Polar Codes for Arbitrary DMCs and Arbitrary MACs

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1 Polar Codes for Arbitrary DMCs and Arbitrary MACs Rajai Nasser and Emre Telatar, Fellow, IEEE, School of Computer and Communication Sciences, EPFL Lausanne, Switzerland rajainasser, arxiv:333v [csit] 3 Nov 03 Abstract Polar codes are constructed for arbitrary channels by imposing an arbitrary quasigroup structure on the input alphabet Just as with usual polar codes, the block error probability under successive cancellation decoding is o N / ɛ, where N is the block length Encoding and decoding for these codes can be implemented with a complexity of ON log N It is shown that the same technique can be used to construct polar codes for arbitrary multiple access channels MAC by using an appropriate Abelian group structure Although the symmetric sum capacity is achieved by this coding scheme, some points in the symmetric capacity region may not be achieved In the case where the channel is a combination of linear channels, we provide a necessary and sufficient condition characterizing the channels whose symmetric capacity region is preserved by the polarization process We also provide a sufficient condition for having a maximal loss in the dominant face I INTRODUCTION Polar coding, invented by Arıkan [], is the first low complexity coding technique that achieves the capacity of binary-input symmetric memoryless channels Polar codes rely on a phenomenon called polarization, which is the process of converting a set of identical copies of a given single user binary-input channel, into a set of almost extremal channels, ie, either almost perfect channels, or almost useless channels The probability of error of successive cancellation decoding of polar codes was proven to be equal to o N / ɛ by Arıkan and Telatar [] Arıkan s technique was generalized by Şaşoğlu et al for channels with an input alphabet of prime size [3] Generalization to channels with arbitrary input alphabet size is not simple since it was shown in [3] that if we use a group operation in an Arıkan-like construction, it is not guaranteed that polarization will happen as usual to almost perfect channels or almost useless channels Şaşoğlu [4] used a special type of quasigroup operation to ensure polarization Park and Barg [5] showed that polar codes can be constructed using the group structure Z r Sahebi and Pradhan [6] showed that polar codes can be constructed using any Abelian group structure The polarization phenomenon described in [5] and [6] does not happen in the usual sense, indeed, it was previously proven by Şaşoğlu et al that it is not the case It is shown in [5] and [6] that while it is true that we don t always have polarization to almost perfect channels or almost useless channels if a general Abelian operation is used, we always have polarization to almost useful channels ie, channels that are easy to be used for communication The proofs in [5] and [6] rely mainly on the properties of Battacharyya parameters to derive polarization results In this paper, we adopt a different approach: we give a direct elementary proof of polarization for the more general case of quasigroups using only elementary information theoretic concepts namely, entropies and mutual information The Battacharyya parameter is used here only to derive the rate of polarization In the case of multiple access channels MAC, we find two main results in the literature: i Şaşoğlu et al constructed polar codes for the two-user MAC with an input alphabet of prime size [7], ii Abbe and Telatar used matroid theory to construct polar codes for the m-user MAC with binary input [8] The This paper was presented in part at the IEEE International Symposium on Information Theory, Istanbul, Turkey, July 03 This paper is submitted to the IEEE Transactions on Information Theory

2 generalization of the results in [8] to MACs with arbitrary input alphabet size is not trivial even in the case of prime size since there is no known characterization for non-binary matroids We have shown in [9] that the use of matroid theory is not necessary; we used elementary techniques to construct polar codes for the m-user MAC with input alphabet of prime size In this paper, we will see how we can construct polar codes for an arbitrary MAC where the input alphabet size is allowed to be arbitrary, and possibly different from one user to another In our construction, as well as in both constructions in [7] and [8], the symmetric sum capacity is preserved by the polarization process However, a part of the symmetric capacity region may be lost in the process We study this loss in the special case where the channel is a combination of linear channels this class of channels will be introduced in section 8 In section, we introduce the preliminaries for this paper We describe the polarization process in section 3 The rate of polarization is studied in section 4 Polar codes for arbitrary single user channels are constructed in section 5 The special case of group structures is discussed in section 6 We construct polar codes for arbitrary MAC in section 7 The problem of loss in the capacity region is studied in section 8 II PRELIMINARIES We first recall the definitions for multiple access channels in order to introduce the notation that will be used throughout this paper Since ordinary channels one transmitter and one receiver can be seen as a special case of multiple access channels, we will not provide definitions for ordinary channels A Multiple access channels Definition A discrete m-user multiple access channel MAC is an m + -tuple P = X, X,, X m, Y, f P where X,, X m are finite sets that are called the input alphabets of P, Y is a finite set that is called the output alphabet of P, and f P : X X X m Y [0, ] is a function satisfying x, x,, x m X X X m, f P x, x,, x m, y = y Y Notation We write P : X X X m Y to denote that P has m users, X, X,, X m as input alphabets, and Y as output alphabet We denote f P x, x,, x m, y by P y x, x,, x m which is interpreted as the conditional probability of receiving y at the output, given that x, x,, x m is the input Definition A code C of block length N and rate vector R, R,, R m is an m + -tuple C = f, f,, f m, g, where f k : W k =,,, e NR k } X N k is the encoding function of the k th user and g : Y n W W W m is the decoding function We denote f k w = f k w,, f k w N, where f k w n is the n th component of f k w The average probability of error of the code C is given by: P e w,, w m = P e w,, w m P e C = W W m, w,,w m W W m y,,y N Y N n= gy,,y N w,,w m N P y n f w n,, f m w m n Definition 3 A rate vector R = R,, R m is said to be achievable if there exists a sequence of codes C N of rate vector R ɛ,n, R ɛ,n,, R m ɛ m,n and of block length N such that the sequence P e C N } N and the sequences ɛ i,n } N for all i m tend to zero as N tends to infinity The capacity region of the MAC P is the set of all achievable rate vectors

