On the Sum Capacity of A Class of Cyclically Symmetric Deterministic Interference Channels

On the Sum Ca pacity of A Class o f Cyclically Symmetric Deterministic Interfer ence Channels Bernd Bandemer , Gonzalo V azquez -V ilar , and Abbas El Gamal Stanford University , Inform ation Systems Laborato ry 350 Serra Mall, Stanford , CA 943 05, USA email: {bandemer, gvazquez}@stanford.ed u, abbas@ee.stanfo rd.edu Abstract — Certain deterministic interference channels hav e been shown t o accurately model Gaussian interference channels in the asymptotic low-noise regime. Motivated by this corre- spondence, we inv estigate a K user-pair , cyclically symmetric, deterministic interference channel in wh ich each receiv er experi- ences interference only from its neighboring t ransmitters (W yner model). W e establish the sum capacity for a large set of chann el parameters, thu s generalizing previous results for the 2-pair case. I . I N T R O D U C T I O N The Gaussian in terference cha nnel (G-I C) is o ne o f th e most im portant and practically relev ant mod els in multiple user informa tion theory . Altho ugh the cap acity region of this channel is no t known in gen eral, significant prog ress has been made recently toward finding capacity und er weak interferen ce [1 ]–[3] and bou nds th at are provably close to capacity [4]–[ 6]. I n [4], the capacity region for the two user- pair G-IC is established to within on e bit using new ou ter bound s a nd a simplified Han- K obayashi achiev ability sch eme. The same asymp totic result is d eriv ed in [ 5] by making a correspo ndence betwe en the G-IC in low-noise regime and a c lass of deterministic, finite-field inter ference cha nnels [7 ]. Some p rogress toward generalizing this result to mo re than two user-pairs has be en mad e in [6], where the solu tion is found for the fully symmetric case. Motiv ated by the se r ecent results, we consider a class of K user-pair ( K ≥ 3 ), cyclically symmetric, deterministic, finite- field interfere nce channels in which each receiver experiences interferen ce only from its two nearest neighb ors as in th e W yner m odel [8]. W e determin e the sum cap acity o f this channel for a wide range of in terference p arameters. Because of sy mmetry in t he c hannel and in the data rates, it su ffices to con sider the K = 3 c ase depicted in Fig. 1. W e f ocus our discussion on this case wh ile keep ing in mind that all results generalize immediately to the K user -pair W yner case. I I . C H A N N E L D E FI N I T I O N Let F 2 denote the binary finite field and let I be the identity ma trix. The zeropad ding o perator Z ∈ F 2 N × N 2 is W e gratefully acknowl edge the contrib ution of Dr . A ydin Sezgin, who origi- nally suggested the channe l that we consider in this work. Bernd Bandemer is supported by an Eric and Ille ana Benhamou Stanford Graduate Fello wship and US Army grant W911NF-07-2-0027-1. Gonzal o V azquez-V ilar is supported by Fundaci on Pedro-Barrie de la Maza Graduate