c onstant wher e as β ∗ ( H ) is unb ounde d. W e discu ss applications of the r esults to Net wo rk and In dex cod ing in w h at follo w s . 1.2 Related W ork Our w ork is a generaliz ation of Index Co ding, whic h w as first stud ied by Birk and K ol [ 6 ]. This problem deals with a sender, who wishes to send n blocks of d ata to n r eceiv er s , where eac h receiv er kno ws a subset of the bloc ks, an d is in terested in a single b lo c k (differen t receiv ers are in terested i n differen t blo cks). The sender can o nly utiliz e a broadcast c hannel, and w e wish to minimize the n umb er of bits h e has to send. Birk and Kol presente d a class of enco dings, b ased on erasure Reed Solomon cod es. They also dealt with some of the practical issues of this scheme, su c h 4 as synchronizatio n b etw een the clien ts and the server. Finally they ga ve examples for scenarios where their co des were not optimal, and p r esen ted the question of fi nding b etter co des. The fir st impro vemen t to the original cod es w as done by Birk, Bar-Y ossef, Ja yram and Kol, w ho found a lo w er b ound to the minimal length of linear codes, called the min-r ank . They also conjectured that linear codes are o ptimal for index coding, a conjecture that w as l ater refuted b y Lub etzky and Sta v. Ho wev er, the pro of by Lub eztky and S ta v w as limited, in th e sense that they constru cted an index co ding problem, for wh ic h linear co des o ver any fi eld were not optimal, and y et a com bination of linear co d es o ver several fields ma y w ell b e optimal. Theorem 1.2 refu tes the conjecture in a stronger sense, by s ho wing an index co din g pr oblem for whic h the optimal solution is not linear for an y fi eld size or ev en an y combination of sev eral fields. Net w ork Co ding deals w ith a scenario in which s ev eral sources wish to pass inform ation to sev eral targets, when the comm unication net wo rk is mo deled by a graph. Eac h edge h as a capacit y , and the goal is to see if the n et w ork is satisfiable, i.e. if it is p ossible to meet all the demand s of the clien ts. This v ery intuitiv e model of comm un ication is motiv ated by the In ternet, wh ere routers pass information f rom differen t sites to users. It has b een b eliev ed that routers need only store and forw ard d ata (Multi Commo dity Flo w), without p ro cessing it at all. This int uition w as p ro ve d false b y Ahlswede, Cai, Li, and Y eung [ 3 ], who s ho we d a v ery simple n et w ork (the Butterfly Net w ork), whic h w as only satisfiable if one of the no des pro cessed the data wh ich en tered it. The enco d ing done in th is example was linear, and for some time it w as not clear if non linearit y is b eneficial in constructing optimal net work cod es. The w ork of Doughert y , F reiling, and Z eger [ 8 ] answe red this in the affirmativ e, giving an example of a n et w ork in whic h non linear co des are essential in order to ac hiev e the required netw ork capacit y . Their construction relies on the parit y of th e c haracteristic of th e und erlying field, and give s a ratio of 1 . 1 b et ween the co ding capacit y and the linear co d in g capacit y . Another wa y to ac hiev e a gap b et w een linear and non linear co des w as pr esen ted in [ 7 ]. Improvi n g this ratio, as we ll as fin ding new w a ys to create suc h gaps are op en p roblems in the field of Net wo rk Co ding (see [ 20 ] for a surv ey). T o see that our mo del is indeed a sp ecial case of net work co ding, we presen t the follo wing simple reduction b et we en a directed h yp ergraph whic h describ es a b roadcast net w ork H = ( V , E ) to a netw ork co d ing problem. W e build a net wo rk of n sources s 1 , . . . , s n , and m sinks t 1 , . . . , t m . There are also tw o sp ecial v ertices u and w . Letting E ∞ = { ( s i , u ) : i ∈ [ n ] } ∪ { ( w, t e ) : e ∈ E } ∪ { ( s j , t e ) : e = ( i, J ) ∈ E , j ∈ J } , the net wo rk has an edge with infin ite capacit y for eac h e ∈ E ∞ . In addition to that, the net wo rk has a single edge with finite capacit y , ( u, w ). If eac h sour ce receiv es an input of t bits, the demand of the net work can b e satisfied if and only if the capacit y of ( u, w ) is at least β t ( H ). Moreo ver, this reduction main tains linearit y of cod es. This reduction enables us to translate some o f our results to the net wo r k co ding mod el. In particular, we p ro ve the follo win g corollary of our results, impr o ving the resu lts of [ 8 ]: Corollary 1.5. Ther e exists a ne twork with 48 vertic es such that the r atio b etwe en the c o ding c ap acity and the line ar c o ding c ap acity in it is at le ast 1 . 324 . The corollary is based on the results of App endix A.3 , where w e sh o w that for a certain dir ected h yp ergraph, H 23 , any lin ear co d e requires 3 bits, while β ∗ ( H 23 ) ≤ 2 . 265. 