3 Definition 4 Given a MAC P and a collection of independent random variables X,, X m taking values in X,, X m respectively, we define the polymatroid region J X,,X m P in R m by: J X,,X m P := R = R,, R m R m : 0 RS I X,,X m [S]P for all S,, m} }, where RS := l S k= R k, XS := X s,, X sls for S = s,, s ls } and I X,,X m [S]P := IXS; Y XS c The mutual information is computed for the probability distribution P y x,, x m P X,,X m x,, x m on X X m Y Theorem Theorem 536 [0] The capacity region of a MAC P is given by the closure of the convex hull of the union of all information theoretic capacity regions of P for all the input distributions, ie, ConvexHull J X,,X m P X,,X m are independent random variables in X,,X m resp Definition 5 I X,,X m P := I X,,X m [,, m}]p is called the sum capacity of P for the input distributions X,, X m It is equal to the maximum value of R + + R m when R,, R m belongs to the information theoretic capacity region for input distributions X,, X m The set of points of the information theoretic capacity region satisfying R + + R m = I X,,X m P is called the dominant face of this region Notation When X,, X m are independent and uniform random variables in X,, X m respectively, we will simply denote J X,,X m P, I X,,X m [S]P and I X,,X m P by J P, I[S]P and IP respectively J P is called the symmetric capacity region of P, and IP is called the symmetric sum capacity of P B Quasigroups Definition 6 A quasigrous a pair Q,, where is a binary operation on the set Q satisfying the following: For any two elements a, b Q, there exists a unique element c Q such that a = b c We denote this element c by b\ a For any two elements a, b Q, there exists a unique element d Q such that a = d b We denote this element d by a/ b Remark If Q, is a quasigroup, then Q, / and Q, \ are also quasigroups Notation 3 Let A and B be two subsets of a quasigroup Q, We define the set: A B := a b : a A, b B} If A and B are non-empty, then A B max A, B } Definition 7 Let Q be any set A partition H of Q is said to be a balanced partition if and only if all the elements of H have the same size We denote the common size of its elements by H The number of elements in H is denoted by H Clearly, Q = H H for such a partition Definition 8 Let H be a balanced partition of a set Q We define the projection onto H as the mapping Proj H : Q H, where Proj H x is the unique element H H such that x H Lemma Let H be a balanced partition of a quasigroup Q, Define: H := A B : A, B H}

4 3 If H is a balanced partition, then H H Proof: Let A, B H then A B H, we have: H = A B max A, B } = H Definition 9 Let Q, be a quasigroup A balanced partition H of Q is said to be a stable partition of period n of Q, if and only if there exist n different balanced partitions H,, H n of Q such that: H = H H i+ = H i = A B : A, B H i } for all i n H = H n It is easy to see that if H is a stable partition of period n, then H i = H for all i n from lemma, we have H = H H H n H Remark Stable partitions always exist Any quasigroup Q, admits at least the following two stable partitions of period : Q} and x} : x Q }, which are called the trivial stable partitions of Q, It is easy to see that when Q is prime, the only stable partitions are the trivial ones Example Let Q = Z n Z n, define x, y x, y = x +y +x +y, y +y For each j Z n and each i n, define H i,j = j + i k, k : k Z n } Let H i = H i,j : j Z n } for i n It is easy to see that H i = H i+ for i n and H n = H Therefore, H := H is a stable partition of Q, whose period is n Note that the operation in the last example is not a group operation when n > Lemma If H is a stable partition and A is an arbitrary element of H, then H = A A : A H} Proof: We have: Q = A Q = A A H A = A A Therefore, A A : A H} covers Q and is a subset of H which is a partition of Q that does not contain the empty set as an element We conclude that H = A A : A H} Definition 0 For any two partitions H and H, we define: A H H H = A B : A H, B H, A B ø} Lemma 3 If H and H are stable then H H is also a stable partition of Q,, and H H = H H Proof: Since H and H are two partitions of Q, it is easy to see that H H is also a partition of Q Now let A, A H and B, B H If A B ø and A B ø, we have: A B A B A A B B H H Let A H and B H be chosen such that A B is maximal Lemma implies that H = A A : A H } and H = B B : B H } Therefore, Q = A A B B, A,B H H

5 4 which implies that Q A A B B A,B H H A B ø A,B H H A B ø where 3 follows from Now if A B ø, we must have A B A B, 3 A B A B A B A B 4 Therefore, we have: A B A B A B A B 5 A,B H H A B ø A,B H H A B ø Now since H and H are two partitions of Q, we must have A,B H H A B ø A B = Q We conclude A,B H H A B ø that all the inequalities in, 3, 4 and 5 are in fact equalities Therefore, for all A H and B H such that A B ø, we have A B = A B ie, H H is a balanced partition, and A B A B = A A B B Now implies that A B A B = A A B B Therefore, H H = H H If H and H are of periods n and n respectively, then H H is a stable partition whose period is at most lcmn, n III POLARIZATION PROCESS In this section, we consider ordinary channels having a quasigroup structure on their input alphabet Definition Let Q, be an arbitrary quasigroup, and let P : Q Y be a single user channel We define the two channels P : Q Y Y and P + : Q Y Y Q as follows: P y, y u = P y u u P y u, Q u Q P + y, y, u u = Q P y u u P y u For any s = s,, s n, +} n, we define P s := P s s sn Remark 3 Let U and U be two independent random variables uniformly distributed in Q Set X = U U and X = U, then X and X are independent and uniform in Q since is a quasigroup operation Let Y and Y be the outputs of the channel P when X and X are the inputs respectively It is easy to see that IP = IU ; Y, Y and IP + = IU ; Y, Y, U We have: IP + IP + = IU ; Y, Y + IU ; Y, Y, U = IU, U ; Y, Y = IX, X ; Y, Y = IX ; Y + IX ; Y = IP It is clear that IP + = IU ; Y, Y, U IU ; Y = IX ; Y = IP We conclude that IP IP IP +

6 5 Definition Let H be a balanced partition of Q, /, we define the channel P [H] : H Y by: P [H]y H = P y x H x Q, Proj H x=h Remark 4 If X is a random variable uniformly distributed in Q and Y is the output of the channel P when X is the input, then it is easy to see that IP [H] = IProj H X; Y Definition 3 Let B n } n be iid uniform random variables in, +} We define the channel-valued process P n } n 0 by: P 0 := P, P n := P Bn n n The main result of this section is that almost surely P n becomes a channel where the output is almost equivalent to the projection of the input onto a stable partition of Q, / : Theorem Let Q, be a quasigroup and let P : Q Y be an arbitrary channel Then for any δ > 0, we have: s, +} n : H s a stable partition of Q, /, lim n IP s log H s < δ, IP s [H s ] log H s < δ } = Remark 5 Theorem can be interpreted as follows: in a polarized channel P s, we have IP s IP s [H s ] log H s for a certain stable partition H s of Q, / Let X s and Y s be the channel input and output of P s respectively IP s [H s ] log H s means that Y s almost determines Proj Hs X s On the other hand, IP s IP s [H s ] means that there is almost no information about X s other than Proj Hs X s which can be determined from Y s In order to prove theorem, we need several lemmas: Lemma 4 Let Q, be a quasigroup If A, B and C are three non-empty subsets of Q such that A = B = C = A C = B C, then either A B = ø or A = B Proof: Suppose that A B ø and let a A B The fact that A C = C implies that A C = a C Similarly, we also have B C = a C since a B Therefore, A B C = a C, and so A B C = C = A By noticing that A A B A B C = A, we conclude that A B = A, which implies that A = B since A = B Definition 4 Let Q be a set, and let A be a subset of Q, we define the distribution I A on Q as I A x = A if x A and I A x = 0 otherwise Lemma 5 Let X be a random variable on Q, and let A be a subset of Q Suppose that there exist δ > 0 and an element a A such that P X x P X a < δ for all x A and P X x < δ for all x / A Then P X I A < Q δ Proof: We have: A PX a = x A x Q P X x A P X a = PX x P X a + x A x Ac P X x P X a + P X x < Q δ x A c P X x