Scholarship. P S f r a g r e p l a c e m e n t s M 1 → X 1 Y 1 → ˆ M 1 W 1 V 1 V 1 M 2 → X 2 Y 2 → ˆ M 2 W 2 V 2 M 3 → X 3 Y 3 → ˆ M 3 W 3 W 3 V 3 α α α β β β Fig. 1. Cyclic ally symmetric deterministic interferen ce channel, K = 3 . defined as Z = [ 0 N × N , I N ] T . F urther, let U , D ∈ F 2 N × 2 N 2 be the upshift an d downshift matrix , respectively , such that U [ x 1 , x 2 , . . . , x 2 N − 1 , x 2 N ] T = [ x 2 , x 3 , . . . , x 2 N , 0] T and D [ x 1 , x 2 , . . . , x 2 N − 1 , x 2 N ] T = [0 , x 1 , . . . , x 2 N − 2 , x 2 N − 1 ] T . W e also use the standar d notation A n = ( A 1 , A 2 , . . . , A n ) . W e refe r to a K user-pair interferen ce ch annel as cyclica lly symmetric if the cha nnel is in variant to cyclic relabeling o f the pairs, i.e., renaming i as i + 1 for i < K , and K as 1 . W e inves tigate the class o f cyclically symmetric, d etermin- istic, fin ite-field interference channels with K = 3 user-pairs depicted in Fig. 1. Th e channel is stationary and mem ory- less across multip le chann el uses. The channel inpu ts ar e X 1 , X 2 , X 3 ∈ F N 2 and its o utputs are Y 1 , Y 2 , Y 3 ∈ F 2 N 2 , wh ere N is the nu mber of inp ut b it pipes at eac h sen der . The outputs of the channe l are given b y Y 1 = Z X 1 + V 2 + W 3 , Y 2 = Z X 2 + V 3 + W 1 , Y 3 = Z X 3 + V 1 + W 2 , where + is the mod ulo-2 add ition oper ator , and V k = U ( α − 1) N Z X k , W k = D (1 − β ) N Z X k for ev ery k . The chan nel is param eterized b y the trip le ( N , α, β ) , which we constrain to α ∈ [1 , 2] , β ∈ [0 , 1 ] , and αN , β N ∈ Z . The parameters α an d β chara cterize the amo unt of u p/down- shift on the cross links and thus loosely cor respond to ch annel gains. Since by the definition of o ur c hannel, for each user- pair ther e is always exactly one interfe rer being up-shifted and o ne being down-shifted, our chann el is a special case of the class of cyclically symmetric, determ inistic, finite field, W yner connected ch annels. No te th at the up -shifted V k retains the complete information of X k , while the down-shifted W k incurs clipping at the low end of the vector . T ransmitter k ∈ { 1 , 2 , 3 } wishes to conv ey an independe nt message M k at data rate R k to its cor respondin g re cei ver . W e define a (2 nR 1 , 2 nR 2 , 2 nR 3 , n ) c ode, pro bability of error, and achiev ability of a given rate triple ( R 1 , R 2 , R 3 ) in the stan dard way [ 9]. The capacity region C of th e ch annel is the closure of the set of all achiev able rate triples. Define the sum ca pacity as R Σ : = sup { R 1 + R 2 + R 3 | ( R 1 , R 2 , R 3 ) ∈ C } and the symmetric capacity as R sym : = sup { R | ( R , R , R ) ∈ C } . By symmetry of th e ch annel an d co n vexity of the capacity region, R Σ = 3 R sym . Furtherm ore, defin e th e symmetric generalized de gr ees of fr eed om d sym : = R sym / N , i.e., the symmetric capacity n ormalized with re spect to the in terference- free c ase. Before we state our main result, define the function V ( x ) : = 