5 2 Optimal co des for a disjoin t union of directed h yp ergraphs The size of an optimal co de for a giv en directed h yp ergraph describing a br oadcast n etw ork ma y b e restated as a pr ob lem of determining the chromatic n umb er of a graph , as observ ed by Bar-Y ossef et al. for the In dex Co ding mo del [ 4 ]. Consider the blo ck-l ength t = 1, and defin e the follo w in g: Definition 1 ( Confusion graph ) . L et H = ([ n ] , E ) b e a dir e cte d hyp er gr aph describing a b r o adc ast network. The confu sion graph of H , C ( H ) , is the undir e cte d gr aph on the vertex set { 0 , 1 } n , wher e two vertic es x 6 = y ar e adjac ent iff for some e = ( i, J ) ∈ E , x i 6 = y i and yet x j = y j for al l j ∈ J . In other words, C ( H ) is the graph wh ose v ertex set is all p ossible inpu t-w ords, an d tw o v ertices are adjacent iff they are confusable, meaning they cannot b e enco d ed by the same co deword for H (otherwise, the deco din g of at least one of the receiv ers w ould b e am b iguous). Hence, a co de for H is equiv alen t to a legal v ertex coloring of C ( H ), w h ere eac h color class corresp on d s to a distinct co dew ord. C onsequen tly , if C is an optimal co de for H , then |C | = χ ( C ( H )). Similarly , one can define C t ( H ), the co nf usion graph corresp ondin g to H with b lo c k-length t . F rom now on, throughout this section, th e length t of th e blo c ks considered will b e 1. Pro of of Theorem 1.1 . Th e OR graph pro d uct is equiv alen t to the complemen t o f the str ong pr o duct 3 , which wa s thoroughly studied in the inv estigation of the Shannon capacit y of a graph, a notoriously c hallenging graph parameter introd uced by Sh an n on [ 18 ]. Definition 2 ( OR graph product ) . The OR graph pro duct of G 1 and G 2 , denote d by G 1 · ∨ G 2 , is the gr aph on the vertex set V ( G 1 ) × V ( G 2 ) , wher e ( u, v ) and ( u ′ , v ′ ) ar e adjac ent iff either uu ′ ∈ E ( G 1 ) or v v ′ ∈ E ( G 2 ) (or b oth). L et G · ∨ k denote the k -fold OR pr o duct of a g r aph G . Let H 1 and H 2 denote directed hyp ergraphs (as b efore) on th e vertex-se ts [ m ] an d [ n ] resp ec- tiv ely , and consider an encod ing sc heme for their disjoint union, H 1 + H 2 . As th ere are n o edges b et ween H 1 and H 2 , suc h a co din g sc heme cannot enco de t w o input-words x, y ∈ { 0 , 1 } m + n b y the same cod ew ord iff this forms an am biguit y either with resp ect to H 1 or with resp ect to H 2 (or b oth). Hence: Observ ation 2.1. F or any p air H 1 , H 2 of dir e cte d hyp er g r aphs as ab ove, the gr aphs C ( H 1 + H 2 ) and C ( H 1 ) · ∨ C ( H 2 ) ar e isomorphic. Th u s, th e num b er of co dewo rd s in an optimal co de for k · H is equal to χ ( C ( H ) · ∨ k ). The c hromatic n umb er s of strong pow ers of a graph, as w ell as those of OR graph p ow ers , h a v e b een studied in tensiv ely . I n the form er case, they corresp ond to the Witsenhausen rate of a graph (see [ 19 ]). In the latter case, the follo wing w as pro ved b y McEliece and P osner [ 16 ], and also by Berge and Simonovit s [ 5 ]: lim k →∞  χ ( G · ∨ k )  1 /k = χ f ( G ) , (3) where χ f ( G ) is the fr actional chr omatic numb er of the graph G , defined as follo ws. A legal v ertex coloring corresp onds to an assignment of { 0 , 1 } -w eigh ts to indep endent-se ts, such that ev ery vertex will be “co v ered” b y a total w eigh t of at least 1. A fractional co loring is the relaxation of th is 3 Namely , th e OR p rod u ct of G 1 and G 2 is the complement of the strong pro du ct of G 1 and G 2 . 6 problem where the w eigh ts b elong to [0 , 1], and χ f is the min im um p ossible sum of weig hts in suc h a fractional coloring. T o obtain an estimate on the rate of the con ve rgence in ( 3 ), we w ill use the follo wing w ell-kno wn prop erties of the fractional c hromatic num b er an d OR graph p ro ducts (cf. [ 2 ],[ 14 ],[ 12 ] and also [ 9 ]): (i) F or an y graph G , χ f ( G · ∨ k ) = χ f ( G ) k . (ii) F or an y graph G , χ f ( G ) ≤ χ ( G ) ≤ ⌈ χ f ( G ) log | V ( G ) | ⌉ . [This is pro v ed b y selecting ⌈ χ f ( G ) log | V ( G ) | ⌉ indep endent sets, eac h c hosen randomly and ind ep endently according to the weig ht distrib ution, dicta ted b y the optimal w eigh t-function ac h ieving χ f . O ne can sho w that the exp ected num b er of un co v ered v ertices is less than 1.] (iii) F or any v ertex transitiv e graph G (that is, a graph whose automorphism group is transitive ), χ f ( G ) = | V ( G ) | /α ( G ) (cf., e.g., [ 10 ]), where α ( G ) is the indep endence n umb er of G . In ord er to translate ( ii ) to the statemen t of ( 2 ), notice that γ , as defined in Th eorem 1.1 is precisely α ( C ( H )), the indep end en ce num b er of the confu sion graph. In addition, the graph C ( H ) is indeed v ertex transitiv e, as it is a Ca yley graph of Z n 2 . Com bining the ab o v e facts, w e obtain that: χ f  C ( H ) · ∨ k  1 /k = 2 n α ( C ( H )) = 2 n γ . Plugging the a b o ve equation into ( ii ), while recalling that χ  C ( H ) · ∨ k  is the s ize of the optimal co de for k · H , completes the pro of of the theorem.  Remark 2.2: The r igh t-hand-side of ( 2 ) can be replaced by  2 n γ  k ⌈ 1 + k log γ ⌉ . T o see this, com bine the simp le fact that α ( G · ∨ k ) = α ( G ) k with the b ound χ ( G ) ≤ ⌈ χ f ( G )(1 + log α ( G )) ⌉ giv en in [ 14 ] (whic h can b e p ro ve d by choosing ⌈ χ f ( G ) log α ( G ) ⌉ indep endent sets randomly as b efore, lea v in g at most ⌈ χ f ( G ) ⌉ unco v ered v ertices, to b e co vered separately). 