The Compound Capacity of Polar Codes

1 The Compound Capaci ty of Polar Codes S. Hamed Hass ani, Satish Babu K orada and R ¨ udiger Urbanke Abstract — W e consider the compound capacity of p olar codes under successive cancellation deco ding for a collection of binary- input memoryless output-symmetric channels. By deriv ing a sequence of upper and lower boun ds, we show that i n general the compound capacity un der successiv e decoding is strictly smaller than th e unrestricted compound capacity . I . H I S T O RY A N D M OT I V AT I O N Polar cod es, recently introduced by Arık an [ 1], are a family of codes that achiev e the c apacity of a large class of channels using low-complexity en coding and decod ing algo rithms. The complexity of these algorithm s scales as O ( N log N ) , whe re N is th e blo cklength of the co de. Recently , it h as been shown that, in addition to being capac ity-achieving for channel coding, polar-like codes are also optimal for lossy source coding as well as multi-termin al pro blems like the W yner-Zi v and the Gelfand-Pinsker problem [2]. Polar c odes a re clo sely related to Reed-Muller (RM) cod es. The ro ws of the generator matrix of a polar code of length N = 2 n are c hosen from th e rows of th e matrix G ⊗ n =  1 0 1 1  ⊗ n , where ⊗ denotes the Kronecker product. The crucial difference of polar cod es to RM codes is in the choice o f the rows. For RM codes the rows of largest weight ar e cho sen, where as for polar codes the cho ice is dependen t on the chann el. W e refer the reader to [1] for a detailed d iscussion o n the construction of polar codes. Th e d ecoding is d one u sing a su ccessi ve cancellation (SC) decoder . This algorithm dec odes the b its one-by -one in a pre-chosen order . Consider a communication scenario where the transmitter and the receiver do not know the chann el. The only kn owledge they have is th e set of ch annels to which the ch annel belongs. This is kn own as the co mpoun d channel scen ario. Let W denote th e set of channels. The compou nd capacity of W is d efined as the rate at which we can reliably transmit irrespective of the particular chann el (out of W ) that is ch osen. The compo und cap acity is given by [3] C ( W ) = max P inf W ∈W I P ( W ) , where I P ( W ) denotes the mutual information between the input and th e outpu t of W , with the inpu t distrib ution being P . Note th at the compoun d capacity of W can be strictly smaller than the infimu m of the individual capacities. Th is h appens if th e capacity- achieving input d istribution f or the individual channels ar e different. On the other hand , if the cap acity- achieving inpu t distribution is the same fo r all cha nnels in W , th en the com pound capacity is equal to the in fimum of the individual cap acities. T his is indee d the case sin ce we EPFL, School of Computer & Communicati on Science s , Lausanne, CH-1015, Switzerla nd, { seyedha med.hassani, satish.k orada, rudiger .urbank e } @epfl.ch. restrict our attention to the class of binar y-inpu t m emoryless output- symmetric (BMS) cha nnels. W e ar e interested in the maxim um ach iev able rate using polar codes and SC decodin g. W e refer to this as the compou nd capacity using p olar codes and d enote it as C P , SC ( W ) . More precisely , given a collection W of BMS channels we are interested in constructing a po lar code of rate R wh ich works well (under SC decoding ) fo r e very chann el in this co llection. This