1+ | x − 1 | 2 = ( x 2 if x ≥ 1 1 − x 2 if x < 1 . Remark 1: T his definition is usef ul in the context of de- terministic fin ite-field inter ference chann els. For example, the symmetric gener alized degrees of freedom of the two u ser-pair symmetric deterministic in terference chan nel with param eters ( N , α ) , with α ∈ [0 , ∞ ) , ar e k nown [4], [5] to be d sym = min  1 , V ( α ) , V (2 α )  . I I I . M A I N R E S U LT Our main result establishes the sy mmetric gen eralized d e- grees of fr eedom f or the class o f in terference ch annels d efined above for a large set o f ( α, β ) parameters. Theor em 1: Th e sy mmetric generalized degrees of free- dom of the class of three user-pair , cyclically symmetric, deterministic, fin ite field interfer ence channel with pa rameters ( α, β ) ∈ [1 , 2] × [0 , 1] , where α ≥ 2 β or α ≥ β 2 + 1 , is d sym = min  1 , V ( α ) , V ( β ) , V (2 β ) , V ( α − β )  . Fig. 2 depicts our r esult. The c laimed d sym is piecewise linear in ( α, β ) , and th e figure shows the line ar regions in the parameter plane with their respective min imum an d ma ximum values o f d sym . Some of the linear pieces are subd i vided (for example, “Ea” and “Eb”) to indicate that different ach ie vability schemes are needed ev en within a sing le linea r pie ce (see Section V). Remark 2: T he theorem implies th at d sym is independen t of N . For fixed α and β , all v alid values of N (satisfying αN , β N ∈ Z ) y ield the same d sym . I V . C O N V E R S E P RO O F The upp er b ounds 1 , V ( α ) , V ( β ) , an d V (2 β ) follow in a straightfor ward way from the known degree of freed om result of the two user -pair case [4], [10]. This can be sho wn by gi v ing the c omplete sign al X n k of one o f th e inter ferers as gen ie P S f r a g r e p l a c e m e n t s α β 0 0 . 5 1 1 1 . 5 2 Aa Ab Ba Bb Bc Bd Be Bf Bg C a C b Da Db Dc D d D e Df Ea Eb Ec Ed Ee X 3 / 5 2 / 3 2 / 3 1 Legend / Boundary between linea r pieces Local minimum ( d sym = 1 / 2 ) Local maximum (with d sym v alue) d s y m i s u n k n o w n Boundary between achie ving schemes Fig. 2. Il lustrati on of Theorem 1 in the ( α, β ) parameter plane. The result applie s ev erywhere except in region “X”. P S f r a g r e p l a c e m e n t s Z X 1 W 3 V 2 W 2 W 2 αN β N ( α − β ) N N (a) Interferers do not ov erlap. P S f r a g r e p l a c e m e n t s Z X 1 W 3 V 2 T 3 T 3 W 2 W 2 (1 − ( α − β )) N (b) Interferer s ov erlap. Fig. 3. Components of recei ve d s ignal Y 1 for the con ve rse proof. The component s are shown side way s, with the bottom pipe on the left. The dotted vert ical line symbolizes the “noise le vel ”, i.e., the lower end of the vector where further down-sh ifts cause loss of information. The recei ve d signal Y 1 is the modulo- 2 sum of the three components. informa tion to the r eceiv ers, thus effecti vely d egenerating the three user-pair case to the two-pair case. Hence we fo cus on pr oving the bo und V ( α − β ) b y gen - eralizing the meth ods introdu ced in [ 10] to the case at hand . First note tha t Fano’ s inequality implies (with som e abuse of notation for brevity) for every k nR k ≤ I ( X n k ; Y n k ) . A. W ithout overlap b etween interfer ers First consider α − β ≥ 1 , wh ich c orrespon ds to the first line in the d efinition of V ( α − β ) . In this case, the two interfe ring signals do no t overlap within the rece i ved signal, as sh own in Fig. 3 ( a). For example, at receiver 1, the sparsity pattern s of V 2 and W 3 are disjoint. W e can write I ( X n 1 ; Y n 1 ) ( a ) = I ( X n 1 ; Y n 1 W n 2 ) = I ( X n 1 ; W n 2 ) + I ( X n 1 ; Y n 1 | W n 2 ) ( b ) = H ( Y n 1 | W n 2 ) − H ( Y n 1 | X n 1 , W n 2 ) , where (a) is with equality since W 2 is not inte rfered with in Y 1 , and (b) uses the in depend ence between X 1 and W 2 . Now consider the last term. H ( Y n 1 | X n 1 , W n 2 ) = H ( Z X n 1 + V n 2 + W n 3 | X n 1 , W n 2 ) = H ( V n 2 + W n 3 | W n 2 ) ( a ) = H ( W n 3 ) + H ( V n 2 | W n 2 ) = H ( W n 3 ) + H ( W n 2 | W n 2 ) , where ( a) f ollows from the fact th at V 2 and W 3 do not overlap and different transmitters’ signals are independ ent, and W 2 is the p art of X 2 that is not co ntained in W 2 (see Fig. 3 (a )). W e conclud e that I ( X n 1 ; Y n 1 ) = H ( Y n 1 | W n 2 ) − H ( W n 3 ) − H ( W n 2 | W n 2 ) . Writing an analogous equation for I ( X n 2 ; Y n 2 ) and I ( X n 3 ; Y n 3 ) , and adding all three of them, we arrive at n P k R k ≤ H ( Y n 1 | W n 2 ) + H ( Y n 2 | W n 3 ) + H ( Y n 3 | W n 1 ) − H ( W n 1 ) − H ( W n 1 | W n 1 ) − H ( W n 2 ) − H ( W n 2 | W n 2 ) − H ( W n 3 ) − H ( W n 3 | W n 3 ) = H ( Y n 1 | W n 2 ) + H ( Y n 2 | W n 3 ) + H ( Y n 3 | W n 1 ) − H ( X n 1 ) − H ( X n 2 ) − H ( X n 3 ) Considering that nR k ≤ H ( X n k ) , we conclud e that 2 n P k R k ≤ H ( Y n 1 | W n 2 ) + H ( Y n 2 | W n 3 ) + H ( Y n 3 | W n 1 ) ≤ nH ( Y 1 | W 2 ) + nH ( Y 2 | W 3 ) + nH ( Y 3 | W 1 ) , where single-letterizatio n is p erformed by using the chain rule and omittin g part of th e condition ing. Th e right han d side of the last equation is maximized b y letting each input b it pip e be independ ent Bern(1 / 2 ) . T hus 2 P k R k ≤ 3 N ( α − β ) , and finally , d sym = R sym N ≤ α − β 2 . B. W ith overlap b etween interfer ers Now consider the case where α − β < 1 , i.e., the two interfering signals at each receiver overlap in signal space, see Fig. 3 (b). Define the top (1 − ( α − β )) N part o f X k as T k . W e will a ugment th e gen ie info rmation W n 2 of th e pr e vious subsection by T n 3 . This is exactly the part of the X 3 -based interferen ce that overlaps with the X 2 -based interferen ce. Similar to the previous section , we c onclude I ( X n 1 ; Y n 1 ) = I ( X n 1 ; Y n 1 , W n 2 , T n 3 ) = I ( X n 1 ; W n 2 , T n 3 ) + I ( X n 1 ; Y n 1 | W n 2 , T n 3 ) = H ( Y n 1 | W n 2 , T n 3 ) − H ( Y n 1 | X n 1 , W n 2 , T n 3 ) , The last term becomes H ( Y n 1 | X n 1 , W n 2 , T n 3 ) = H ( Z X 1 + V n 2 + W n 3 | X n 1 , W n 2 , T n 3 ) = H ( V n 2 + W