7 Therefore, P X a < Q δ Q δ Let x A, then A A P X x PX x P X a PX + a < Q δ A A On the other hand, if x / A we have P X x < δ Q δ Thus, P X I A < Q δ Definition 5 Let Q and Y be two arbitrary sets Let H be a set of subsets of Q Let X, Y be a pair of random variables in Q Y We define: } A H,δ X, Y = y Y : H y H, P X Y =y I Hy < δ, P H,δ X; Y = P Y AH,δ X, Y If P H,δ X; Y > δ for a small enough δ, then Y is almost equivalent to Proj H X The next lemma shows that if IP is close to IP, then the output Y of P is almost equivalent to Proj H X, where X is the input to the channel P and H is a certain balanced partition of Q Lemma 6 Let Q and Y be two arbitrary sets with Q Let X, Y be a pair of random variables in Q Y such that X is uniform Let H be a set of disjoint subsets of Q that have the same size If P H, X; Y >, then H is a balanced partition of Q Q Q Proof: We only need to show that H covers Q Suppose that there exists x Q such that there is no H in H such that x H Then for all y A H, X, Y, P Q X Y x y < We have: Q P X x = P X Y x yp Y y + P X Y X Y P Y Y y A H, X,Y y A Q H, X,Y c Q < Q P Y AH, Q X, Y + P Y AH, Q X, Y c < Q + Q = Q Q which is a contradiction since X is uniform in Q Therefore, H covers Q and so it is a balanced partition of Q Lemma 7 Let Q and Y be two arbitrary sets with Q, and let H and H be two balanced partitions of Q Let X, Y be a pair of random variables in Q Y such that X is uniform If P H, X; Y > Q Q and P H, X; Y >, then H = H Q Q Proof: Define H = H H Let y A H, X, Y A Q H, X, Y, choose H H and H H Q such that P X Y =y I H < and P Q X Y =y I H <, then Q which implies that H = H and y A H, I H I H < Q Q Q P H, X; Y P Q Y AH, X, Y Therefore, X, Y A Q H, X, Y > Q Q From lemma 6 we conclude that H is a balanced partition Therefore, H = H = H Lemma 8 Let Q, be a quasigroup with Q, and let Y be an arbitrary set For any δ > 0, there exists ɛ δ > 0 depending only on Q such that for any two pairs of random variables X, Y and X, Y that are independent and identically distributed in Q Y with X and X being uniform in Q, then HX X Y, Y < HX Y + ɛ δ implies the existence of a balanced partition H of Q such that P H,δ X ; Y > δ Moreover, H H = H = H for every H, H H 6

8 Proof: Choose δ > 0, and let δ = min δ, Q Q 4 } Define: p y x := P X Y x y and p y,x x := p y x/ x q y x := P X Y x y and q y,x x := q y x \ x Note that q y x = p y x since X, Y and X, Y are identically distributed Nevertheless, we choose to use q y x to denote P X Y x y for the sake of notational consistency We have: P X X Y,Y x y, y = x Q p y x q y,x x 6 7 = x Q q y x p y,x x 7 Due to the strict concavity of the entropy function, there exists ɛ δ > 0 such that: If x, x Q such that p y x δ, p y x δ and q y,x q y,x δ then see 6 HX X Y = y, Y = y HX Y = y + ɛ δ, 8 If x, x Q such that q y x δ, q y x δ and p y,x p y,x δ then HX X Y = y, Y = y HX Y = y + ɛ δ, 9 see 7 Define: C = y, y Y Y : x, x Q, p y x δ, p y x } δ q y,x q y,x < δ, C = From 8 we have: y, y Y Y : x, x Q, q y x δ, q y x δ p y,x p y,x < δ } HX X Y, Y HX Y + ɛ δ P Y,Y C c = HX Y + ɛ δ P Y,Y C c Similarly, from 9 we have Let ɛ δ = ɛ δ δ, and suppose that HX X Y, Y HX Y + ɛ δ P Y,Y C c HX X Y, Y < HX Y + ɛ δ, then we must have P Y,Y C c < δ and P Y,Y C c < δ, which imply that P Y,Y C > δ, where C = C C Now for each a, a, x Q, define: π a,a x := x a/ a, and γ a,a x := a \ a x And for each y, y Y Y, define: A y := x Q, p y x δ } B y := x Q, q y x δ } a y := arg max p y x b y := arg max q y x x x } H y,y = x Q : b, b, b, b,, b n, b n B y, x = π bn,b π n b,b a y } K y,y = x Q : a, a, a, a,, a n, a n A y, x = γ an,a γ n a,a b y

9 8 Suppose that y, y C Let x H y,y, and let n be minimal such that there exists b, b, b, b,, b n, b n B y satisfying x = π bn,b n π b,b a y Define a := a y, and for i n define a i+ = π bi,b i a i, so that a n+ = x We must have a i a j for i j since n was chosen to be minimal Therefore, n + Q For any i n, we have a i+ = a i b i / b i Let x = a i b i, then a i+ = x/ b i and a i = x/ b i We have y, y C, q y b i δ and q y b i δ, so we must have p y,b i p y,b i < δ, and p y,b i x p y,b i x < δ, which implies that p y a i+ p y a i < δ Therefore: p y x p y a y = p y a n+ p y a n p y a i+ p y a i i= < nδ Q δ Q Q 4 < Q Q 0 Since p y a y Q, we have p y x > Q > δ for every x H y,y Therefore, H y,y A y y, y C A similar argument yields K y,y B y y, y C Fix two elements b, b B y We have x b/ b H y,y and so x b H y,y b for any x H y,y Therefore, H y,y b H y,y b But this is true for any two elements b, b B y, so H y,y b = H y,y b b, b B y, and H y,y B y = H y,y Similarly, we have A y K y,y = K y,y If we also take into consideration the fact that H y,y A y and K y,y B y we conclude: B y H y,y B y = H y,y A y, A y A y K y,y = K y,y B y Therefore, A y = H y,y = B y = K y,y We conclude that H y,y = A y and K y,y = B y Moreover, we have A y B y = A y = B y Recall that p y x p y a y < Q δ for all x A y see 0 and p y x < δ Q δ for x / A y It is easy to deduce that p y I Ay < Q Q δ < Q δ Therefore, p y I Ay < δ and p y I Ay < Similarly, q Q y I By < δ and q y I By < Q Now define C Y = y Y : P Y y, Y C > δ }, and for each y C Y, define K y = } y Y : y, y C Then we have: δ < P Y,Y C P Y C Y δ + P Y C Y, from which we conclude that P Y C Y > δ And by definition, we also have P Y K y > δ for all y C Y Define H y = B y : y K y } Fix y C Y Since A y B = A y B = A y = B = B for every B, B H y, we conclude that the elements of H y are disjoint and have the same size lemma 4 Now since P Y K y > Q 4 and since X is uniform in Q, it is easy to see that H y covers Q and so it is a balanced partition of Q for all y C Y Moreover, since P Y K y >, we can also conclude that all the balanced partitions Q 4 H y are the same Let us denote this common balanced partition by H We have A B = A = B for all A H and all B H, where H = A y : y C Y } By using a similar argument as in the previous paragraph, we can deduce that H is a balanced partition of Q