means, gi ven a target block erro r pr obability , call it P B , we ask whether there exists a po lar code of ra te R such th at its b lock error proba bility is at most P B for any ch annel in W . In particular, how large can we make R so that a c onstruction exists for any P B > 0 ? W e consider the compou nd capacity with respec t to igno - rance at the transmitter but we allow the d ecoder to have knowledge of the actual channel. I I . B A S I C P O L A R C O D E C O N S T RU C T I O N S Rather than describ ing the stan dard construction of polar codes, let us give here an alternative but entirely equiv ale nt formu lation. For the standard v iew we r efer the read er to [1]. Binary p olar co des have leng th N = 2 n , w here n is an integer . Un der successiv e d ecoding, there is a BMS channel associated to each bit U i giv en the o bservation vector Y N − 1 0 as well as the values of the previous bits U i − 1 0 . This cha nnel has a fairly simple description in term s of the und erlying BMS channel W . 1 Definition 1 (T r ee Channe ls of Heig ht n ): Consider the following N = 2 n tree chann els o f height n . Let σ 1 . . . σ n be the n -bit binary expansion o f i . E.g., we have for n = 3 , 0 = 000 , 1 = 0 01 , . . . , 7 = 11 1 . Let σ = σ 1 σ 2 . . . σ n . Note that for our purpo se it is sligh tly more convenient to denote the least (most) significant bit as σ n ( σ 1 ). Each tree chan nel consists of n + 1 levels, n amely 0 , . . . , n . It is a c omplete binary tree. The r oot is at level n . At lev el j we h av e 2 n − j nodes. For 1 ≤ j ≤ n , if σ j = 0 then all no des on level j are ch eck no des; if σ j = 1 th en all no des on level j ar e variable nod es. All nodes a t level 0 co rrespond to ind ependen t observations of th e ou tput of the ch annel W , assuming that the input is 0 . An examp le for W 011 (that is n = 3 and σ = 011 ) is sh own in Figure 1. Let us call σ = σ 1 . . . σ n the type of the tr ee. W e have σ ∈ { 0 , 1 } n . Let W σ be the chann el associated to the tre e of type σ . T hen I ( W σ ) denotes the cor respondin g ca pacity . Further, by Z ( W σ ) we mean the corr esponding Bhattacharyya function al (see [4, Chap ter 4]). 1 W e note that in order to arriv e at this descript ion we crucial ly use the fac t that W is symmetric. T his allo ws us to assume that U i − 1 0 is the all-zero vec tor . 2 W 011 W W W W W W W W Fig. 1. T ree represent ation of the channel W 011 . The 3 -bit binary expansio n of 3 is σ 1 σ 2 σ 3 = 011 . Consider the channels W ( i ) N introdu ced by Arık an in [1 ]. The ch annel W ( i ) N has in put U i and o utput ( Y N − 1 0 , U i − 1 0 ) . W ithou t proof we note that W ( i ) N is equ i valent to the channel W σ introdu ced above if we let σ be the n -bit binary expan sion of i . Giv e n the d escription of W σ in ter ms of a tree channel, it is clear that we can use density evolution [4] to co mpute the channel law o f W σ . Indeed , assuming that infinite- precision density ev olu tion has unit cost, it was shown in [5] th at the total cost of co mputing all channel laws is linear in N . When using density ev olutio n it is co n venien t to represent the channel in th e log -likelihood dom ain. W e refer the reader to [4 ] for a d etailed description o f density ev olu tion. The BMS W is r epresented a s a probability d istribution over R ∪ { ±∞} . The probability distribution is the distribution o f the variable log( W ( Y | 0) W ( Y | 1) ) , where Y ∼ W ( y | 0) . Density evolution starts at th e leaf no des which ar e the channel observations and proc eeds up the tree. W e ha ve two types of conv o lutions, namely th e variable conv olution (denoted by ⊛ ) and the ch eck conv o lution (d enoted by  ). All the den sities correspo nding to n odes which are at the same lev el are id entical. Each n ode in th e j -th le vel is con nected to two no des in the ( j − 1) -th level. Hence the conv o lution (depen ding on σ j ) of two id entical d ensities in th e ( j − 1) -th lev el yields the d ensity in the j -th lev el. If σ j = 0 , then we use a ch eck conv olution (  ) , and if σ j = 1 , then we use a variable conv olution ( ⊛ ). Example 2 (Density Evolu tion): Con sider the channel shown in Figu re 1. By som e abuse of n otation, let W also denote the in itial density co rrespon ding to th e chann el W . Recall that σ = 011 . Then the density correspon ding to W 011 (the root nod e) is g i ven by  ( W  2 ) ⊛ 2  ⊛ 2 = ( W  2 ) ⊛ 4 . ♦ I I I . M A I N R E S U LT S Consider two BMS chann els P and Q . W e are inte rested in co nstructing a comm on polar co de o f rate R (of arb itrarily large b lock length) wh ich allows reliab le transmission over both chann els. T r i vially , C P , SC ( P, Q ) ≤ min { I ( P ) , I ( Q ) } . (1) W e will see shortly that, prop erly applied, this simp le fact can be used to give tigh t b ound s. For th e lower bou nd we claim that C P , SC ( P, Q ) ≥ C P , SC ( BEC ( Z ( P )) , BEC ( Z ( Q ))) = 1 − max { Z ( P ) , Z ( Q ) } . (2) T o see th is claim , we p roceed as follows. Consider a particular computatio n tree of heigh t n with observations at its leaf nodes f rom a BMS channel with Battachary ya constant Z . What is the largest value that the Bhattacharyy a constant of the r oot n ode can take on? From the extremes of inform ation combinin g framework ([ 4, Chap ter 4 ]) we can d educe th at we get the largest value if we take the BMS ch annel to be the BEC ( Z ) . This is tr ue, sinc e at variable nodes the Bhattacharyy a constant acts multiplica ti vely for any channel, and at check nod es the worst in put distribution is known to be the on e from the family of BEC channe ls. Furth er , BEC densities stay preserved within the compu tation g raph. The above consideration s g iv e rise to the following trans- mission scheme. W e signa l on those chann els W σ which are reliable for the BEC (max { Z ( P ) , Z ( Q ) } ) . A f ortiori these channels are also re liable for the actua l in put distribution. In this way we can ach iev e a r eliable transmission at rate 1 − max { Z ( P ) , Z ( Q ) } . Example 3 (BSC and BE C): L et us apply the above men - tioned b ounds to C P , SC ( P, Q ) , wh ere P = BEC (0 . 5) an d Q = BSC (0 . 11002 ) . W e I ( P ) = I ( Q ) = 0 . 5 , Z ( BEC (0 . 5)) = 0 . 5 , Z ( BSC (0 . 1100 2)) = 2 p 0 . 1102 (1 − 0 . 110 02) ≈ 0 . 6258 . The up per bou nd (1) and the lower b ound (2) then translate to C P , SC ( P, Q )) ≤ min { 0 . 5 , 0 . 5 } = 0 . 5 , C P , SC ( P, Q )) ≥ 1 − max { 0 . 62 58 , 0 . 5 } = 0 . 3742 . Note that the upper bound is trivial, but th e lower bou nd is not. ♦ In som e special cases the best achiev able rate is ea sy to determine. This happens in p articular if the two chan nels are ordered by degradation . Example 4 (BSC and BE C Or dered by Degradation): Let P = BEC (0 . 220 04) and Q = BSC ( 0 . 1100 2) . W e ha ve I ( P ) = 0 . 770 098 and I ( Q ) = 0 . 5 . Further , one can check that the BSC (0 . 