n 3 | W n 2 , T n 3 ) ( a ) = H ( T n 3 | T n 3 ) + H ( W n 2 | W n 2 ) , where T 3 denotes the part of W 3 that is not included in T 3 . Its size is N ( α − 1) . W e are allowed to separate the term s in (a) because the overlapping part is resolved b y T 3 . Again, repeating the same for all three rates, we arriv e at n P k R k ≤ H ( Y n 1 | W n 2 , T n 3 ) − H ( T n 1 | T n 1 ) − H ( W n 1 | W n 1 ) + H ( Y n 2 | W n 3 , T n 1 ) − H ( T n 2 | T n 2 ) − H ( W n 2 | W n 2 ) + H ( Y n 3 | W n 1 , T n 2 ) − H ( T n 3 | T n 3 ) − H ( W n 3 | W n 3 ) . Since T k and T k together for m W k , which tog ether with W k forms X k , we can write nR 1 ≤ H ( X n 1 ) = H ( T n 1 ) + H ( T n 1 | T n 1 ) + H ( W 1 | T n 1 , T n 1 | {z } = W n 1 ) Using this expr ession and its equiv alent for R 2 and R 3 with the previous inequa lity , we o btain 2 n P k R k ≤ H ( Y n 1 | W n 2 , T n 3 ) + H ( T n 1 ) + H ( Y n 2 | W n 3 , T n 1 ) + H ( T n 2 ) + H ( Y n 3 | W n 1 , T n 2 ) + H ( T n 3 ) ≤ n  H ( Y 1 | W 2 , T 3 ) + H ( T 1 ) + H ( Y 2 | W 3 , T 1 ) + H ( T 2 ) + H ( Y 3 | W 1 , T 2 ) + H ( T 3 )  . Again, the r ight h and side is max imized b y cho osing all X k compon ents indepen dently acco rding to B ern(1 / 2) , yielding 2 P k R k ≤ 3 N + 3 N (1 − ( α − β )) , d sym ≤ 1 − α − β 2 , which matches the definition of V ( α − β ) for α − β < 1 . V . A C H I E V A B I L I T Y P RO O F The set of interest { ( α, β ) } is divided into regions “ Aa” to “Df ” as shown in Fig. 2. In eac h region, we use th e following coding scheme. For every send er k , we set X k = G D k , where G ∈ F N × d sym N 2 is the a ssignment ma trix , and D k ∈ F d sym N 2 is a vector of i.i. d. Bern(1 / 2) message bits. W e constrain th e coding scheme in sev eral ways, namely , (a) the re is no coding acr oss mu ltiple chan nel uses, (b) a ll transmitters use th e same G , and (c ) the pro posed G matrices will hav e at most one non-zero element per row , i.e., each pipe in X k is assigned either an informatio n bit or a zero. While these assumptions may seem overly r estrictiv e, th ey ar e sufficient for ou r p urposes. In deed, it is surprising that such a constrained set o f codes is able to meet the upp er boun d of Section IV. Remark 3: I f the number of input pipes N is small, it can se verely limit our optio ns in terms of assignment matrices. The following argument can circum vent this p roblem by expanding a gi ven chann el to o ne with mor e input pipes. T o th is en d, co nsider L ≥ 2 subsequ ent cha nnel uses, with channel in puts X k, 1 , . . . , X k,L . By inter leaving these vector s into a supersymb ol e X k = P L l =1 ( I N ⊗ e l ) X k,l , and likewise for the outpu ts e Y k , it c an be shown that the r esulting c hannel { e X 1 , e X 2 , e X 3 } → { e Y 1 , e Y 2 , e Y 3 } is in fact ( LN , α, β ) as defined in Section II. (Here, ⊗ denotes the Kro necker prod uct, and e l is the l th column of I L .) Throug h this method , a ch annel with a given N can be expan ded to one with LN inpu t X k : Bottom T op 1 3 − δ ε − 1 2 δ 1 3 − δ − ε + 2 δ ε − 1 2 δ 1 3 − δ − ε + 2 δ Fig. 4. Proposed G assignment for region “Df ”, expressed in terms of the transmit vector . T he block lengths are gi ven as fractions of N , such that the sum of all block lengths is 1 . Z X 1 V 2 W 3 6 4 3 4 5 3 2 1 2 1 4 3 4 3 Fig. 5. Re cei ved signal Y 1 in “Df ”, at α = 1 . 