3 The p ossible gaps b et w een the parameters β , β ∗ and β 1 As noted in ( 1 ), β ≤ β ∗ ≤ β 1 . In this section, we describ e net works where the gap b et wee n any t w o of these parameters can b e unboun ded. The first constru ction is of a directed h yp ergraph on n v ertices wh ere β = 2 while β ∗ and β 1 are b oth Θ(log n ), for an y n = 2 k . W e then describ e a more surprising, general construction whic h pro vides a family of directed h yp ergraph s , f or wh ic h β ∗ < 3 wh ile β 1 = Θ(log log n ). Finally , we d escrib e simple scenarios where ev en in the restricted Index Co ding m o del, taking disjoint co pies of the net wo rk can b e encod ed strictly b etter than concatenati n g the enco dings of eac h of the copies. These constructions also app ly to n etw ork co ding, where for the latter ones it is also p ossible to pr ov e a lo w er bou n d on the length of th e optimal linear enco din g sc heme. Throughout th is sec tion, we use th e follo wing notations. F or b inary ve ctors u and v , let | u | denote the Hamming weigh t of u , and let u ⊕ v b e the bit wise xor of u an d v . 7 3.1 Pro of of Theorem 1.4 Consider a scenario in wh ic h the inp ut w ord consists of a blo ck x i for eac h n onzero elemen t i ∈ Z k 2 , th us th e n umb er of blo c ks is n = 2 k − 1. F or an y pair of distinct nonzero elemen ts i, j ∈ Z k 2 , there exists a receiv er wh ich is intereste d in the blo ck x i and kn o ws all other b lo c ks except for x j . This scenario corresp onds to a directed hyp ergraph H in w hic h the v ertices are th e nonzero elemen ts of Z k 2 , and for ev er y i, j ∈ Z k 2 w e h a v e a directed edge ( i, Z k 2 \ { 0 , i, j } ). Let C = C ( H ) b e the confus ion graph of H for blo c k-length t = 1. Since eac h r eceiv er is miss in g precisely tw o blo cks, for any pair of distinct co d ew ords u, v ∈ { 0 , 1 } n , w henev er | u ⊕ v | ≥ 3, ev ery receiv er ca n distinguish b et w een the co d ew ords. On the other hand, if | u ⊕ v | ≤ 2, there is some receiv er wh o ma y confu se h is blo c k in u an d v . Thus, C is exactly the Cayle y graph of Z n 2 whose generators are all element ary unit vect ors e i , as we ll as all v ectors e i ⊕ e j . Claim 3.1. The ab ove define d gr aph C satisfies χ ( C ) ≤ n + 1 . Pr o of. S ince n = 2 k − 1 there is a Hamming code of le ngth n , and the requ ir ed coloring of any v ector u is giv en by computing its syndr ome. More explicitly , define c : V ( C ) 7→ Z k 2 as follo w s. F or a v ector u = ( u 1 , . . . , u n ) ∈ Z n 2 , pu t c ( u ) := P i ∈ Z k 2 −{ 0 } u i · i . Let u, v ∈ Z n 2 b e a p air of adjacent v ertices in C . Th us 1 ≤ | u ⊕ v | ≤ 2, and c ( u ) ⊕ c ( v ) is a sum (in Z k 2 ) of one or tw o distinct nonzero elemen ts of Z k 2 , whic h is not zero. Thus c ( u ) 6 = c ( v ), as needed.  Claim 3.2. The ab ove define d gr aph C satisfies χ f ( C ) ≥ n + 1 , and thus χ ( C ) = χ f ( C ) = n + 1 . Pr o of. Recall th at the clique n umb er of th e graph, n amely the size of the largest clique in the graph, p ro vides a lo we r b oun d on the fractional c hr omatic num b er. Th us it suffices to show that C con tains a clique of size n + 1. Defin e the v ertex set S = { 0 } ∪ { e i } n i =1 of size n + 1. F or an y u, v ∈ S , | u ⊕ v | ≤ 2 and hence S induces a clique in C completing the pro of of th e inequality . The equalit y follo ws from the p revious claim.  Corollary 3.3. The p ar ameters β 1 ( H ) , β ∗ ( H ) satisfy β ∗ ( H ) = β 1 ( H ) = log 2 ( n + 1) . Pr o of. Recall that β 1 ( H ) = ⌈ log 2 χ ( C ( H )) ⌉ , and that in Theorem 1.1 we h av e actually sho wn that β ∗ ( H ) = log 2 χ f ( C ( H )). The pro of therefore follo w s from th e fact that χ ( C ) = χ f ( C ) = n + 1.  Let C t = C t ( H ) denote the confusion graph for blo c k-length t . Thus, C t is the Cayle y graph of Z nt 2 whose generators are all ve ctors { ( w 1 , . . . , w n ) | w i ∈ Z t 2 , 1 ≤ |{ i | w i 6 = 0 }| ≤ 2 } . In o ther w ords, tw o v ertices are connected in th e confusion graph if they d iffer at no more than 2 blo c ks. Claim 3.4. F or 2 t ≥ n , χ ( C t ) = χ f ( C t ) = 2 2 t . Pr o of. F or a lo w er b ound , it suffices to sho w a set o f size 2 2 t whic h is a cli qu e in C t . Consider the v ertex set { ( u 1 , u 2 , 0 , . . . , 0) | u 1 , u 2 ∈ Z t 2 } in C t , whic h consists of 2 2 t v ertices. Ev ery pair of v ertices in this set is connected in C t since th ey differ in at most t wo blo c ks, and therefore this is a clique in C t . This sho ws that χ ( C t ) ≥ 2 2 t . T o complete the pr o of, we describ e a prop er coloring of C which uses 2 2 t colors, using a simple Reed-Solomon cod e. Let α 1 , . . . , α n b e pairwise d istinct elements in the finite field GF 2 t , and 8 define the coloring c : ( GF 2 t ) n → GF 2 t × GF 2 t as follo ws. F or a ve ctor u = ( u 1 , . . . , u n ), let c ( u ) := ( P n i =1 u i , P n i =1 α i · u i ). C learly , if u, v ∈ ( GF 2 t ) n differ in exactly one blo ck then the first co ordinate of c ( u ) and c ( v ) is different . Moreo v er, if u and v differ in exact ly tw o blo cks i, j , then either u i + u j 6 = v i + v j or α i u i + α j u j 6 = α i v i + α j v j (or b oth inequalities hold), and aga in they will h a v e d ifferen t colors as needed. This sho ws that the coloring c is indeed prop er and completes the pro of of the claim.  