10 9 Moreover, since X, Y and X, Y are identically distributed, we can see that H = H We conclude the existence of a balanced partition H of Q satisfying A B = A = B for all A, B H and P Y y Y : H y H, P X Y =y I Hy < δ} P Y C Y > δ > δ Lemma 9 Let X and X be two independent random variables in Q such that there exists two sets A, B Q satisfying P X I A < δ, P X I B < δ and A B = A = B, then P X X I A B < δ + Q δ Proof: The fact that A B = A = B implies that for every x A B, we have x/ b A for every b B, and x/ b A c for every b B c For every a Q define ɛ,a = P X a if a A, and ɛ A,a = P X a if a / A Similarly, for every b Q define ɛ,b = P X b if b B, and ɛ A,b = P X b if b / B Let x A B, we have: Therefore, Now let x / A B, we have: P X X x = b B = b B P X X x = b B P X x/ bp X b + b B c P X x/ bp X b = A + ɛ,x/ b A + ɛ,b + ɛ,x/ bɛ,b b B b B c = A + ɛ,x/ A b + ɛ,b + ɛ,x/ bɛ,b b Q P X x/ bp X b + b/ B ɛ,x/ b A + ɛ,b b B P X X x < δ + Q δ A x/ b A + b/ B x/ b A P X x/ bp X b + b/ B x/ b/ A P X x/ bp X b A + ɛ,x/ b ɛ,b + ɛ,x/ bɛ,b δ + Q δ b/ B x/ b/ A Lemma 0 Let Q, be a quasigroup with Q, and let Y be an arbitrary set For any δ > 0, there exists ɛδ > 0 depending only on Q and δ such that for any channel P : Q Y, IP IP < ɛδ implies the existence of a balanced partition H of Q such that H / = H/ H : H, H H} is also a balanced partition of Q, P H,δ X ; Y > δ, P H,δ U ; Y, Y, U > δ and P H /,δu ; Y, Y > δ Where U and U are two independent random variables uniformly distributed in Q, X = U U, X = U, and Y resp Y is the output of the channel P when X resp X is the input Proof: Let δ = minδ, δ, }, where δ > 0 is a small enough number that will be specified 6 Q later Let ɛδ = ɛ δ, where ɛ is given by lemma 8 Let P : Q Y be a channel as in the hypothesis Then from lemma 8 we conclude the existence of two balanced partitions H and H such that P H,δ X ; Y > δ and P H,δ U ; Y, Y > δ Moreover, we have H / H = H = H for every H, H H Given H H, define: } } A H = y Y : P X Y =y I H < δ = y Y : P X Y =y I H < δ,

11 0 note that X, Y and X, Y are identically distributed Let x H, we have: Q = P X x = P X Y x yp Y y + P X Y x yp Y y + P X Y x yp Y y y A H,δ X ;Y \A H y A h y / A H,δ X ;Y δ P H,δ X ; Y + H + δ P Y A h + P H,δ X ; Y < δ + P Y A H Therefore, 8 Q + P Y A H < Q + P Y A H P Y A H = P Y A H > 4 Q Now for each H, H H, define: A H,H = y,y Y Y : P U Y =y,y =y I H / H < } Q Let y, y A H A H, then P X Y =y I H < δ and P X Y =y I H < δ Lemma 9 implies that P U Y =y,y =y I H / H = P X / X Y =y,y =y I H / H < δ + Q δ 8 Q + Q 6 Q < 4 Q Therefore, A H A H A H,H and so P Y,Y A H,H P Y A H P Y A H > δ see We 6 Q recall that P Y,Y AH,δ U ; Y, Y = P H,δ U ; Y, Y > δ, so A H,δ U ; Y, Y A H,H ø Let y, y A H,δ U ; Y, Y A H,H, then there exists H H such that P U Y =y,y =y I H < δ < Now since y Q, y A H,H, we have P U Y =y,y =y I H / H <, so Q I H I H / H <, we conclude that Q H = H / H and H / H H But this is true for any H, H H Therefore, H / H, which implies that H / = H since both H and H / are partitions of Q whose all elements are non-empty Thus, P H,δ X ; Y P H,δ X ; Y > δ δ, P H /,δu ; Y, Y P H /,δ U ; Y, Y > δ δ It remains to prove that P H,δ U ; Y, Y, U > δ Define: K = A H /,δ U ; Y, Y A H,δ X ; Y A H,δ X ; Y We have: P Y A H,δ X ; Y = P Y A H,δ X ; Y = P H,δ X ; Y P H,δ X ; Y > δ δ Thus, P Y,Y A H,δ X ; Y A H,δ X ; Y > δ On the other hand, we have: P Y,Y A H /,δ U ; Y, Y = P H /,δ U ; Y, Y = P H,δ U ; Y, Y P H,δ U ; Y, Y > δ δ,

12 we conclude that P Y,Y K > 3δ Define: B = y, y, u Y Y Q : y, y K, and H H /, P U Y =y,y =y I H < δ and u H } If y, y K, then y, y A H /,δ U ; Y, Y and so there exists H y,y H / such that P U Y =y,y =y I Hy,y < δ, which implies that y, y, u B for all u H y,y Now since P U Y =y,y =y I Hy,y < δ, it is easy to see that P U Y =y,y =y H y,y H y,y δ Q δ Therefore, P Y,Y,U B > P Y,Y K Q δ > 3δ Q δ > Q + 3δ Therefore, if δ δ, then P Q +3 Y,Y,U B > δ Now let y, y, u B There exists H, H H and H H / such that: u H, P U Y =y,y =y I H < δ, P X Y =y I H < δ, P X Y =y I H < δ Since U = X / X, lemma 9 implies that P U Y =y,y =y I H / H < δ + Q δ, and I H I H / H < 3δ + Q δ Therefore, if δ, then I 4 Q H I H / H < and H = H Q / H Now we have: u H implies PU Y,Y u y, y H < δ, ie, H δ < P U Y,Y u y, y < + H δ If u H, then u u H which implies that PX Y u u y H < δ and PX Y u y < δ H If u / H, then u u / H, so P X Y u u y < δ and P X Y u y < δ By noticing that P U Y,Y,U u y, y, u = P U,U Y,Y u, u y, y P U Y,Y u y, y we conclude that: If u H, we have: If u / H, we have: δ H + < P U H δ Y,Y,U u y, y, u < P U Y,Y,U u y, y, u < Consequently, there exists βδ > 0 such that if δ βδ we get = P X Y u u y P X Y u y, P U Y,Y u y, y δ H δ H + δ H δ P U Y =y,y =y,u =u I H < δ By setting δ δ = min, },, βδ we get y Q +3 4 Q, y, u A H,δ U ; Y, Y, U for every y, y, u B, ie, B A H,δ U ; Y, Y, U and P H,δ U ; Y, Y, U P Y,Y,U B > δ Now we are ready to prove theorem In fact, we will prove a stronger theorem:

