On the Polarization Levels of Automorphic-Symmetric Channels

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1 On the Polarization Levels of Automorphic-Symmetric Channels Rajai Nasser arxiv: v2 [cs.it] 5 Nov 208 Abstract It is known that if an Abelian group operation is used in an Arıkan-style construction, we have multilevel polarization where synthetic channels can approach intermediate channels that are neither almost perfect nor almost useless. An open problem in polarization theory is to determine the polarization levels of a given channel. In this paper, we discuss the polarization levels of a family of channels that we call automorphic-symmetric We show that the polarization levels of an automorphic-symmetric channel are determined by characteristic subgroups. In particular, if the group that is used does not contain any non-trivial characteristic subgroup, we only have two-level polarization to almost perfect and almost useless I. INTRODUCTION Polar codes are the first family of low-complexity capacityachieving codes. Polar codes were first introduced by Arıkan for binary-input channels []. The construction of polar codes relies on a phenomenon that is called polarization: A collection of independent copies of the channel is transformed into a collection of synthetic channels that are almost perfect or almost useless. The transformation of Arıkan for binary-input channels uses the XOR operation. The polarization phenomenon was generalized to channels with non-binary input by replacing the XOR operation with a binary operation on the input-alphabet [2], [3], [4], [5], [6], [7], [8]. Note that if the input alphabet size is not prime, we may have multilevel polarization where the synthetic channels can polarize to intermediate channels that are neither almost perfect nor almost useless. In this paper, we are interested in the multilevel polarization phenomenon when an Abelian group operation is used. More precisely, we are interested in determining the polarization levels of a family of channels that we call automorphic-symmetric In Section II, we introduce the preinaries of this paper. In Section III we introduce H-polarizing and stronglypolarizing families of We show that if W is an H-polarizing family of channels, then the polarization levels of every channel in W are determined by subgroups in H. In Section IV we show that the family of -ary erasure channels is strongly polarizing. This implies that every - ary erasure channel polarizes to almost perfect and almost useless In Section V we introduce -symmetric channels and generalized -symmetric -symmetric channels generalize binary symmetric channels to arbitrary input alphabets. Generalized -symmetric channels are a generalization of binary-input memoryless symmetric-output (BMS) In Section VI, we introduce the family of automorphic-symmetric We show that generalized - symmetric channels are automorphic-symmetric. We show that the polarization levels of an automorphic-symmetric channel are determined by characteristic subgroups. This implies that if the group that is used does not contain any non-trivial characteristic subgroup, we only have two-level polarization to almost perfect and almost useless II. PRELIMINARIES Throughout this paper,(g, +) denotes a fixed finite Abelian group, and = G denotes its size. Let Y be a finite set. We write W : G Y to denote a discrete memoryless channel (DMC) of input alphabet G and output alphabet Y. We write I(W) to denote the symmetric capacity of W. Define the two channels W : G Y 2 and W + : G Y 2 G as follows: W (y,y 2 u ) = W(y u +u 2 )W(y 2 u 2 ), and W + (y,y 2,u u 2 ) = W(y u +u 2 )W(y 2 u 2 ). Furthermore, for every n and every s = (s,...,s n ),+} n, define the channel W s = (...(W s ) s2...) sn. Now let H be a subgroup of (G,+). We denote by G/H the uotient group of G by H. Define the channel W[H] : G/H Y as follows: W[H](y A) = A W(y x) = W(y x). It is easy to see that if X is a uniformly distributed random variable in G and Y is the output of W when X is the input, then I(W[H]) = I(X mod H;Y). Definition. Let δ > 0. A channel W : G Y is said to be δ-determined by a subgroup H of G if I(W) log G/H < δ and I(W[H]) log G/H < δ. The ineualities I(W) log G/H < δ and I(W[H]) log G/H < δ can be interpreted as follows: Let X be a uniformly distributed random variable in G and let Y be the output of the channel W when X is the input. If δ > 0 is small and I(W[H]) log G/H < δ, then from Y we can The symmetric capacity is the mutual information between a uniformly distributed input and its corresponding output.