6 , β = 0 . 9 , with d sym = 0 . 55 . Blocks in differe nt rows carry diffe rent data. pipes. N ote that d sym is unaffected by this tran sformation since it is nor malized by the num ber of p ipes. In light o f this tran sformation, w e assume f rom now on that N is (or has been mad e) large en ough such tha t any fraction of N that we incur correspo nds to an integer n umber of pipes. Optimal assignment ma trices G for all regions in Fig. 2 are listed in T able 1. An interactive online animatio n is a lso av ailab le at [1 1]. Each row in th e table co ntains the d efinition of a region in terms of affine constrain ts in ( α, β ) and a representatio n of G by mean s o f the resulting transmit vector X k . In the following we discuss the d etails fo r on e particu lar example, wh ich is representative fo r all other c ases. Example (Region “Df ”): T his region is p arameterized by ( α, β ) = (4 / 3 + ε, 2 / 3 + δ ) with ε ≤ 2 δ, ε ≥ 1 2 δ, δ ≤ 1 3 . Fig. 4, co pied from T able 1, represents an optimal assignment G by means of the resultin g transmit vector . The vector X k is subdivided into data blocks (hatched) that correspond to non-ze ro rows of G , and zer o blocks (gray) that corresp ond to all-zero r ows o f G . Some data blo cks oc cur twice. W e d enote such block pairs as twins . T wins carry the same data bits, albeit in reverse order as discussed later . The length of each b lock as a fraction of N is a nnotated in the figure. T o prove achie v ability of Theorem 1, we require the transmit vector to be both va lid and decoda ble . By valid we mean (a) all block len gths are no n-negative fo r the ran ge of ( ε, δ ) that constitute the region, (b) the sum of the block lengths is 1 , and (c ) adding the sizes of a ll data bloc ks, counting twins only once, results in the desired d sym as claimed in Theore m 1 ( 2 / 3 − δ/ 2 in o ur example). By decodab le, we mean tha t using this tr ansmit vector assignmen t, the receiver can r ecover all desired data blocks from the received sig nal. T o verify d ecodability , consider Fig. 5, which u ses th e same conv entions as Fig. 3. The receiver sees the sum o f data blocks from dif ferent tran smitters, e ach characterized by its length and shift location . Different b locks may or may no t overlap. Decoding is per formed seq uentially , b lock by block. In each step, o ne o f thr ee rules is a pplied in order to decode a dditional data block s, wh ich are then r emoved fro m the received signal. The three decoding rules are as follows. 