13 Theorem 3 Let Q, be a quasigroup and let P : Q Y be an arbitrary channel Then for any δ > 0, we have: lim s, +} n : H s a stable partition of Q, /, n IP s [H ] log H s H s H < δ for all stable partitions H of Q, / } H = Proof: Due to the continuity of the entropy function, there exists γδ > 0 depending only on Q such that if X, Y is a pair of random variables in Q Y where X is uniform, and if there exists a stable partition of H such that P H,γδ X; Y > γδ, then IX; Y log H < δ and I Proj H X; Y log H H H H < δ for all stable partitions H of Q, / remember that H H is a stable partition by lemma 3 Let P n be as in definition 3 From remark 3 we have: E IP n+ P n = IP n + IP n + = IP n This implies that the process IP n } n is a martingale, and so it converges almost surely Let m be the number of different balanced partitions of Q, choose l > m and let 0 i l + Almost surely, IP n l+i+ IP n l+i converges to zero Therefore, we have: where A n,l,i := lim n l+i A n,l,i = } s, s, +} n l, +} i : IP s,s, IP s,s < ɛδ, and ɛδ is given by lemma 0 Now for each s, +} i, define: } A n,l,s := s, +} n l : IP s,s, IP s,s < ɛδ It is easy to see that A n,l,i = A n,l,s Therefore, s,+} i ie, i s,+} i lim A n l n,l,s s,+} i lim = lim A n,l,i =, n l+i A n l n,l,s = i On the other hand, it is obvious that A n,l,s n l, and so lim A n l n,l,s for all s, +} i We can now use to conclude that lim A n l n,l,s for all s, +} i Therefore, we must have lim A n,l =, where n l A n,l : = = A n,l,s 0 i l+ s,+} i s, +} n l : IP s,s, IP s,s < ɛδ, s, +} i, 0 i l + }

14 3 Now define: C l := } s, +} l : s contains the sign at least m times, B n,l := A n,l C l = s = s, s, +} n l, +} l : s A n,l, s C l }, D n := s, +} n : H s a stable partition of Q, /, IP s [H ] log H s H s H } < δ for all stable partitions H of Q, / H 3 Now let s A n,l, let n l j n, let s = s, s, +} j for some s, +} j n+l, let X s be the input to the channel P s and Y s be the output of it Since j n+l l, both s and s, have lengths of at most l + Therefore, we have IP s,s, IP s,s < ɛδ and IP s,s,, IP s,s, < ɛδ Lemma 0 implies the existence of a balanced partitions H s such that P X Hs,δ s; Y s > δ, P / H X s,δ s, ; Y s, > δ and P X Hs,δ s,+; Y s,+ > δ for all s, +} j n l j n having s as a prefix Since δ <, lemma 7 implies that H Q s, = H s / and H s,+ = H s for all s, +} j n l j < n having s as a prefix Let s C l, and let l be the number of signs in s we have m l l, then there exist l + balanced partitions H i 0 i l such that H 0 = H s, H l = H s,s, and H i+ = H / i for each 0 i l Since m is the number of different balanced partitions of Q, there exist two indices i and j such that i < j l and H i = H j We conclude that H l = H s,s is a stable partition of Q, / Moreover, since δ γδ, s, s belongs to D n Therefore, B n,l D n for any l m Thus: lim inf But this is true for any l m, we conclude: which implies that D n lim n B n,l = lim n A n,l n l lim inf D n lim n l C l =, l lim n D n = l C = l C l l IV RATE OF POLARIZATION In this section, we are interested in the rate of polarization of P n to deterministic projection channels Definition 6 The Battacharyya parameter of an ordinary channel P with input alphabet X and output alphabet Y is defined as: ZP = P y xp y x X X x,x X X x x if X > And by convention, we take ZP = 0 if X = It s known that P e P X ZP see [3], where P e P is the probability of error of the maximum likelihood decoder of P y Y

15 4 Definition 7 Let Q, be a quasigroup with Q, and Y be an arbitrary set Let P : Q Y be an arbitrary channel, and H be a stable partition of Q, / We define the channels P [H] : H / Y Y and P [H] + : H Y Y H / by: P [H] + y, y, H H = H P [H]y H H P [H]y H, P [H] y, y H = H H H P [H]y H H P [H]y H Lemma P [H] + is degraded with respect to P + [H], and P [H] is equivalent to P [H / ] Proof: Let H, H, y, y H / H Y Y, we have: P [H] + y, y, H H = H P [H]y H H P [H]y H = P y x P y x Q H x Q x Q Proj H x =H H Proj H x =H = P y x x P y x Q H x Q x Q Proj H x =H = H = Proj H / x =H P x Q x Q Proj H / x =H Proj H x =H P x Q Proj H / x =H + [H]y, y, x H + y, y, x x Therefore, P [H] + is degraded with respect to P + [H] Now let H, y, y H / Y Y, we have: P [H] y, y H = P [H]y H H P [H]y H H H H = P y x P y x Q H H H x Q x Q Proj H x =H H Proj H x =H = Q H = Q H = H Therefore, P [H] is equivalent to P [H / ] H H x Q Proj H / x =H x Q x Q Proj H / x =H P x Q Proj H / x =H P y x x P y x x Q Proj H x =H P y x x P y x y, y x = P [H / ]y, y H Definition 8 Let H be a stable partition of Q, /, we define the stable partitions H and H +, by H / and H respectively