2 2 determine X mod H with high probability. If we also have I(W) log G/H < δ, then X mod H is almost the only information about X which can be reliably deduced from Y. This is why we can say that if W is δ-determined by H for a small δ, then W behaves similarly to a deterministic homomorphism channel projecting its input onto G/H. It was proven in [6] that as the number n of polarization steps becomes large, the synthetic channels (W s ) s,+} n polarize to deterministic homomorphism channels projecting their input onto uotient groups. More precisely, for every δ > 0, we have 2 n s,+} n : H s a subgroup of G, } =. III. H-POLARIZING FAMILIES OF CHANNELS Definition 2. Let H be a set of subgroups of (G,+). We say that a channel W : G Y H-polarizes if for every δ > 0, we have 2 n s,+} n : H s H, } =. If H = 0}, G} and W H-polarizes, we say that W strongly polarizes. If W H-polarizes, then the levels of polarization are determined by subgroups in H. W strongly polarizes if and only if its synthetic channels (W s ) s,+} n polarize only to almost useless and almost perfect Let W : G Y be a given channel, and assume that after simulating enough polarization steps, we are convinced that W H-polarizes for some family H of subgroups. How can we prove that this is indeed the case? Characterizing H- polarizing channels seems to be very difficult. In this paper, we aim to provide sufficient conditions for H-polarization. Our approach to show the H-polarization of a channel, is to show that it belongs to what we call H-polarizing family of channels: Definition 3. A family W of channels with input alphabet G is said to be H-polarizing if it satisfies the following conditions: If W W, then W W and W + W. 2 There exists δ W,H > 0 such that W does not contain any channel that isδ W,H -determined by a subgroup other than those in H. Proposition. Let H be a family of subgroups and let W be an H-polarizing family of Every channel W W H-polarizes. Proof. Fix W W and let 0 < δ < δ W,H. For every n, define A n,δ = s,+} n : H s a subgroup of G, }. 2 This implies that W s W for every n and every s,+} n. 2 n A n,δ =. Let s A n,δ. There exists a We have subgrouph s of G such that. Since W W, then W s W, which implies that W s cannot be δ-determined by a subgroup other than those in H. Therefore, H s H. We conclude that 2 n s,+} n : H s H, which means that W H-polarizes. } =, IV. -ARY ERASURE CHANNELS Our first example of a strongly polarizing family of channels is the family of -ary erasure Definition 4. Let e be a symbol that does not belong to G. We say that a channel W : G G e} is a -ary erasure channel with parameter ǫ (denoted W = EC(ǫ)) if ǫ if y = x, W(y x) = ǫ if y = e, 0 otherwise. We also call -ary erasure channel any channel that is euivalent to EC(ǫ) in the following sense: Definition 5. A channel W : G Y is said to be degraded from another channel W : G Y if there exists a channel V : Y Y such that W(y x) = W (y x)v (y y ). y Y W and W are said to be euivalent if they are degraded from each other. Denote by W EC the family of all -ary erasure Lemma. If W W EC, then W W EC and W + W EC. Proof. It is easy to check that if W is euivalent to EC(ǫ), then W is euivalent to EC(2ǫ ǫ 2 ) and W + is euivalent to EC(ǫ 2 ). Lemma 2. There exists δ EC > 0 such that there is no - ary erasure channel that is δ EC -determined by a non-trivial 3 subgroup. Proof. Define δ EC = (log2)2. Let W = EC(ǫ) be a -ary log(2) erasure channel and assume there exists a non-trivial subgroup H ofgsuch that I(W[H]) log G/H < δec. It is easy to check that W[H] is a -erasure channel of input alphabet G/H and of parameter ǫ. Moreover, we have I(W[H]) = (log G/H )( ǫ). Now since I(W[H]) log G/H < δ EC = (log2)2 log(2), we have ǫlog G/H < (log2)2 log(2) 3 The trivial subgroups of (G,+) are 0} and G. (log2) 2 ǫ < log(2)log G/H log2 log(2),