Recalling that β t ( H ) = ⌈ log 2 χ ( C t ) ⌉ we obtain the follo w ing corollary , whic h together w ith Corollary 3.3 completes the pro of of Theorem 1.4 . Corollary 3.5. F or the hyp er gr aph H define d ab ove, β ( H ) = lim t →∞ 1 t log 2 χ ( C t ( H )) = 2 . 3.2 Pro of of Theorem 1.3 The basic in gred ien t of the construction is a Ca yley graph G of an Ab elian group K = { k 1 , . . . , k n } of size n , for whic h there is a large gap b et ween the c hromatic num b er and the fractional chromatic n umb er. In our conte xt, w e u se K = Z k 2 , though suc h a Ca yley graph of any Ab elian group will d o. Lemma 3.6. F or any n = 2 k ther e e xists an explicit Cayley gr aph G o ver th e A b elian gr oup Z k 2 for which χ ( G ) > 0 . 01 √ log n and yet χ f ( G ) < 2 . 05 . Pr o of. L et n = 2 k , and consider the graph G on the set of v ertices Z k 2 as follo ws. F or an y i, j ∈ Z k 2 , the edge ( i, j ) ∈ G if | i ⊕ j | ≥ k − √ k 100 . W e will no w sho w that this graph has a large gap χ ( G ) /χ f ( G ) > Ω( √ k ) = Ω( √ log n ). Claim 3.7. The chr omatic numb er of G satisfies χ ( G ) ≥ √ k 100 + 2 . Pr o of. T he indu ced subgraph of G on the ve rtices { i ∈ Z k 2 | | i | = s } where s = k 2 − √ k 200 , is the Kneser graph K ( k , s ) whose c hromatic num b er is precisel y k − 2 s + 2, as prov ed in [ 13 ], using the Borsuk-Ulam Theorem. It is w orth noting that one can giv e a sligh tly simpler, self-conta ined (top ologica l) pro of of th is claim, based on the approac h of [ 11 ].  Claim 3.8. The fr actional chr omatic numb er of G satisfies χ f ( G ) < 2 . 05 Pr o of. S ince G is a Ca yley graph, it is w ell kno wn that χ f ( G ) = | V ( G ) | /α ( G ) (c.f. e.g. [ 17 ]), and therefore it s u ffices to show it co ntains an ind ep endent set of size at least 2 k 2 . 05 . Let I = { i ∈ Z k 2 | | i | < k 2 − √ k 200 } . Obvio usly the set I i s an indep end en t set as the Hamming weigh t of i ⊕ j is b elo w k − √ k / 100 for any i, j ∈ I , and therefore ( i, j ) 6∈ E ( G ). Hence, χ f ( G ) ≤ 2 k | I | = 2 k P i< k 2 − √ k 200  k i  < 2 . 05 .  This completes the pro of of Lemma 3.6 .  9 Let H be the directed hyp ergraph on the ve rtices V = { 1 , 2 , . . . , n } defined as follo ws. F or eac h pair of vertic es i, j suc h that k i , k j are adjacen t in G (i.e. k i − k j is a generator in th e defining set of G ), H con tains the d ir ected edges ( i, V \ { i, j } ) and ( j, V \ { i, j } ). As b efore, every receiv er misses precisely tw o blo c ks. Let C = C 1 ( H ) b e the confu sion graph of H (for blo ck-l ength t = 1). Thus, C is the Cayley graph of Z n 2 whose ge n er ators are all v ectors e i , as well as all v ectors e i ⊕ e j so that k i , k j are adjacen t in G . Claim 3.9. The chr omatic numb er of C satisfies χ ( G ) ≤ χ ( C ) ≤ 3 · χ ( G ) . Pr o of. T hat fac t that χ ( G ) ≤ χ ( C ) follo ws from the observ ation that the induced subgraph of C on the vertic es e i is precisely G (and hence, w e similarly ha ve χ f ( G ) ≤ χ ( C )). It remains to pr o v e that χ ( C ) ≤ 3 χ f ( G ). Let c b e some optimal colo r in g of G w ith d = χ ( G ) colors. Define a coloring of C with 3 d colors as follo ws. F or a ve rtex x = x 1 . . . x n assign the color c ′ ( x ) = ( | x | mo d 3 , P i x i · c ( i )) w h ere the sum is in Z d . Clearly , c ′ uses 3 d colo rs . It r emains to sho w that this is in deed a legal coloring. Consider x, y whic h are adjacen t in C , thus either x ⊕ y = e i ( ⇒ | x | 6≡ | y | mo d 3) or x ⊕ y = e i ⊕ e j . If | x | 6≡ | y | mo d 3 they will h a v e differen t colors, otherwise, x ⊕ y = e i ⊕ e j , | x | = | y | and there exists z ∈ Z d so t h at c ′ ( x ) = ( | x | , z + c ( i )) and c ′ ( y ) = ( | y | , z + c ( j )). Since x, y are adjacen t, we kno w i, j are adjacent in G , th us c ( i ) 6 = c ( j ) which implies that c ′ ( x ) 6 = c ′ ( y ) as r equ ired. Note that in the sp ecial case where G is a C a yley graph of the group Z k 2 , the ab o ve u pp er b ound on χ ( C ) can b e m o dified in to the smallest p o we r of 2 that is str ictly larger than χ ( G ).  Notice that in the ab o ve claim we did use an y of the prop erties of G , hence they hold for an y graph. Th is shows that regardless of the c hoice of G , for the confusion graph defined ab o v e, the gap b etw een χ and χ f is at most 3 times the corresp onding gap in the original graph. Claim 3.10. The fr actional chr omatic numb er of C satisfies χ f ( G ) ≤ χ f ( C ) ≤ 3 · χ f ( G ) . Pr o of. As ment ioned ab ov e, the lo wer b ound on χ f ( C ) follo ws from the induced cop y of G in C , and it remains to sho w that χ f ( C ) ≤ 3 χ f ( G ). Since C is a Ca yley graph, it su ffi ces to sho w it con tains an in dep end ent set of size at least 2 n 3 · χ f ( G ) . Let I ⊂ K be a maxim um indep enden t set in G . As G is a C a yley graph, χ f ( G ) = n / | I | . F or a vect or u = ( u 1 , . . . , u n ) ∈ Z n 2 , defin e s ( u ) ∈ K