16 5 Lemma Let B n and P n be defined as in definition 3 For each stable partition H of Q, /, we define the stable partition-valued process H n by: H 0 := H, H n := H Bn n n Then IP n [H n ] converges almost surely to a number in L H := log d : d divides H } Proof: Since P n [H n ] is equivalent to Pn [H n / ] and P n [H n ] + is degraded with respect to P n + [H n ] lemma, we have: E IP n+ [H n+ ] P n = IP n [H n / ] + IP n + [H n ] IP n[h n ] + IP n[h n ] + = IP n [H n ] This implies that the process IP n [H n ] is a sub-martingale and therefore it converges almost surely Let δ > 0, and define D l,δ as in 3, we have shown that lim n D n,δ = It is easy to see that almost n surely, for every δ > 0 and for every n 0 > 0 there exists n > n 0 such that B,, B n D l,δ Let B n be a realization in which IP n [H n ] converges to a limit x, and in which for every δ > 0 and for every n 0 > 0 there exists n > n 0 such that B,, B n D n,δ Let δ > 0 and let n 0 > 0 be chosen such that IP n [H n ] x < δ for every n > n 0 Choose n > n 0 such that B,, B n D n,δ, this means that there exists a stable partition H of Q, / such that IP n [H n ] log H H H n < δ H n Therefore, x log H n H H n < δ, which implies that x log H H H n since Q = H H n H H = H n H n By noticing that Hn H H n divides H H n = H, we conclude that dx, L H < δ for every δ > 0 Therefore, x L H Lemma 3 Let P : Q Y be an ordinary channel where Q is a quasigroup with Q For any stable partition H of Q, /, we have: s, +} n : H a stable partition of Q, /, lim n for any 0 < ɛ < log and any 0 < β < IP s [H] > log H ɛ, ZP s [H] nβ} = 0, Proof: Let 0 < ɛ < log and 0 < β <, and let H be a stable partition of Q, / IP n [H n ] converges almost surely to an element in L H Due to the relations between the quantities IP and ZP see proposition 33 of [] we can see that ZP n [H n ] converges to 0 if and only if IP n [H n ] converges to log H, and there is a number z 0 > 0 such that lim inf ZP n [H] > z 0 whenever IP n [H] converges to a number in L H other than log H Therefore, we can say that almost surely, we have: lim ZP n [H n ] = 0 or lim inf ZP n [H] > z 0 ZP + n [H + n ] ZP n [H n ] + since P n [H n ] + is degraded with respect to P + n [H + n ], and ZP n [H n ] = ZP n [H n ] since P n [H n ] and P n [H n ] are equivalent see lemma From lemma 35 of [] we have: ZP n [H n ] H H + ZP n [H n ] ZP n [H n ] + H ZP n [H n ]

17 6 Therefore, we have ZP n [H n ] KZP n [H n ] and ZP + n [H n ] KZP n [H n ], where K is equal to H H + By applying exactly the same techniques that were used to prove theorem 35 of [] we get: lim Pr IP n [H n ] > log H ɛ, ZP n [H n ] nβ = 0 But this is true for all stable partitions H Therefore, s, +} n : H a stable partition of Q, /, lim n IP s [H s ] > log H ɛ, ZP s [H s ] nβ} = 0 By noticing that for each s, +} n, there exists a stable partition H s such that H = Hs, s we conclude: s, +} n : H a stable partition of Q, /, lim n IP s [H] > log H ɛ, ZP s [H] nβ} = 0 Theorem 4 The convergence of P n to projection channels is almost surely fast: s, +} n : H a stable partition of Q, /, lim n IP s log H < ɛ, IP s [H] log H < ɛ, ZP s [H] < βn } =, for any 0 < ɛ < log, and any 0 < β < Proof: Let 0 < ɛ < log, and 0 < β < Define: E 0 = s, +} n : H a stable partition of Q, /, IP s [H] > log H ɛ, ZP s [H] βn}, E = s, +} n : H a stable partition of Q, /, IP s log H < ɛ, IP s [H] log H < ɛ }, E = s, +} n : H a stable partition of Q, /, IP s log H < ɛ, IP s [H] log H < ɛ, ZP s [H] < βn } It is easy to see that E \ E 0 E and E E E 0 By theorem and lemma 3 we get: lim n E lim n E E 0 = 0 =

18 7 V POLAR CODE CONSTRUCTION Choose 0 < ɛ < log and 0 < β < β <, let n be an integer such that Q n β n < βn and E ɛ n > n log Q, where E n = s, +} n : H a stable partition of Q, /, IP s log H < ɛ, IP s [H] log H < ɛ }, ZP s [H] < β n Such an integer exists due to theorem 4 A polar code is constructed as follows: If s / E n, let U s be a frozen symbol, ie, we suppose that the receiver knows U s On the other hand, if s E n, there exists a stable partition H s of G, such that IP s log H s < ɛ, IP s [H s ] log H s < ɛ, and ZP s [H s ] < n β Let f s : H s G be a frozen mapping in the sense that the receiver knows f s such that f s H H for all H H s, we call such mapping a section mapping We choose U s uniformly in H s and we let U s = f s U s Note that if the receiver can determine Proj Hs U s = U s accurately, then he can also determine U s since he knows f s Since we are free to choose any value for the frozen symbols and for the section mappings, we will analyse the performance of the polar code averaged on all the possible choices of the frozen symbols and for the section mappings Therefore, U s are independent random variables, uniformly distributed in Q If s / E n, the receiver knows U s and there is nothing to decode, and if s E n, the receiver has to determine Proj Hs U s in order to successfully determine U s We associate the set, +} n with the strict total order < defined as s,, s n < s,, s n if and only if there exists i,, n} such that s i =, s i = + and s j = s j j > i A Encoding Let P s } s,+} n be a set of n independent copies of the channel P P s should not be confused with P s : P s is a copy of the channel P and P s is a polarized channel obtained from P as before Define U s,s for s, +} l, s, +} n l, 0 l n, inductively as: U ø,s = U s if l = 0, s, +} n U s ;,s = U s,s ;+ U s,s ; if l > 0, s, +} l, s, +} n l U s ;+,s = U s,s ;+ if l > 0, s, +} l, s, +} n l We send U s,ø through the channel P s for all s, +} n Let Y s be the output of the channel P s, and let Y = Y s } s,+} n We can prove by induction on l that the channel U s,s Ys } s has s as a prefix, U s,s } s <s is equivalent to the channel P s In particular, the channel U s Y, Us } s <s is equivalent to the channel P s Figure is an illustration of a polar code construction for n = ie, the block-length is N = = 4 B Decoding If s / E n then the receiver knows U s, there is nothing to decode Suppose that s E n, if we know U s } s <s then we can estimate Proj Hs U s from Y, U s } s <s by the maximum likelihood decoder of P s [H s ] After that, we estimate U s = f s Proj Hs U s This motivates the following successive cancellation decoder: Û s = U s if s / E n Û s = D s Y, Ûs } s <s if s E n Where D s Y, U s } s <s is the estimate of U s obtained from Y, U s } s <s by the above procedure