110 02) is degraded with respect to the BEC (0 . 2200 4) . This implies that any sub-chan nel of type σ which is good for the BSC (0 . 1100 2) , is also go od f or the BEC (0 . 22 004) . Hence , C P ,SC ( BEC (0 . 22004) , BSC (0 . 11002)) = I ( Q ) = 0 . 5 . ♦ More gene rally , if the c hannels W a re such th at there is a channe l W ∈ W which is degraded with r espect to ev ery channel in W , then C P ,SC ( W ) = C ( W ) = I ( W ) . M oreover , the sub-chan nels σ tha t a re goo d fo r W ar e good also for all channels in W . So far we have looked at seemingly trivial upper and lower bound s o n the compo und capacity of two c hannels. As we 3 will see now , it is quite simple to consider ably tighten the result by conside ring ind i vidual b ranches of the comp utation tree separately . Theor em 5 (Boun ds on P airwise Compound Rate): Let P and Q be two BMS chan nels. Then for any n ∈ N C P , SC ( P, Q ) ≤ 1 2 n X σ ∈{ 0 , 1 } n min { I ( P σ ) , I ( Q σ ) } , C P , SC ( P, Q ) ≥ 1 − 1 2 n X σ ∈{ 0 , 1 } n max { Z ( P σ ) , Z ( Q σ ) } . Further, the u pper as w ell as the lower boun ds conver ge to the compou nd capacity as n tends to infinity and the bound s a re monoto ne with re spect to n . Pr oof: Con sider all N = 2 n tree cha nnels. Note that there are 2 n − 1 such channels th at hav e σ 1 = 0 an d 2 n − 1 such channels th at have a σ 1 = 1 . Recall that σ 1 correspo nds to the type of nod e at le vel n . This level tra nsforms the orig inal channel P into P 0 and P 1 , respectively . Consider first the 2 n − 1 tree chan nels that correspo nd to σ 1 = 1 . I nstead of think ing of each tree as a tree of height n with obser vations f rom the channel P , think of each of them as a tree of height n − 1 with observations coming from the chann el P 1 . By apply ing our previous argum ent, we see that if we let n tend to infinity then the comm on capacity for this half of channels is at most 0 . 5 min { I ( P 1 ) , I ( Q 1 ) } . Clearly the same argument can be made f or the second half of channe ls. T his improves the trivial upper bound (1) to C P , SC ( P, Q ) ≤ 0 . 5 min { I ( P 1 ) , I ( Q 1 ) } + 0 . 5 min { I ( P 0 ) , I ( Q 0 ) } . Clearly the same argument can be ap plied to trees of any height n . This explain s the up per b ound on th e co mpoun d capacity of the for m min { I ( P σ ) , I ( Q σ ) } . In the same way we can app ly this argument to the lower bound (2). From the basic po larization ph enomen on we know tha t f or ev ery δ > 0 there exists an n ∈ N so that 1 2 n |{ σ ∈ { 0 , 1 } n : I ( P σ ) ∈ [ δ, 1 − δ ] }| ≤ δ / 4 . Equiv alent statemen ts hold for I ( Q σ ) , Z ( P σ ) , and Z ( Q σ ) . In words, except for at most a fra ction δ , all channel pair s ( P σ , Q σ ) have “p olarized. ” For each p olarized pair bo th the upper as well as the lower boun d are loo se by at most δ . Therefo re, th e gap be tween the up per and lower bou nd is at most (1 − δ )2 δ + δ . T o see that the boun ds are mon otone consid er a pa rticular type σ of len gth n . Then we have min { I ( P σ ) , I ( Q σ ) } = min { 1 2 ( I ( P σ 0 ) + I ( P σ 1 )) , 1 2 ( I ( Q σ 0 ) + I ( Q σ 1 )) } ≥ 1 2 min { I ( P σ 0 ) , I ( Q σ 0 ) } + 1 2 min { I ( P σ 1 ) , I ( Q σ 1 ) } . A similar argume nt applies to the lower bound . Remark: In ge neral there is no finite n so that either upper or lower boun d ag ree exactly with the com pound capacity . On the positive side, the lower bo unds are constructive and g iv e an actual strategy to construc t polar co des of this rate. Example 6 (Compo und Ra te of BSC ( δ ) an d BEC ( ǫ ) ): Let us compute upper and lo wer bounds on C P , SC ( BSC (0 . 