1. Dir ect readout: Consider the situation in Fig. 6 (b) . If a d ata block (i) d oes not overlap with any o ther data block and (ii) is located above the no ise level, the n its data conten t can be read out directly from the rec ei ved signal. 1 A blo ck that has been read out is then rem oved from the r eceiv ed signal. I f the b lock has a twin, it is removed as well. 2. Overlapping twins scenario (A): Consider Fig. 6 ( c). If two twin pairs exist such that (i) they have the same block len gth, b 1 = b 2 , (ii) they h av e the same separation, s 1 = s 2 , (iii) the relative shift between th e pair s is less than the separatio n, c < s 1 , and (iv) the dashed sections of ( A) in Fig. 6 (c) do not overlap with any othe r data blo ck and are a bove the noise floor , th en both twin pairs can be decoded and canceled from th e r eceiv ed signal. 2 T o see this, consider th e f ollowing suc cessi ve decodin g argu ment [6]. Let the two copies with in a twin be in r ev erse o rder of each o ther . First, th e leftmost part of the left blue twin is read o ut. Its data reapp ears on the right side of the right blue twin, thus revealing a chu nk of data on the rig ht side of th e righ t yellow twin. This data in tu rn is rep licated o n th e left side of the le ft yellow twin, whic h exposes a new part o f the left blue twin. The process repeats until both twins ar e completely decoded . 3. Overlapping twins scenario (B): This rule is a variant of th e previous o ne, where p attern (B) replaces pattern (A) in Fig. 6 (c). In o ur examp le, the sequence of steps that com pletely decodes X 1 is annotated in Fig. 5: First, block 1 is d ecoded via direct readou t (rule 1). Th e now-known data block and its twin are removed from the rec ei ved signal Y 1 . The same rule allows block 2 to be d ecoded, which is then removed from Y 1 . Each removal step makes mor e r oom for subsequen t r ule applications. Next, r ule 2 is applied to the two pair s of twins 3. Continu ing in the same fashion, the rem oval o f blocks 1, 2 1 It is crucial that both (i) and (ii) hold for all ( α, β ) in the regi on, since the length and location of the blocks in Fig. 5 change when α and β vary . 2 Again, conditions (i)–(i v) must hold for all ( α, β ) in the regio n. P S f r a g r e p l a c e m e n t s Legend no other data blocks allo wed other data blocks allo wed (a) Legend. (b) Direct readout. P S f r a g r e p l a c e m e n t s (A) (B) s 1 b 1 b 1 s 2 b 2 b 2 c (c) Overlappi ng twins. Fig. 6. Rules for verifyin g decodabilit y . Legend (a) applies to “direct readout ”, shown in (b), and two va riants of “ov erla pping twins”, shown in (c). and 3 enab les th e two twin pairs 4 to be deco ded using rule 3. Finally , da ta blocks 5 and 6 can be recovered by direct r eadout (rule 1 ), wh ich co mpletes th e d ecoding p rocess. By symm etry , the sign als at th e other two r eceiv ers can be similar ly dec oded. 