19 8 Fig : Polar code construction for n = C Performance of polar codes If s E n, the probability of error in estimating U s is the probability of error in estimating Proj Hs U s using the maximum likelihood decoder, which is upper bounded by H s ZP s [H s ] < Q β n Note that D s Y, U s } s <s = U s s E n D s Y, Ûs } s <s = U s s E n Therefore, the probability of error of the above successive cancellation decoder is upper bounded by P D s Y, U s } s <s U s < En Q β n Q n β n < βn s E n This upper bound was calculated on average over a random choice of the frozen symbols and of the section mappings Therefore, there exists at least one choice of the frozen symbols and of the section mappings for which the upper bound of the probability of error still holds We should note here that unlike the case of binary input symmetric memoryless channels where the frozen symbols can be chosen arbitrarily, the choice of the frozen symbols and section mappings in our construction of polar codes cannot be arbitrary The code designer should make sure that his choice of the frozen symbols and section mappings actually yields the desirable probability of error The last thing to discuss is the rate of polar codes The rate at which we are communicating is R = log H n s On the other hand, we have IP s log H s < ɛ for all s E n And since we have s E n IP s = n IP, we conclude: s,+} n IP = IP s = IP s + IP s < n n n n s,+} n s E n s E c n < R + n E n ɛ + ɛ log Q log Q R + ɛ + ɛ = R + ɛ s E n log H s + ɛ + n Ec n log Q To this end we have proven the following theorem which is the main result of this paper: Theorem 5 Let P : Q Y be a channel where the input alphabet has a quasigroup structure For every ɛ > 0 and for every 0 < β <, there exists a polar code of length N having a rate R > IP ɛ and a probability of error P e < N β VI THE CASE OF GROUPS Lemma 4 Let G, be a group, and let H be a stable partition of G, / There exists a normal subgroup of G such that H is the quotient group of G by H also denoted by G/H, and Proj H x = x mod H for all x G

20 9 Proof: Let H be the element of H containing the neutral element e of G For any H H, we have H = H / e H / H Now because of the stability of H, we have H / H = H and so H / H = H for all H H This implies that H / = H Now for any H H = H / and H H, there exists H 3 H such that H = H 3 / H, and so H H = H 3 H Therefore, we also have H = H Now for any H H, we have H = e H H H H, H = H e H H H, and H = H H = H H, from which we conclude that H H = H H = H This implies that H H = H, and k H = H k for any k G Therefore, H is a normal subgroup of G, and H is the quotient subgroup of G by H By combining the last lemma with theorem 4, we get: Theorem 6 Let P : G Y be a channel where the input alphabet G has a group structure P n converges almost surely to homomorphism channels Moreover, the convergence is almost surely fast: s, +} n : H a normal subgroup of G, lim n IP s log G/H < ɛ, IP s [H] log G/H < ɛ, ZP s [H] < βn } =, for any 0 < ɛ < log, and any 0 < β < Where P [H] : G/H Y is defined as: P [H]y a = P y x H x G x mod H=a VII POLAR CODES FOR ARBITRARY MULTIPLE ACCESS CHANNELS In this section, we construct polar codes for an arbitrary multiple access channel, where there is no constraint on the input alphabet sizes: they can be arbitrary, and possibly different from one user to another If we have X k = p r p r p rn k n k, where p,, p nk are prime numbers, we can assume that X k = F r p F r p F rn k p nk, and so we can replace the k th user by r +r + +r nk virtual users having F p, F p,, or F pnk as input alphabet respectively Therefore, we can assume without loss of generality that X k = F qk for all k, where q k is a prime number Let p, p,, p l be the distinct primes which appear in q,, q m, and for each i l let m i be the number of times appears in q,, q m We adopt two notations to indicate the users and their inputs: The first notation is the usual one: we have an index k taking value in,, m}, and the input of the k th user is denoted by X k F qk In the second notation, the m i users having their inputs in F pi will be indexed by i,,, i, j,, i, m i, where i l and j m i The input of the i, j th user is denoted by X i,j F pi The vector X i,,, X i,mi F m i is denoted by X i m m Definition 9 Let P : F qk Y be a discrete m-user MAC We define the two channels P : F qk Y and P + : k= m F qk Y k= P y, y u,, u m = m k= F qk q q m as: P + y, y, u,, u m u,, u m = u,,u m m k= Fq k k= P y u + u,, u m + u mp y u,, u m, q q m P y u + u,, u m + u mp y u,, u m,

21 0 where the addition u k + u k takes place in F q k P and P + can be constructed from two independent copies of P as follows: The k th user chooses independently and uniformly two symbols Uk and U k in F q k, then he calculates Xk = U k + U k and Xk = U k, and he finally sends X k through the first copy of P and X k through the second copy of P Let Y and Y be the output of the first and second copy of P respectively P is the conditional probability distribution of Y Y given U Um, and P + is the conditional probability distribution of Y Y U Um given U Um Note that the transformation U,, Um, U,, Um X,, Xm, X,, Xm is bijective and therefore it induces uniform and independent distributions for X,, Xm, X,, Xm which are the inputs of the P channels Definition 0 Let B n } n be iid uniform random variables on, +} We define the MAC-valued process P n } n 0 by: P 0 := P, P n := P Bn n n Proposition [7] [8] The process I[S]P n } n 0 is a bounded super-martingale for all S,, m} Moreover, it s a bounded martingale if S =,, m} Proof: I[S]P = I[S]P + I[S]P = IX S; Y X S c + IX S; Y X S c = IX SX S; Y Y X S c X S c = IU SU S; Y Y U S c U S c = IU S; Y Y U S c U S c + IU S; Y Y U S c U S c U S IU S; Y Y U S c + IU S; Y Y U U mu S c = I[S]P + I[S]P + Thus, E I[S]P n+ Pn = I[S]P n + I[S]P + n I[S]P n, and I[S]P n log q i for all i S S,, m}, which proves that I[S]P n } n 0 is a bounded super-martingale If S =,, m}, the inequality becomes equality, and I[S]P n } n 0 is a bounded martingale From the bounded super-martingale convergence theorem, we deduce that the sequences I[S]P n } n 0 converge almost surely for all S,, m} Since I[S]P + I[S]P + I[S]P S,, m}, then J P + J P + J P, but this subset relation can be strict if one of the inequalities is strict for a certain S,, m} Nevertheless, for S =,, m}, we have IP + IP + = IP, so at least one point of the dominant face of J P is present in J P + J P + since the capacity region is a polymatroid Therefore, the symmetric sum capacity is preserved, but the dominant face might lose some points Definition In order to simplify our notation, we will introduce the notion of generalized matrices: A generalized matrix A = A,, A l F m i l i is a collection of l matrices F m i l i denotes the i= set of m i l i matrices with coefficients in F pi If l i = 0 in A = A,, A l F m i l i, we write A i = ø In case A i = ø for all i, we write A = ø i= A generalized vector x = x,, x l i= F m i is a collection of l vectors Addition of generalized vectors is defined as component-wise addition