3 3 where follows from the fact that H is non-trivial (and hence G/H 2). Thus, I(W) log G/H = (log)( ǫ) log = log ǫlog log2 ǫlog > log2 (log)(log2) log2+log = (log2)2 log(2) = δ EC, where follows from the fact that H is non-trivial (and hence 2). Therefore, we cannot have I(W) log G/H < δ EC. We conclude that if W is a channel with input alphabet G such that there exists a non-trivial subgroup H of G satisfying I(W) log G/H < δec and I(W[H]) log G/H < δ EC, then W / W EC. Proposition 2. W EC is a strongly polarizing family of Proof. The proposition follows from Lemmas and 2. Corollary. Every -ary erasure channel with input alphabet G strongly polarizes. Proof. The corollary follows from Propositions and 2 V. -SYMMETRIC CHANNELS AND GENERALIZED -SYMMETRIC CHANNELS Definition 6. Let 0 ǫ. The -symmetric channel of parameter ǫ (denoted SC(ǫ)) is the channel W : G G defined as ( )ǫ if y = x, W(y x) = ǫ otherwise. -symmetric channels generalize the binary symmetric channels to non-binary input alphabets. We are interested in showing the strong polarization of -symmetric More generally, we are interested in showing the strong polarization of a more general family of channels: Definition 7. We say that a channel W : G Y is a generalized -symmetric channel if there exist a set Y W and a bijection π W : G Y W Y such that: There exists a mapping p W : Y W [0,] such that y Y W p W (y ) =, i.e., p W is a probability distribution on Y W. For every y Y W, there exists 0 ǫ y such that for every x,x G, we have: W(π W (x,y ) x) p W (y ) ( ( )ǫ y ) if x = x, = p W (y ) ǫ y otherwise. Generalized -symmetric channels generalize binary memoryless symmetric-output (BMS) Example. SC(ǫ) is a generalized -symmetric channel: Let Y W = 0}, define π W : G Y W G as π W (x,0) = x, and define p W (0) =. Example 2. Every -ary erasure channel is euivalent to a generalized -symmetric channel. Remark. A generalized-symmetric channel can be thought of as a combination of -symmetric channels indexed by y Y W : The channel picks y Y W with probability p W (y ) and independently from the input. The channel sends the input x through a channel SC(ǫ y ) and obtains x. The channel output is y = π W (x,y ). Since π W is a bijection, the receiver can recover (x,y ) from y. In other words, the receiver knows which SC from the collection SC(ǫ y ) : y Y W } was used. Moreover, the receiver knows the SC output x. The reader can check that ifw is a generalized-symmetric channel, then W is a generalized -symmetric channel as well. Unfortunately, W + is not necessarily a generalized - symmetric channel. Therefore, generalized -symmetric channels do not form a strongly polarizing family of In the next section, we will see that under some condition on the group (G, +), generalized -symmetric channels form a subfamily of a strongly polarizing family of VI. AUTOMORPHIC-SYMMETRIC CHANNELS Definition 8. An automorphism of G is an isomorphism 4 from G to itself. Definition 9. A channel W : G Y is said to be automorphic-symmetric with respect to (G, +) if for every automorphism f : G G there exists a bijection π f : Y Y such that W(π f (y) f(x)) = W(y x). Example 3. If G Z, then the identity is the only automorphism of G. This means that every channel is (trivially) automorphic-symmetric with respect to Z. Another example of automorphic-symmetric channels is generalized -symmetric channels: Proposition 3. Every generalized -symmetric channel is automorphic-symmetric with respect to (G, +). Proof. Let W : G Y be a generalized -symmetric channel. Let Y W, π W and p W be as in Definition 7. Let f : G G be an automorphism. Define π f : Y Y as π f = π W g f π W, where g f : G Y W G Y W is defined as g f (x,y ) = (f(x ),y ). 4 An isomorphism is a bijective homomorphism.