1100 2 ) , BEC (0 . 5)) . Note that both the BSC (0 . 11002) as well as the BEC (0 . 5 ) ha ve capac ity one-ha lf. Applyin g th e bounds of The orem 5 we get: n=0 1 2 3 4 5 6 0.500 0.48 2 0.482 0 .482 0.482 0.482 0.48 2 0.374 0.40 7 0.427 0 .440 0.449 0.456 0.46 1 These results suggest that the nu merical v alue of C P , SC ( BSC (0 . 1100 2 ) , BEC (0 . 5)) is close to 0 . 48 2 . ♦ Example 7 (Bou nds o n Compoun d Rate of BMS Channels): In the p revious example we conside red the compou nd capacity of two BMS channels. H ow does the result ch ange if we consider a wh ole family of BMS chan nels. E.g., what is C P , SC ( { BMS ( I = 0 . 5 ) } ) ? W e currently do no t kn ow o f a pro cedure (even n umerical) to comp ute th is rate. But it is easy to giv e some upp er and lower bo unds. In particular we have C P , SC ( { BMS ( I = 0 . 5 ) } ) ≤ C ( BSC (0 . 1 1002 ) , BEC (0 . 5)) ≤ 0 . 4817 , C P , SC ( { BMS ( I = 0 . 5 ) } ) ≥ 1 − Z ( BSC ( I = 0 . 5)) ≈ 0 . 3 74 . (3) The upper bound is tri vial. The compo und rate of a whole class cannot be larger than the compou nd rate of two of its members. For the lower bound note that fro m Th eorem 5 we know that the ac hiev able rate is at least as large as 1 − max { Z } , where the maxim um is over all chan nels in the class. Since the BSC has the largest Bhattach aryya parameter of all channels in the class of channe ls with a fixed cap acity , the result follows. ♦ I V . A B E T T E R U N I V E R S A L L O W E R B O U N D The univ ersal lower boun d expressed in (3) is rather we ak. Let us therefo re show how to stren gthen it. Let W d enote a class of BMS channels. Fro m T heorem 5 we k now that in order to evaluate the lower boun d we h av e to optimize the term s Z ( P σ ) over the class W . T o be specific, let W be BMS ( I ) , i. e., the space o f BMS channels that have capacity I . Expr essed in an alternative w ay , this is the space o f distributions th at have entr opy eq ual to 1 − I . The ab ove op timization is in general a difficult pr oblem. The fir st difficulty is that the spac e { BMS ( I ) } is infinite dimensiona l. Thus, in order to use numerical procedures we h av e to ap proximate th is space by a finite dimension al space. Fortuna tely , as the space is compact, this ta sk can be accomplished . E.g., loo k at the den sities correspo nding to the class { BMS ( I ) } in the | D | -do main. In this dom ain, each BMS channel W is represented by the den sity cor respond ing to the probab ility distribution of | W ( Y | 0) − W ( Y | 1 ) | , where Y ∼ W ( y | 0 ) . For example, the | D | -d ensity cor respondin g to BSC ( ǫ ) is ∆ 1 − 2 ǫ . W e quan tize the in terval [0 , 1 ] using r eal values 0 = p 1 < p 2 < · · · < p m = 1 , m ∈ N . Th e m -d imensional p olytope 4 approx imation of { BMS ( I ) } , denote d by W m , is the space of all the de nsities which a re of the f orm P m i =1 α i ∆ p i . Let α = [ α 1 , · · · , α m ] ⊤ . Th en α must satisfy the fo llowing linear constraints: α ⊤ 1 m × 1 = 1 , α ⊤ H m × 1 = 1 − I , α i ≥ 0 , (4) where H m × 1 = [ h 2 ( 1 − p i 2 )] m × 1 and 1 m × 1 is the all-one vector . Due to quantizatio n, there is in g eneral an approx imation error . Lemma 8 ( m versus δ ) : Let a ∈ BMS ( I ) . Assume a uni- form quantization of the inter val [0 , 1] with m points 0 = p 1 < p 2 < · · · < p m = 1 . If m ≥ 1 + 1 1 − 4 √ 1 − δ 2 , then th ere exists a density b ∈ W m such that | Z ( a  a ) − Z ( b  b ) | ≤ δ . Pr oof: For a giv en density a , let Q u ( a )( Q d ( a )) de- note the qu antized density obtained by mapping the mass in the inter val ( p i , p i +1 ] ( [ p i , p i +1 ) ) to p i +1 ( p i ). No te that Q u ( a ) ( Q d ( a ) ) is u pgrad ed ( d egraded) with respect to a . Thus, H ( Q u ( a )) ≤ H ( a ) ≤ H ( Q d ( a )) . The Bhattachary ya parameter Z ( a  a ) is giv en b y Z ( a  a ) = Z 1 0 Z 1 0 q 1 − x 2 1 x 2 2 a ( x 1 ) dx 1 a ( x 2 ) dx 2 . Since √ 1 − x 2 is decr easing o n [0 , 1 ] , we hav e Z ( Q d ( a )  Q d ( a )) − Z ( a  a ) ≤ m − 1 X i,j =1 Z p i +1 p i Z p j +1 p j  q 1 − p 2 i p 2 j − p 1 − x 2 y 2  a ( x ) dxa ( y ) dy , Z ( a  a ) − Z ( Q u ( a )  Q u ( a )) ≤ m − 1 X i,j =1 Z p i +1 p i Z p j +1 p j  p 1 − x 2 y 2 − q 1 − p 2 i +1 p 2 j +1  a ( x ) dxa ( y ) dy . Now n ote that the maximu m ap proxim ation erro r , call it δ , happen s when xy is close to 1 . Th is m aximum error is equal to r 1 −  1 −  1 m − 1  4 − p 1 − 1 4 . Solving for m we see that the qu antization error can b e made smaller than δ by choosing m such that m ≥ 1 + 1 1 − 4 √ 1 − δ 2 . (5) Note that if a ∈ W then in general ne ither Q d ( a ) nor Q d ( a ) are elem ents of W m , since their en tropies do not match. In fact, as discussed ab ove, the entropy of Q d ( a ) is too high, and the entropy of Q u ( a ) is too lo w . But by taking a suitable conv ex com bination we can find an element b ∈ W m for which Z ( b  2 ) differs f rom Z ( a  2 ) by at most δ . In mo re detail, co nsider the fu nction f ( t ) = H ( tQ u ( a ) + (1 − t ) Q d ( a )) , 0 ≤ t ≤ 1 . Clear ly , f is a continuo us function on its domain. Since e very density of the form of tQ u ( a )+ (1 − t ) Q d ( a ) is upg raded with re spect to Q d ( a ) a nd degraded with respect to Q u ( a ) , we have Z (( Q u ( a ))  2 ) ≤ Z (( tQ u ( a ) + (1 − t ) Q d ( a ))  2 ) ≤ Z (( Q d ( a ))  2 ) . As a re sult: | Z (( tQ u ( a ) + (1 − t ) Q d ( a ))  2 ) − Z ( a  2 ) | ≤ δ . W e f urther h av e f (0) = H ( Q u ( a )) ≤ H ( a ) ≤ H ( Q d ( a )) = f (1) . T hus there exists a 0 ≤ t 0 ≤ 1 suc h th at f ( t 0 ) = H ( a ) = I . Hence, t 0 Q u ( a ) + (1 − t 0 ) Q d ( a ) ∈ BMS ( I ) and t 0 Q u ( a ) + (1 − t 0 ) Q d ( a ) ∈ W m . Therefo re t 0 Q u ( a ) + (1 − t 0 ) Q d ( a ) is th e desired density . Example 9 (Impr oved Bo und for BMS ( I = 1 2 ) ): Let us de- riv e an impr oved bou nd for the class W = BMS ( I = 1 2 ) . W e pick n = 1 , i.e., we con sider tre e channels of height 1 in Theorem 5. For σ = 0 the imp lied operatio n is ⊛ . It is well known that in this case th e maximum o f Z ( a ⊛ a ) over all a ∈ W is ach iev ed for a = BSC (0 . 11 002) . The cor respondin g maximum Z value is 0 . 3916 . Next c onsider σ = 1 . This corr esponds to the convolution  . Motiv ated b y Lemma 8 con sider a t first the max imization of Z with in the class W m : maximize : X i,j α i α j Z (∆ p i  ∆ p j ) = X i,j α i α j q 1 − ( p i p j ) 2 subject to : α ⊤ 1 m × 1 = 1 , α ⊤ H m × 1 = 1 2 , α i ≥ 0 . (6) In the a bove, since the p i s a re fixed, the terms p 1 − ( p i p j ) 2 are also fixed. Th e task is to optim ize the quadra tic form α ⊤ P α over th e correspon ding α polytop e, where the m × m matrix P is defined as P ij = p 1 − ( p i p j ) 2 . W e claim that this is a conve x optimizatio n p roblem. T o see th