by s ( u ) = P u i · k i . F or any j ∈ K , put I j = { u ∈ Z n 2 | s ( u ) + j ∈ I } . Let u, v b e a pair of v ertices so that u ⊕ v = e i ⊕ e j and | u | = | v | . Hence s ( u ) = x + k i and s ( v ) = x + k j (here w e r ely on K b eing Ab elian). If u, v b oth b elong to I l for some l ∈ K , it must b e that k i and k j are not adjacent in G , and thus u and v are n ot adjacen t in C . It no w follo ws that if u and v are v ectors in I j and | u | ≡ | v | (mod 3), then u and v are not adjacen t in C . Therefore, I j is a un ion of three indep endent sets in C , and hence at lea st one of them is of size at le ast | I j | / 3. Th is holds for every j ∈ K . When j ∈ K is c h osen rand omly and 10 uniformly , then, by linearit y of exp ectation, the exp ected num b er of elemen ts in I j is exactly 2 n · | I | n = 2 n χ f ( G ) . Th u s, there is some choice of j for whic h I j is of size at least 2 n /χ f ( G ), and C con tains an indep end en t set of size at least | I j | / 3 = 2 n / 3 χ f ( G ), as n eeded.  Corollary 3.11. F or any Cayley gr aph G of an Ab elian gr oup, ther e exists a c onfusion gr aph C and some c ∈ [ 1 3 , 3] such that χ ( C ) χ f ( C ) = c · χ ( G ) χ f ( G ) . Plugging the g rap h which is guarant eed by Lemm a 3.6 in to this construction c ompletes the pro of of Th eorem 1.3 . Remark 3.12: If we set G = K n whic h is indeed a Ca yley graph o ver an Ab elian group, w e get the example from Section 3.1 . Some of the claims in this section generalize claims fr om Section 3.1 . 3.3 Applications t o Index and Net w ork Co ding W e are no w consid ering the m ore restricted mo del where there is a single receiv er whic h is interested in eac h blo ck. In the directed hyper grap h notation, this is equiv alent to having precisely m = n directed edges where eac h directed edge has a d ifferen t origin v ertex. F or easier notations, we can describ e s uc h a scenario (as done in [ 4 ], [ 15 ]) by a dir ected graph. Eac h directed edge ( i, J ) will b e translated into | J | directed edges ( i, j ) f or all j ∈ J . W e can also consider an undirected graph for the case where the receiv er who is in terested in x i kno ws x j iff the receiv er who is in terested in x j kno ws x i . W e u se similar notations to the directed h yp ergraphs. Clearly , β 1 ( k · G ) ≤ k · β 1 ( G ), as o n e can alw a ys obtain an index co d e f or k · G b y taking the k -fold concatenation of an optimal index co de f or G . F urthermore, it is not difficult to see that this b ound is tigh t for all p erfect graphs. Hence, the smallest graph wh ere β 1 ( k · G ) may p ossib ly b e smalle r than k · β 1 ( G ) is C 5 , the cycle on 5 ve rtices - the smallest non-p erfect graph. Ind eed, in this case index co d es for k · C 5 can b e significan tly b etter than those obtained b y treating eac h cop y of C 5 separately . Th is is stated in Th eorem 1.2 which we no w pr o v e. Pro of of Theorem 1.2 . On e can v erify th at the follo wing is a maxim um indep endent set of size 5 in the confusion graph C ( C 5 ): { 00000 , 011 00 , 00011 , 1101 1 , 11101 } . In the formulat ion of Theorem 1.1 , γ = 5, and th is theorem no w implies that β 1 ( k · C 5 ) /k tends to 5 − log 2 5 as k → ∞ . On the other hand, one can v erify 4 that χ ( C ( C 5 )) = 8, hence β 1 ( C 5 ) = 3.  This s ho ws that there is a graph G with an optimal index cod e C , so th at muc h less than |C | k w ords suffice to esta b lish an ind ex code for k · G , although eac h of th e k copies of G has no side information on any of the bits corresp onding to the remaining copies. 4 This fact can b e veri fied by a computer assisted pro of, as stated in [ 4 ]. 11 Remark 3.13: Usin g the u pp er b ound of ( 2 ) in its alternate form , as stated in Remark 2.2 , w e obtain that β 1 ( k · C 5 ) < k · β 1 ( C 5 ) already for k = 15. The example of C 5 can b e extended to other examples by lo oking at all the complement of od d cycles, i.e. C k for an y o d d k ≥ 5. All graph s in this family ha ve a gap b et we en th e optimal co de for disjoin t union in comparison t o the conca tenation of the optima l co de for a single cop y . In App end ix A.2 w e pro v e the follo win g p r op erties of the complements of o dd cycles: Claim 3.14. Ther e exists a c onstant c > 1 so that for any n ≥ 2 , χ ( C ( C 2 n +1 )) > c · χ f ( C ( C 2 n +1 )) . Theorem 3.15. L et H 2 n +1 = ([2 n + 1] , E ) , wher e for e ach i ∈ [2 n + 1] ther e is a d ir e cte d e dge ( i, N C 2 n +1 ( i )) in E , and N C 2 n +1 ( i ) ar e the neighb ors of i in C 2 n +1 . Then any line ar c o de for H 2 n +1 r e q u ir es 3 letters. Theorem 3.15 implies that a b roadcast net w ork based on the complemen t of an y o dd cycle h as linear co d e of m in imal length 3, regardless of the blo c k length. Sp ecifically for C 23 , we kno w that χ f ( C ( C 23 )) ≤ 4 . 809 (as can b e seen in App endix A.3 ). Th erefore, the ab o v e mentio ned reduction to Net work Co ding pro vides us with an explicit net work (of size 48 ) w here the linear code m us t b e of length at least 3 w hereas the optimal co de can b e of length β ≤ β ∗ ≤ log 2 4 . 809 ≈ 2 . 26 5, yielding a r atio of 1 . 