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A v ailabl e: http:/ /www .stanford.edu/~ban demer/detic/isit09/ ( α, β ) Regi on constraint s d sym Assignment (shown graphica lly and as correspondi ng list of block lengths) Aa (2 + ε, δ ≥ 0 , δ ≤ 1 + ε, 1 + 1 2 ε − 1 2 δ δ ) ε ≤ − δ ( δ | − 1 2 ε − 1 2 δ | 1 + ε | − 1 2 ε − 1 2 δ ) Ab ( 2 + ε , ε ≤ 0 , δ ≤ 1 2 , 1 − δ δ ) ε ≥ − δ ( δ | 1 − δ ) Ba (6 / 5 + ε, ε ≥ 3 δ, ε ≤ 1 5 + δ, 3 5 − 1 2 ε + 1 2 δ 2 / 5 + δ ) ε ≥ 1 10 − 1 2 δ ( 1 5 − ε + δ | 1 5 − 1 2 ε − 1 2 δ | 1 5 − 1 2 ε − 1 2 δ | − 1 5 + 2 ε + δ | 1 5 − 1 2 ε − 1 2 δ | 1 5 − 1 2 ε − 1 2 δ | 1 5 + ε ) Bb (6 / 5 + ε, ε ≥ − 1 3 δ, ε ≤ 1 5 + δ, 3 5 − 1 2 ε + 1 2 δ 2 / 5 + δ ) ε ≤ 1 10 − 1 2 δ, ε ≥ 3 δ ( 1 5 − ε + δ | 1 5 − 1 2 ε − 1 2 δ | 3 2 ε + 1 2 δ | 1 5 − 2 ε − δ | 3 2 ε + 1 2 δ | 1 5 − 1 2 ε − 1 2 δ | 1 5 + ε ) Bc (6 / 5 + ε, ε ≥ 1 2 δ, ε ≤ 1 5 + δ, 3 5 − 1 2 ε + 1 2 δ 2 / 5 + δ ) ε ≤ − 1 3 δ ( 1 5 + 1 2 δ | 1 5 + ε | − 3 2 ε − 1 2 δ | 1 5 + ε | − 3 2 ε − 1 2 δ | 1 5 + ε | 1 5 + 1 2 δ ) Bd (6 / 5 + ε, ε ≤ 1 2 δ, ε ≤ − 2 δ, 3 5 + 1 2 ε 2 / 5 + δ ) ε ≥ − 1 5 ( − ε + 1 2 δ | 1 5 + ε | − ε + 1 2 δ | 1 5 + ε | − 1 2 ε − δ | 1 5 + ε | − 1 2 ε − δ | 1 5 + ε | − ε + 1 2 δ | 1 5 + ε | − ε + 1 2 δ ) Be (6 / 5 + ε, ε ≤ 1 2 δ, ε ≥ − 2 δ, 3 5 − δ 2 / 5 + δ ) δ ≤ 1 10 ( − ε + 1 2 δ | 1 5 + ε | − ε + 1 2 δ | 1 5 + ε | 1 5 − 2 δ | 1 5 + ε | − ε + 1 2 δ | 1 5 + ε | − ε + 1 2 δ ) Bf (6 / 5 + ε , ε ≥ 1 2 δ, ε ≤ 3 δ, 3 5 − δ 2 / 5 + δ ) ε ≤ 1 10 − 1 2 δ ( 1 5 − ε + δ | 1 5 − ε + δ | 2 ε − δ | 1 5 − 2 ε − δ | 2 ε − δ | 1 5 − ε + δ | 1 5 + ε ) Bg (6 / 5 + ε, ε ≤ 3 δ, δ ≤ 1 10 , 3 5 − δ 2 / 5 + δ ) ε ≥ 1 10 − 1 2 δ ( 1 5 − ε + δ | 1 5 − ε + δ | 1 5 − 2 δ | − 1 5 + 2 ε + δ | 1 5 − 2 δ | 1 5 − ε + δ | 1 5 + ε ) Da (4 / 3 + ε, ε ≥ 2 δ, ε ≥ − δ, 2 3 − 1 2 ε + 1 2 δ 2 / 3 + δ ) ε ≤ 1 3 + δ ( 1 3 − δ | 1 2 ε + 1 2 δ | 1 3 − δ | 1 2 ε + 1 2 δ | 1 3 − ε + δ ) Db ( 4 / 3 + ε, ε ≤ − δ, ε ≥ 1 2 δ, 2 3 + δ 2 / 3 + δ ) δ ≥ − 1 6 ( 1 3 + 1 2 δ | 1 3 − δ | 1 3 + 1 2 δ ) Dc (4 / 3 + ε, ε ≤ 1 2 δ, ε ≥ 2 δ, 2 3 + δ 2 / 3 + δ ) δ ≥ − 1 6 ( − ε + 1 2 δ | 1 3 + ε | − ε + 1 2 δ | 1 3 + 2 ε − 2 δ | − ε + 1 2 δ | 1 3 + ε | − ε + 1 2 δ ) Df (4 / 3 + ε, ε ≤ 2 δ, ε ≥ 1 2 δ, 2 3 − 1 2 δ 2 / 3 + δ ) δ ≤ 1 3 ( 1 3 − δ | ε − 1 2 δ | 1 3 − δ | − ε + 2 δ | ε − 1 2 δ | 1 3 − δ | − ε + 2 δ ) Ea (2 + ε, ε ≤ 0 , δ ≥ − 1 15 , 2 3 + δ 2 / 3 + δ ) ε ≥ 3 δ ( − 3 δ | 1 3 + 2 δ | − 3 δ | 1 3 + 5 δ | − 3 δ | 1 3 + 2 δ ) Eb (2 + ε, ε ≤ 0 , δ ≤ − 1 15 , 2 3 + δ 2 / 3 + δ ) δ ≥ − 1 6 , ε ≥ 3 δ ( − 3 δ | 1 3 + 2 δ | 1 3 + 2 δ | − 1 3 − 5 δ | 1 3 + 2 δ | 1 3 + 2 δ ) Ec (2 + ε, ε ≤ 3 δ, ε ≥ − 1 3 + δ, 2 3 + 1 2 ε − 1 2 δ 2 / 3 + δ ) ε ≤ − 1 3 − 2 δ ( − 1 2 ε − 3 2 δ | 1 3 + 1 2 ε + 1 2 δ | 1 3 + 1 2 ε + 1 2 δ | − 1 3 − ε − 2 δ | 1 3 + 1 2 ε + 1 2 δ | 1 3 + 1 2 ε + 1 2 δ | − 1 2 ε + 3 2 δ ) Ed (2 + ε, ε ≤ 3 δ, ε ≥ − 1 3 − 2 δ, 2 3 + 1 2 ε − 1 2 δ 2 / 3 + δ ) ε ≥ − 1 3 + δ, ε ≤ − 3 δ ( − 1 2 ε − 3 2 δ | 1 3 + 1 2 ε + 1 2 δ | − 1 2 ε − 3 2 δ | 1 3 + ε +2 δ | − 1 2 ε − 3 2 δ | 1 3 + 1 2 ε + 1 2 δ | − 1 2 ε + 3 2 δ ) Ee (2 + ε, ε ≤ 0 , δ ≤ 1 3 + ε, 2 3 + 1 2 ε − 1 2 δ 2 / 3 + δ ) δ ≥ − 1 3 ε ( 1 3 − δ | 1 3 − δ | 1 2 ε + 3 2 δ | 1 3 − δ | 1 2 ε + 3 2 δ | − ε ) T able 1. Assignments that achie ve d sym as stated in Theorem 1.