is, expand √ 1 − x 2 as a T aylo r series in the fo rm p 1 − x 2 = 1 − X l ≥ 0 t l x 2 l , (7) where the t l ≥ 0 . W e fu rther hav e α ⊤ P α = X i,j α i α j q 1 − ( p i p j ) 2 = 1 − X l ≥ 0 t l  X i α i p i 2 l  2 . (8) Thus, since t l ≥ 0 and the p i s are fixed, e ach of the terms − t l ( P i α i p i 2 l ) 2 in the above sum represen ts a con cav e function . As a result the whole func tion is co ncave. T o fin d a bou nd, let u s relax the conditio n 0 ≤ α i ≤ 1 and admit α ∈ R . W e are thus faced with solving the co n vex optimization prob lem maximize : α ⊤ P α subject to : α ⊤ 1 m × 1 = 1 , α ⊤ H m × 1 = 1 2 . The Kuhn-T u cker conditions for this pro blem yield   2 P 1 H 1 ⊤ 0 0 H ⊤ 0 0            α 1 α 2 . . . α n λ 1 λ 2          =          0 0 . . . 0 1 1 2          . (9) As P is no n-singular, the an swer to the above set o f line ar equations is unique . W e can now num erically comp ute this upper boun d and from Lem ma 8 we have an upp er boun d on th e estimation 5 error due to qu antization. W e g et an app roximate value of 0 . 799 . W e conclude th at C P , SC ( { BMS ( I = 0 . 5) } ) ≥ 1 − 1 2 (0 . 392 + 0 . 79 9) = 0 . 404 . This slightly impr oves on the value 0 . 3 74 in (3). In pr inciple ev en better b ounds can b e derived b y consid ering values of n beyond 1 . But the imp lied optimization prob lems that need to be solved are non-trivial. ♦ V . C O N C L U S I O N A N D O P E N P RO B L E M S W e proved that the comp ound capacity o f po lar codes u nder SC decod ing is in general strictly less than the com pound capacity itself. It is n atural to inquir e why polar codes com - bined with SC decoding fail to achie ve the compoun d capacity . Is this du e to th e codes them selves or is it a result of th e sub-optim ality of the decoding algorith m? W e pose this as an interesting open question . In [6] polar codes based on g eneral ℓ × ℓ matrices G were considered . It was shown that suitab ly chosen such codes have an improved error exponent. Perhaps this generalization is also useful in o rder to incr ease th e comp ound capacity of pola r codes. A C K N O W L E D G M E N T The work presented in th is paper is partially supp orted by the Nation al Co mpetence Cente r in Research o n Mo bile Inform ation and Commun ication Systems ( NCCR-MICS), a center supp orted by th e Swiss Natio nal Science Foun dation under gran t num ber 50 05-67 322 an d by the Swiss Natio nal Science Foundation under grant numb er 2 00021 -1219 03. R E F E R E N C E S [1] E. Arıkan, “Channel polarizatio n: A method for construc ting capac ity- achie ving codes for symm etric binary-input memoryle ss channels, ” sub- mitted to IEEE T rans. Inform. Theory , 2008. [2] S. B. Korada and R. Urbanke, “Polar codes are optimal for lossy source coding, ” submitte d to IEEE T rans. Inform. Theory , 2009. [3] D. Blackwell, L. Breiman, and A. J. Thomasian, “The capacity of a class of channel s, ” The Annals of Mathemati cal Statisti cs , vol . 3, no. 4, pp. 1229–1241, 1959. [4] T . Richa rdson and R. Urbanke, Modern Coding Theory . Cambridge Uni versity Press, 2008. [5] R. Mori and T . T anaka , “Performance and Constructi on of Polar Codes on Symmetric Binary-Input Memoryless Channel s, ” Jan. 2009, av ailable from http:// arxi v .org/pdf/0901.2207. [6] S. B. Korada , E . S ¸ as ¸ o ˘ glu, and R. Urbank e, “Polar codes: Characteriz ation of e xponent , bounds, and const ruction s, ” submitted to IEEE T rans. Inf orm. Theory , 2009.