324 . This prov es Corollary 1.5 .  4 Conclusions and op en problems • In this wo rk, w e hav e shown that for ev ery broadcast net work H with n b lo c ks and m receiv ers, and for large v alues of k , β ∗ k ( H ) = β 1 ( k · H ) = ( n − log 2 α ( C 1 ( H )) + o (1)) k , where th e o (1)- term tends to 0 as k → ∞ . F or ev ery la rge constan t C th ere are examples H suc h that for large k , β ∗ k ( H ) /k < 3 and y et β 1 ( H ) > C . • Ou r results also imply that enco din g the en tire block at once can b e strictly b etter than concatenati n g the optimal co de for H with a single bit blo c k. This justifies the d efinition of the broadcast rate of H , β ( H ), as the optimal asymptotic a ve r age n u m b er of bits required p er a single b it of co ding in eac h blo c k for H . • W e ha ve shown an infinite f amily of graph s (includin g the smallest p ossible non-p erfect graph C 5 ) for which there exists a constant c > 1 so that for eac h of these graphs there is a m ultiplicativ e gap of at lea st c b et wee n the c hromatic num b er and the fr actional c hromatic n umb er of their confusion graphs. How eve r , the gap in all these graphs is b elo w 2, and it is not known if for graphs this gap can b e arbitrarily large. • Generalizing the ab o v e setti n g, allo win g multiple users to request the same blo ck, allo ws us to construct hyp ergraphs whose confusion graph s exhibit bigger gaps. – W e ha ve sho wn a s p ecific f amily of confu sion graph s wh ere the fractional chromatic n umb er is boun ded ( < 7) while the chromatic num b er is un b ound ed (Ω( √ log n )). In these settings, a 1-bit blo c k-length will require us to transmit Θ(log log n ) b its while for large t -bit blo ck-le ngth, the required num b er of bits is linear in t . F or o ther famili es, 12 this ratio can ev en reac h Θ(log n ). More surpr isingly , for the first family , a net work consisting of t indep endent copies of the original one will only require a n u m b er of bits that is linear in t . – With the generalized construction, one can bu ild a h yp ergraph for any C a yley graph of an Ab elian group for which the confu sion graph mainta ins th e same gap as the original graph. Th e maxim um gap that ca n b e obtained in this wa y is O (log n ), since th is is the maxim um p ossible gap b et we en the fractional and in teger c hromatic num b ers of an y n -v ertex graph (c.f., e.g.,[ 17 ]). • Cu rrentl y , th ere is no kno wn b etter upp er b oun d for this gap w hic h is sp ecific for confusion graphs. The examples ab o ve are all exp onenti ally far from the general upp er b oun d Θ(log V ), whic h in our case equals to Θ(log 2 n ) = Θ( n ). • An in teresting prob lem in Net work C o ding is that of deciding w hether or not th er e are net wo rks with an arb itrarily large gap b et w een the optimal linear and non-linear fl o ws. Note that the netw ork is not allo w ed to dep end on the size of the un derlying fi eld. Generalizing our constructions to create suc h examp les could b e interesti ng. References [1] N. Alon, The Shann on capacit y of a un ion, C om binatorica 18 (199 8), 301-310 . [2] N. Alon and A. Or litsky , Rep eated comm un ication and Ramsey graphs, IE EE T ransactions on Information T heory 41 (1995), 1276-128 9. [3] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Y eung, Net work information flo w, IEEE T rans. Inform. Th eory , vol. IT-46, pp. 12041216, 2000. [4] Z. Bar-Y ossef, Y. Birk, T.S. Ja yram and T . 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Theory , v ol. IT-45, pp. 1111-1120 , 19 99. 14 A App endix A.1 Pro of of Theorem 3.15 W e firs t need t o presen t some definitions and t heorems from [ 4 ]. W e say that a matrix A fits a graph G = ( V , E ) if A [ i, i ] 6 = 0 f or all i and A [ i, j ] = 0 for i 6 = j , ( i, j ) 6∈ E (for ( i , j ) ∈ E , A [ i , j ] is not limited). A generalization of a result in [ 4 ] (as noted in [ 15 ]) states that the lengt h of the minimal linear encod ing of G ov er a field F is alw a ys at least the minimal rank o ve r F of a matrix A whic h fits G . It therefore suffi ces to sho w the follo w in g: Claim A.1. L et A b e a matrix that fits the g r aph C 2 n +1 over some field F . Then rank( A ) ≥ 3 . Pr o of. L et A b e a matrix that fits C 2 n +1 . This means that ∀ i . A [ i, i ] 6 = 0 , ∀ i ∈ [2 n ] . A [ i, i + 1] = 0 , ∀ i ∈ [2 n ] . A [ i + 1 , i ] = 0 , A [1 , 2 n + 1] = A [2 n + 1 , 1] = 0 . Let A ( t ) denote the t ’th row of A , when w e lo ok at it as a vecto r. Note that A (1) , A (2) are linearly indep end en t, as A [1 , 1] 6 = 0 but A [2 , 1] = 0 and A [2 , 2] 6 = 0. Assume, to wards a cont radiction, that rank( A ) = 2 and hence for ev ery t : A ( t ) = a t A (1) + b t A (2). W e pro ve b y induction on t that if t is o dd then b t = 0, an d if t is ev en then a t = 0. Note that for an y t it is imp ossible that a t = b t = 0, as eac h line has a nonzero element. F o r t = 1 , 2 this is trivial. F or some o dd t = 2 k + 1, b y assu mption A (2 k ) = b 2 k A (2) and A (2 k − 1) = a 2 k − 1 A (1). This means that A (2 k + 1) = a 2 k + 1 /a 2 k − 1 A (2 k − 1) + b 2 k + 1 /b 2 k A (2 k ) As A [2 k − 1 , 2 k ] = A [2 k + 1 , 2 k ] = 0 but A [2 k, 2 k ] 6 = 0, this means that b 2 t +1 = 0, as required . A similar argumen t w orks for even t , whic h completes the pro of of the induction. Ho wev er, this leads to a con tradiction, as A (2 n + 1) = a 2 n +1 A (1), and this is impossible as A [2 n + 1 , 1] = 0 but A [1 , 1] 6 = 0. Altog ether, the assumption that r ank ( A ) = 2 leads to a co ntradictio n, hence rank( A ) ≥ 3.  