4 4 Let x G and y Y. Define (x,y ) = π W (y). We have W(π f (y) f(x)) = W ( (π W g f π W )(y) ) f(x) = W(π W (g(x,y )) f(x)) = W(π W (f(x ),y ) f(x)) = W(π W (x,y ) x) = W(y x), where follows from the definition of generalized - symmetric Definition 0. Let H be a subgroup of G. We say that H is a characteristic subgroup of G if f(h) = H for every automorphism f of G. A subgroup that is not characteristic is said to be non-characteristic. We denote the family of characteristic subgroups of (G, +) by H ch (G). In the rest of this section, we will show that automorphicsymmetric channels form an H ch (G)-polarizing family of Lemma 3. If W : G Y is automorphic-symmetric, then W and W + are automorphic-symmetric as well. Proof. Let f : G G be an automorphism and let π f : Y Y be a bijection satisfying W(π f (y) f(x)) = W(y x). Define π f : Y2 Y 2 and π + f : Y2 G Y 2 G as follows: π f (y,y 2 ) = (π f (y ),π f (y 2 )), π + f (y,y 2,u ) = (π f (y ),π f (y 2 ),f(u )). Obviously, π f and π+ f W (π f (y,y 2 ) f(u )) are bijections. Moreover, we have: = W (π f (y ),π f (y 2 ) f(u )) = W(π f (y ) f(u )+u 2 )W(π f (y 2 ) u 2 ) = W(π f (y ) f(u )+f(u 2 ))W(π f (y 2 ) f(u 2 )) (b) = = = W (y,y 2 u ), W(π f (y ) f(u +u 2 ))W(π f (y 2 ) f(u 2 )) W(y u +u 2 )W(y 2 u 2 ) where and (b) follow from the fact that f is an automorphism. This shows that W is automorphic-symmetric. On the other hand, we have W + (π + f (y,y 2,u ) f(u 2 )) = W + (π f (y ),π f (y 2 ),f(u ) f(u 2 )) = W(π f(y ) f(u )+f(u 2 ))W(π f (y 2 ) f(u 2 )) = W(π f(y ) f(u +u 2 ))W(π f (y 2 ) f(u 2 )) = W(y u +u 2 )W(y 2 u 2 ) = W + (y,y 2,u u 2 ). This shows that W + is automorphic-symmetric as well. Lemma 4. Let δ > 0. If W is an automorphic-symmetric channel which is δ-determined by a subgroup H of G, then W is δ-determined by f(h) for every automorphism f : G G. Proof. Let W : G Y be an automorphic-symmetric channel, let f : G G be an automorphism, and let H be a subgroup of G. For every coset A G/H, define f(a) = f(x) : x A}. It is easy to see that f(a) G/f(H). Moreover, the reader can check that the mapping f : G/H G/f(H) is an isomorphism of groups. Now let X be a uniformly distributed random variable in G and let Y be the output of the channelw when X is the input. For every (x,y) G Y, we have P X,Y (x,y) = W(y x). Therefore, for every A G/H, we have P f (X mod f(h)),π f (Y )(A,y) = P X mod f(h),y (f(a),π f (y)) = x f(a) P X,Y (x,π f (y)) = P X,Y (f(x),π f (y)) = W(π f(y) f(x)) = W(y x) = P X,Y (x,y) = P X mod H,Y (A,y), () where follows from the fact that W is automorphicsymmetric. We deduce that I(W[f(H)]) = I(X mod f(h);y) (b) = I(f (X mod f(h));π f (Y)) (c) = I(X mod H;Y) = I(W[H]), where (b) follows from the fact that f : G/H G/f(H) and π f : Y Y are bijections. (c) follows from Euation (). Since G/f(H) = G/H and I(W[f(H)]) = I(W[H]), W is δ-determined by H if and only if W is δ-determined by f(h). Proposition 4. There existsδ 0 > 0 such that for every channel W with input alphabetg, ifw isδ 0 -determined by a subgroup H of G, then H is the only subgroup δ 0 -determining W (i.e., there is no subgroup H other than H such that W is δ 0 - determined by H ). Proof. Define δ 0 = log2. Assume that there are two 3 subgroups H and H 2 such that W is δ 0 -determined by both H and H 2. Let X be a random variable uniformly distributed in G and let Y be the output when X is the input. We have I(W[H ]) = I(X mod H ;Y) = H(X mod H ) H(X mod H Y) = log G/H H(X mod H Y).