22 The transposition of a generalized matrix is obtained by transposing each matrix of it: A T = A T,, A T l A generalized matrix operates on a generalized vector component-wise: if A and x i= F m i, then y = A T x ø T x i = 0 i= F l i pi i= F m i l i is defined by y = A T x,, A T l x l By convention, we have A generalized matrix A is said to be full rank if and only if each matrix component of it is full rank l The rank of a generalized matrix A F m i l i is defined by: ranka = ranka i The logarithmic rank of a generalized matrix is defined by: lranka = i= i= l ranka i log If A is a generalized matrix satisfying A i ø and A j = ø for all j i, we say that A is an ordinary matrix and we identify A and A i Definition Let P : Y be an m-user MAC, let A be a full rank generalized i= F m i matrix We define the ranka-user MAC P [A] : P [A]y u = i= l i= pm i l i i i= F l i pi Y as follows: x l i= Fm i A T x= u F m i l i P y x The main result of this section is that, almost surely, P n becomes a channel where the output is almost determined by a generalized matrix, and the convergence is almost surely fast: Theorem 7 Let P : Y be an m-user MAC Then for every 0 < ɛ < log, and for every i= F m i 0 < β < we have: s, +} n : A s lim n i= F m i r i,s, A s is full rank, IP s lranka s < ɛ, IP s [A s ] lranka s < ɛ, ZP s [A s ] < βn} = i= Proof: Since G := i= F m i is an abelian group, we can view P as a channel from the Abelian group G to Y Note that any subgroup of an Abelian grous normal Therefore, from theorem 6 we have: s, +} n : H s subgroup of G, lim n IP s log G/H s < ɛ, IP s [H s ] log G/H s < ɛ, ZP s [H s ] < βn} = Let s, +} n such that that there exists a subgroup H s of G satisfying: IP s log G/H s < ɛ

23 IP s [H s ] log G/H s < ɛ ZP s [H s ] < βn From the properties of abelian groups, there exist l integers: r,s m,, and r l,s m l such that G/H s is isomorphic to Note that r i,s can be zero Therefore, there exists a surjective homomorphism f s : i= F m i vice versa i= i= F r i,s F r i,s, such that for any x For all i l, and all j m i, define the vector e i,j i= F m i, f s x can be determined from x mod H s and i= F m i as having all its components as zeros except the i, j th component which is equal to The order of e i,j is Let y i,j = y i,j, y i,j,, y i,j = f s e i,j F r i,s, if y i,j 0 then the order of y i,j must be equal to l i= If y i,j i 0 for a certain i i, then p i divides the order of y i,j which is a contradiction Therefore, we must have y i,j i = 0 for all i i l m i l m i Now for any x F m i, we have x = x i,j e i,j, therefore, f s x = x i,j y i,j Since y i,j i i= = 0 for all i i, then f s x = A T s x, where A s = A,s,, A l,s i= j= matrix whose components are given by A i,s = [ y i, i y i, i Moreover, we have: Recall that for any x lranka s = i= i= y i,m i i i= i= l r i,s log = log p r i,s i = log G/H s i= F m i r i,s j= is a generalized ] T A s is full rank since f s is surjective F m i, A T s x = f s x can be determined from x mod H s and vice versa, we conclude that P s [H s ] is equivalent to P s [A s ] Therefore: s, +} n : A s F m i r i,s, A s is full rank, lim n i= IP s lranka s < ɛ, IP s [A s ] lranka s < ɛ, ZP s [A s ] < βn} = A Polar code construction for MACs Choose 0 < ɛ < log, 0 < β < β <, and let n be an integer such that n β n < βn p m i i i= E n > n ɛ l m i log i=

24 3 where E n = s, +} n : A s i= F m i r i,s, A s is full rank, IP s lranka s < ɛ, IP s [A s ] lranka s < ɛ, ZP s [A s ] < β n Such an integer exists due to theorem 7 For each s, +} n, if s / E n set F s, i, j = i,, l} j,, m i }, and if s E n choose a generalized matrix A s = A,s,, A l,s that satisfies the conditions in E n For each i l choose a set of r i,s indices S i,s = j, j ri,s },, m i } such that the corresponding rows of A i,s are linearly independent, then set F s, i, j = if j / S i,s, and F s, i, j = 0 if j S i,s F s, i, j = indicates that the user i, j is frozen in the channel P s, ie, no useful information is being sent A polar code is constructed as follows: The user i, j sends a symbol U s,i,j through a channel equivalent to P s If F s, i, j = 0, U s,i,j is an information symbol, and if F s, i, j =, U s,i,j is a certain frozen symbol Since we are free to choose any value for the frozen symbols, we will analyse the performance of the polar code averaged on all the possible choices of the frozen symbols, so we will consider that U s,i,j are independent random variables, uniformly distributed in F pi s, +} n, i,, l}, j,, m i } However, the value of U s,i,j will be revealed to the receiver if F s, i, j =, and if F s, i, j = 0 the receiver has to estimate U s,i,j from the output of the channel We associate the set, +} n with the same strict total order < that we defined earlier Namely, s s n < s s n if and only if there exists i,, n} such that s i =, s i = + and s j = s j j > i Encoding: Let P s } s,+} n be a set of n independent copies of the channel P P s should not be confused with P s : P s is a copy of the channel P and P s is a polarized channel obtained from P as before Define U s,s,i,j for s, +} l, s, +} n l, 0 l n inductively as: U ø,s,i,j = U s,i,j if l = 0, s, +} n U s ;,s,i,j = U s,s ;+,i,j + U s,s ;,i,j if l > 0, s, +} l, s, +} n l U s ;+,s,i,j = U s,s ;+,i,j if l > 0, s, +} l, s, +} n l The user i, j sends U s,ø,i,j through the channel P s for all s, +} n Let Y s be the output of the channel P s, and let Y = Y s } s,+} n We can prove by induction on l that the channel U s,s Ys } s has s as a prefix, U s } s <s is equivalent to P s In particular, the channel U s Y, U s } s <s is equivalent to the channel P s Decoding: If s / E n then F s, i, j = for all i, j, and the receiver knows all U s,i,j, there is nothing to decode Suppose that s E n, if we know U s } s <s then we can estimate U s as follows: If F s, i, j = then we know U s,i,j We have F s, i, j = 0 for r i,s values of j corresponding to r i,s linearly independent rows of A i,s So if we know A T i,su s, we can recover U s,i,j for the indices j satisfying F s, i, j = 0 Since A T s U s Y, U s } s <s is equivalent to P s [A s ], we can estimate A T s U s using the maximum likelihood decoder of the channel P s [A s ] Let D s Y, U s } s <s be the estimate of U s obtained from Y, U s } s <s by the above procedure This motivates the following successive cancellation decoder: ˆ U s = U s if s / E n ˆ U s = D s Y, ˆ Us } s <s if s E n }