Claim A.1 completes the pr o of of Theorem 3.15 . A.2 Complemen t s of odd cycles W e sh o w ed that the cycle of length 5 is the smallest graph where th ere exists a gap b et wee n the fractional and integ er chromatic num b ers of its confu s ion graph . The cycle and its complement on 5 v ertices are isomorphic, ho wev er this is not the case for larger o dd cycles and their complements. W e now sh o w th at there is a gap b etw een those n u mb ers for any complemen t of an od d cycle of 5 or more ve r tices. Throughout this s ection, let C 2 n +1 = C ( C 2 n +1 ). 15 Claim A.2. A ny indep endent set A of C 2 n +1 c an b e extende d to an indep endent set A ′ in C 2 n +3 wher e | A ′ | = 4 | A | . Pr o of. W e first d efine a function f from the v ertices of C 2 n +1 in to sets of size 4 from the vertic es of C 2 n +3 whic h satisfies the follo w ing: • F or eve r y vertex v of C 2 n +1 , f ( v ) is an indep end en t set in C 2 n +3 . • If u and v are not adjacen t in C 2 n +1 , then f ( v ) ∪ f ( u ) is an indep endent set in C 2 n +3 . • f ( v ) ∩ f ( u ) = ∅ f or an y u 6 = v . Giv en s u c h f and an ind ep endent set A of C 2 n +1 , define A ′ = S v ∈ A f ( v ) whic h will b e an indep end en t set of size 4 | A | in C 2 n +3 as n eeded. W e now describ e this f explicitly and p ro v e its prop erties: f ( v = ( x 1 , x 2 , . . . , x 2 n , x 2 n +1 )) =          ( x 1 , x 2 , . . . , x 2 n , x 2 n +1 , 0 , 0) = v ′ ⊕ m 0 ( x 1 , x 2 , . . . , x 2 n , x 2 n +1 , 0 , 1) = v ′ ⊕ m 1 ( x 1 , x 2 , . . . , x 2 n , x 2 n +1 , 1 , 0) = v ′ ⊕ m 2 ( x 1 , x 2 , . . . , x 2 n , x 2 n +1 , 1 , 1) = v ′ ⊕ m 3          where v ′ is v extended to length 2 n + 3 w ith t wo additional zeros at the righ t end and m 0 , m 1 , m 2 , m 3 are 4 appr opriate constan t b inary v ectors of length 2 n + 3. • f ( v ) is an indep enden t set: Since C 2 n +3 is a Ca yley graph o ver Z 2 n +3 2 , we only n eed to sh o w that the su ms of all pairs of the 4 v ectors in f ( v ) are not generators in our graph . All these sums are in { m 1 , m 2 , m 3 } since m 0 = 0 and m 1 ⊕ m 2 ⊕ m 3 = 0 (notice that these sums are indep endent of the c hoice of v ). Since w e are lo oking at the confu sion graph of the complemen t of an o d d cycle, the generators of this graph are ve ctors of hamming w eigh t 1,2 and 3 of consecutiv e ones (i.e. e i , e i ⊕ e i +1 and e i ⊕ e i +1 ⊕ e i +2 for an y i wher e the indices are reduced mo dulo 2 n + 3). Indeed { m 1 , m 2 , m 3 } are not generators, therefore f ( v ) is indep endent. • f ( v ) ∪ f ( u ) is an indep en den t set w hen u and v are n ot adjacen t: Consider x = v ′ ⊕ m i and y = u ′ ⊕ m j for some i, j (where u ′ and v ′ are as b efore). W e wan t to sh ow that x ⊕ y is not a generator. If i = j then x ⊕ y = u ′ ⊕ v ′ and ther efore it is not a generator (t w o additional zero b its at the righ t w ill not turn a non-generator in to a generator). A ssu me n o w i 6 = j , then w e kno w x ⊕ y = ( u ′ ⊕ v ′ ) ⊕ m k = ( u ⊕ v ) ′ ⊕ m k for some k ∈ { 1 , 2 , 3 } . S ince u ⊕ v is not a generator, with claim A.3 w e get that x ⊕ y is not a generator as n eeded. • F or v 6 = u , f ( v ) ∩ f ( u ) = ∅ : Let us assume ther e exists some x ∈ f ( v ) ∩ f ( u ). Since all v ectors in b oth f ( v ) and f ( u ) differ at their t w o right most b its { x 2 n +2 , x 2 n +3 } , there exists i ∈ { 0 , 1 , 2 , 3 } so that x = v ′ ⊕ m i = u ′ ⊕ m i in con tradiction to v 6 = u .  Claim A.3. If x of length 2 n + 1 is not a gener ator, then x ′ ⊕ m k for k ∈ { 1 , 2 , 3 } is not a gener ator as wel l (wher e x ′ is x extende d with two zer o bi ts on the right as b efor e). 16 Pr o of. T he cases of k = 1 and k = 2 are s y m metric so w e will consider only k = 1 and k = 3. W e sho w explicitl y what x can b e in order for the result to b e a generator, and as all suc h x vect ors turn out to b e generators the d esired result follo ws. Since x ′ is the extension of x with t wo zero bits on the r ight, it cannot affect the t w o righ tmost b its of x ′ ⊕ m k . • k = 1: m 1 = 0 2 n − 1 z }| { 0 . . . 0 101 so in ord er to mak e it a generator, we m ust flip the 2 n + 1 bit and then we can at most flip 0,1 or 2 consecutiv e ones at the left most s id e. Hence, x ∈ { 00 0 . . . 0 | {z } 2 n − 2 1 , 10 0 . . . 0 | {z } 2 n − 2 1 , 11 0 . . . 0 | {z } 2 n − 2 1 } whic h are all generators of length 2 n + 1. • k = 3: m 3 = 1 2 n − 1 z }| { 0 . . . 0 111 so in order to mak e it a generator, w e m ust flip at least one of the bits at lo cations { 1 , 2 n + 1 } and no other bit. Thus, x ∈ { 0 0 . . . 0 | {z } 2 n − 1 1 , 1 0 . . . 0 | {z } 2 n − 1 0 , 1 0 . . . 0 | {z } 2 n − 1 1 } whic h are all generators of length 2 n + 1.  