5 5 Therefore, H(X mod H Y) = log G/H I(W[H ]) < δ 0, where follows from the fact that W is δ 0 -determined by H. Similarly, we can show that H(X mod H 2 Y) < δ 0. Now since there is a one-to-one correspondence between (X mod H H 2 ) and (X mod H,X mod H 2 ), we have H(X mod H H 2 Y) Therefore, = H(X mod H,X mod H 2 Y) H(X mod H Y)+H(X mod H 2 Y) < 2δ 0. I(W) = I(X;Y) I(X mod H H 2 ;Y) = H(X mod H H 2 ) H(X mod H H 2 Y) > log G/(H H 2 ) 2δ 0 = log G log H H 2 2δ 0. Now since W is δ 0 -determined by H, we have hence I(W) log G/H < δ 0, (2) I(W) < log G log H +δ 0. (3) By combining Euations (2) and (3), we get log H H H 2 < 3δ 0 = log2, which implies that H H 2 > H. On the other hand, since 2 H H 2 is a subgroup of H, we have either H H 2 = H or H H 2 2 H. Therefore, H = H H 2 and so H H 2. Similarly, we can show that H 2 H. Hence H = H 2. We conclude that W is δ 0 -determined by at most one subgroup of G. Lemma 5. Let δ 0 be as in Proposition 4. Automorphicsymmetric channels cannot be δ 0 -determined by a subgroup that is non-characteristic. Proof. Let W be an automorphic-symmetric channel. Assume that W is δ 0 -determined by a non-characteristic subgroup H. Since H is non-characteristic, there exists an automorphism f of G such that f(h) H. Lemma 4 implies that W is δ 0 - determined by f(h). This contradicts Proposition 4. Theorem. Automorphic-symmetric channels form an H ch (G)-polarizing family of Proof. The theorem follows from Lemmas 3 and 5. Theorem shows that the synthetic channels of an automorphic-symmetric channel polarize to channels that are determined by characteristic subgroups. Corollary 2. If (G, +) does not contain any non-trivial characteristic subgroup, then the family of automorphic-symmetric channels is strongly polarizing. Proof. The corollary follows from Theorem. Corollary 3. If (G, +) does not contain any non-trivial characteristic subgroup, then all automorphic-symmetric channels strongly polarize. In particular, all generalized -symmetric channels strongly polarize. Proof. The corollary follows from Corollary 2, Proposition and Proposition 3. Example 4. If G F r p for a prime p, then every nontrivial subgroup is non-characteristic. In this case, every automorphic-symmetric channel strongly polarizes. Example 5. If G = F ri p i, where p,...,p n are prime numbers, the reader can check that the characteristic subgroups of (G,+) are those of the form H = F li i, where l i = 0 or l i = r i for every i n. Therefore, if G = F ri p i andw is automorphic-symmetric, the polarization levels of W are determined by subgroups of the form H = F li i, with l i = 0 or l i = r i for every i n. VII. DISCUSSION If G Z with composite, then G contains non-trivial characteristic subgroups, so we cannot apply Corollary 3. Nevertheless, the simulations in [9, Section V] suggest that -symmetric channels strongly polarize when the group Z is used. Proving this remains an open problem. ACKNOWLEDGMENT I would like to thank Emre Telatar, Min Ye and Alexander Barg for helpful discussions. REFERENCES [] E. Arıkan, Channel polarization: A method for constructing capacityachieving codes for symmetric binary-input memoryless channels, Information Theory, IEEE Transactions on, vol. 55, no. 7, pp , [2] E. Şaşoğlu, E. Telatar, and E. Arıkan, Polarization for arbitrary discrete memoryless channels, in Information Theory Workshop, ITW IEEE, 2009, pp [3] E. Şaşoğlu, Polar codes for discrete alphabets, in Information Theory Proceedings (ISIT), 202 IEEE International Symposium on, 202, pp [4] W. Park and A. Barg, Polar codes for -ary channels, Information Theory, IEEE Transactions on, vol. 59, no. 2, pp , 203. [5] A. G. Sahebi and S. S. Pradhan, Multilevel channel polarization for arbitrary discrete memoryless channels, IEEE Transactions on Information Theory, vol. 59, no. 2, pp , Dec 203. [6] R. Nasser and E. Telatar, Polar codes for arbitrary DMCs and arbitrary MACs, IEEE Transactions on Information Theory, vol. 62, no. 6, pp , June 206. [7] R. Nasser, An ergodic theory of binary operations, part I: Key properties, IEEE Transactions on Information Theory, vol. 62, no. 2, pp , Dec 206. [8], An ergodic theory of binary operations, part II: Applications to polarization, IEEE Transactions on Information Theory, vol. 63, no. 2, pp , Feb 207. [9] T. C. Gulcu, M. Ye, and A. Barg, Construction of polar codes for arbitrary discrete memoryless channels, IEEE Transactions on Information Theory, vol. 64, no., pp , Jan 208.

On the Convergence of the Polarization Process in the Noisiness/Weak- Topology

1 On the Convergence of the Polarization Process in the Noisiness/Weak- Topology Rajai Nasser Email: rajai.nasser@alumni.epfl.ch arxiv:1810.10821v2 [cs.it] 26 Oct 2018 Abstract Let W be a channel where