Claim A.4. The fr actional chr omatic numb er of C 2 n +1 is monotone de cr e asing with n , that is, χ f ( C 2 n +3 ) ≤ χ f ( C 2 n +1 ) . Pr o of. S ince G is a Cayle y graph, w e kno w χ f ( C 2 n +1 ) = 2 2 n +1 /α ( C 2 n +1 ). Using the pr evious claim w e kn o w α ( C 2 n +3 ) ≥ 4 α ( C 2 n +1 ) and therefore χ f ( C 2 n +3 ) = 2 2 n +3 /α ( C 2 n +3 ) ≤ 2 2 n +1 /α ( C 2 n +1 ) .  Corollary A.5. F or a ny n ≥ 8 , χ f ( C 2 n +1 ) < 4 . 99 ( < 5) . Pr o of. By a computer searc h (see App endix A.3 ) one can see that the fractional c hr omatic n umber of the confusion graph of the co mp lemen t of an odd cycle on 17 v ertices is b elo w 5. Since this prop erty is monotone, this holds for any n ≥ 8.  Pro of of Cla im 3.14 . Th e authors of [ 4 ] sho w ed that the c h romatic n umber of the confus ion graph of an y complemen t of an o dd cycle is strictly bigger than 4 and is at most 8. W e ha v e sho wn that the fractional c hromatic num b er of the confusion graph of a cycle on 5 vertice s is 32/5=6.4 and w e j ust prov ed it is monotone decreasing. S ince the num b er of vertic es in these co n f usion graphs is a p o we r of 2, the fr actional chromatic n umb er cannot b e an in teger betw een 4 and 8 so it will alw a ys b e smaller than the in teger c h romatic num b er. W e kn ow that for an y n ≥ 8 there is a gap of at least 5 /χ f ( C 17 ). F or s m aller v alues of n there is some fixed gap exceeding 1, so taking the minimum b et we en these gaps will giv e a single c for whic h the claim holds.  The limit lim n →∞ χ f ( C 2 n +1 ), which exists b y m on otonicit y , remains u nknown at this time. Claim A.6. The c hr omatic numb er of C 2 n +1 is monotone de cr e asing with n : χ ( C 2 n +3 ) ≤ χ ( C 2 n +1 ) . Pr o of. A coloring of C 2 n +1 with k colo rs is a p artition of the graph in to k indep endent sets. W e can obtain a coloring of C 2 n +3 with the same num b er of colors b y applying the extension describ ed 17 b efore o n eac h of the indep end ent sets. W e a lready pro ved the new sets w ould be indep endent sets in the new graph. W e need to prov e that all the vertic es of th e g raph b elong to one of the indep end en t sets. T his can b e shown by noticing th e size of the new s ets is 4 times their p revious size and since t h ey ha ve empt y int ersection (as w e hav e seen b efore) they must co ver the entire graph (as the size of the graph is p recisely 4 times th at of the pr evious one).  Corollary A.7. F or a ny n ≥ 3 , 5 ≤ χ ( C 2 n +1 ) ≤ 7 . Pr o of. By a computer searc h (see App end ix A.4 ) one can see that for 7 v ertices, the int eger c hro- matic n umber is at m ost 7. By monotonicit y and the fact that is m ust b e at least 5 (as shown in [ 4 ]) the desired result follo w s.  As in th e fractional case, the limit lim n →∞ χ ( C 2 n +1 ) exists an d can only b e 5,6 or 7, how ever it remains unkn o wn at this time. A.3 F ractional c hromatic n um b er upper bounds for C  C n  Here is a table of u pp er b ounds for the fractional c hromatic num b er of the confus ion graphs of the complemen ts of o dd cycles. These up p er b ounds w ere found b y searc hing for a large indep endent set in eac h of these graphs as they are Ca yley graphs. This searc h wa s done by a computer pr ogram whic h do es not assure us for the optimal result, hence it only pro vides the b oun ds stated in the table, wh ich are n ot necessarily tigh t. n α ( C  C n  ) χ f ( C  C n  ) 5 5 2 5 / 5 = 6 . 4 7 ≥ 22 ≤ 2 7 / 22 ≈ 5 . 818 9 ≥ 93 ≤ 2 9 / 93 ≈ 5 . 505 11 ≥ 386 ≤ 2 11 / 386 ≈ 5 . 306 13 ≥ 1586 ≤ 2 13 / 1586 ≈ 5 . 165 15 ≥ 6476 ≤ 2 15 / 6476 ≈ 5 . 060 17 ≥ 26317 ≤ 2 17 / 2631 7 ≈ 4 . 981 19 ≥ 106744 ≤ 2 19 / 1067 44 ≈ 4 . 912 21 ≥ 430592 ≤ 2 21 / 4305 92 ≈ 4 . 870 23 ≥ 1744414 ≤ 2 23 / 1744 414 ≈ 4 . 809 Although the b ounds are not necessarily tigh t, they clearly suggest a m onotone b ehavi or of the fractional c hr omatic num b ers of these graph s . The computer program which found most of these sets and a p rogram that ve r ifi es an indep en- den t set (of a sp ecific form) can b e found at www.math .tau.ac. il/ ~ amitw/br oadcasti ng . A.4 Coloring t he confusion graph of C 7 with 7 colors It was pro ved in [ 4 ] that the in dex co ding for any complement of an o dd cycle is p recisely 3, h o we ver, the minimum n umb er of co dewords can v ary b et we en 5 an d 8. Here w e sho w a legal coloring using 18 7 colo rs for n = 7, whic h wa s found using a co mp uter program. Eac h cell in the follo wing table represent a v ertex out of the 128 v ertices in the graph wh ic h are { 0 , 1 , . . . 127 } . The v ertex is the sum of its tw o indices in the table, e.g. the b olded vertex in the table is 16 + 4 whic h is colored with the seven th color. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 2 7 6 5 3 4 7 2 4 1 5 7 16 2 7 4 3 7 5 1 4 4 3 2 1 5 6 7 2 32 3 4 7 2 4 1 5 7 1 2 3 4 2 7 6 5 48 4 3 2 1 5 6 7 2 2 7 4 3 7 5 1 4 64 5 7 1 6 3 5 2 4 7 6 5 1 6 3 4 2 80 1 5 6 7 2 4 3 6 6 1 7 5 4 2 5 3 96 7 6 5 1 6 3 4 2 5 7 1 6 3 5 2 4 112 6 1 7 5 4 2 5 3 1 5 6 7 2 4 3 6 A computer pr ogram that v erifies this is ind eed a legal coloring can also b e found at www.math .tau.ac. il/ ~ amitw/br oadcasti ng . 19