On the Achievable Rate Region of the K-Receiver Broadcast Channels via Exhaustive Message Splitting

This paper focuses on K-receiver discrete-time memoryless broadcast channels (DM-BCs) with private messages, where the transmitter wishes to convey K private messages to K receivers. A general inner bound on the capacity region is proposed based on an exhaustive message splitting and a K-level modified Marton’s coding. The key idea is to split every message into ∑j=1KKj−1 submessages each corresponding to a set of users who are assigned to recover them, and then send these submessages via codewords chosen from a K-level structure codebooks. To guarantee the joint typicality among all transmitted codewords, a sufficient condition on the subcodebooks’ sizes is derived through a newly establishing hierarchical covering lemma, which extends the 2-level multivariate covering lemma to the K-level case with more intricate dependences. As the number of auxiliary random variables and rate conditions both increase exponentially with K, the standard Fourier–Motzkin elimination procedure becomes infeasible when K is large. To tackle this problem, we obtain a closed form of achievable rate region with a special observation of disjoint unions of sets that constitute the power set of {1,⋯,K}. The proposed achievable rate region allows arbitrary input probability mass functions and improves over previously known achievable (closed form) rate regions for K-receiver (K≥3) BCs.


I. INTRODUCTION
The 2-receiver discrete-time memoryless broadcast channels (DM-BCs) are first introduced by Cover [3], who proposed the prestigious superposition coding that outperforms the traditional time-division strategy.However, superposition coding is optimal only for certain categories of broadcast channels such as degraded, less noisy and more capable BCs [4].The best known inner bound on the capacity region of DM-BCs is achieved by Marton's coding with message splitting [2].The key idea is to split each source message into common and private parts, where the common part is encoded into a cloud-center codeword, and two private parts are encoded into two separate codewords.To enlarge the achievable rate region, the submitted codewords are jointly typical which is guaranteed by a sufficient condition on the sizes of subcodebooks established by the covering lemma [1].
For the general K-receiver DM-BCs, most previous work mainly focused on superposition coding and message splitting (or merging) [5]- [7] wherein they require certain Markov chains for auxiliary random variables (RVs).Especially, [7] gives a general inner bound based on superposition coding and rate-splitting using notions from order theory and lattices wherein each receiver decodes its intended message (common or private) along with the partial interference designated to it through rate-splitting.The Marton's coding with 2-level superposition coding structure (consisting of one cloud-center codebook and K satellite codebooks) can be easily constructed and analyzed by the multivariate covering lemma and packing lemma [1].
In this paper, we consider K-receiver DM-BCs including only private messages.To inherit the characteristics of superposition coding and combine them with Marton's coding, a general inner bound is proposed based on an exhaustive message splitting and a K-level modified Marton's coding.More specifically, every message is split into K j=1 K j submessages, with each corresponding to a set S belonging to the power set of {1, . . ., K}.The submessages respecting to S are encoded into an exclusive codeword and will be decoded by receiver j if j ∈ S. To obtain a potentially larger region, we enlarge the subcodebooks sizes and use a K-level Marton's coding to send all codewords that are jointly typical with each other.
Note that there are mainly two challenges on establishing the final form of achievable rate region.The first one is how to derive rate conditions such that all transmitted codewords are jointly typical.The solution is related to the covering lemma.Unfortunately, the known multivariate covering lemma only deals with sequences which are generated conditionally independently [1], while in our scheme there are (2 K −1) RVs and the sequences are generated under more intricate dependence.We solve this problem by dividing (2 K − 1) RVs into K levels and process them hierarchically, leading to a new lemma called hierarchical covering lemma.The second challenge lies on how to apply Fourier-Motzkin elimination procedure to obtain the final form of the inner bound.As the number of auxiliary RVs and rate conditions increase linearly with (2 K − 1), it's infeasible using the standard Fourier-Motzkin elimination procedure, in particular when K is large.To tackle this problem, we aggregate the rate of submessages based on a special observation of disjoint unions of sets that constitute the power set of {1, . . ., K}, and finally establish the final form of achievable rate region.
Transmitter p(y 1 , . . ., y K x) p(y 1 , . . ., y K x) Fig. 1: Broadcast Channel for K receivers In our proposed scheme, the exhaustive message splitting enables each user to decode different fine particles of the source message according to its observed signal, and the K-level Marton's coding can enlarge the inner bound by allowing arbitrary dependence among the input RVs, rather than satisfying certain Markov chains as in [5]- [7].Compared to all previous work on inner bounds of K-receiver DM-BCs with only private messages, our inner bound includes their regions as special cases.
Let an all-one column vector (1, . . ., 1) with a specified dimension be denoted by 1. Albeit with an abuse of notation, we use a string of elements in a singleton to represent the set with only one element, e.g.,

II. CHANNEL MODEL
Consider a K-receiver DM-BC with only private messages depicted in Fig. 1.The setup is characterized by a input alphabet X , K output alphabets (Y k : k ∈ K), and a collection of channel transition pmfs p y 1 , . . ., y K x .At time i ∈ [1 : n], the transmitter sends the channel input x i ∈ X , receiver k ∈ K observes the output y k,i ∈ Y k .
The goal of the communication is that the transmitter conveys private messages M k to receiver k, for k ∈ K, respectively.Each M k is independently and uniformly distributed over the set M k [1 : 2 nR k ], where R k denotes the communication rate of receiver k.
The encoder maps the messages (M 1 , M 2 , . . ., M K ) to a sequence x i ∈ X : and receiver k ∈ K uses channel outputs y n k to estimate Mk as a guess of messages M k : A rate region (R k : k ∈ K) is called achievable if for every blocklength n, there exists an encoding function f (n) and K decoding functions g K such that the error probability tends to zero as the blocklength n tends to infinity.The closure of the set of achievable rate tuple (R k : k ∈ K) is called the capacity region.

III. PRELIMINARY
In this section, we present decomposition of sets which will be used to in Section IV and V. Given a set S ∈ P(K), k ∈ S and l, l ′ ∈ K with l ≤ l ′ , define To simplify notations, we let where for some Note that k∈T A(i) denotes the subset of P(K) whose elements contains at least one index belonging to T , i.e., k∈T and B π (i) contains all sets which include π(i + 1), and are subsets of K \ {π(1), . . ., π(i)}.Also, we can have the following relation: Eq. ( 4) describes how we decompose i∈T A(i) For example, consider the case K = 3, T = {1, 2, 3} and π = (2, 1, 3), we have We can easily find that (4) is satisfied.Fig. 2 is given for illustration.To simplify the notation, an arbitrary collection of sets, e.g., {1, 12, 123}, is recognized as {{1}, {1, 2}, {1, 2, 3}} With the definitions above, we can also obtain the following decomposition: where Here k∈T A l (k) denotes the subset of P(K) whose element each has cardinality l and contains at least one index belonging to T , i.e., i∈T and B l (i) contains all the sets with cardinalities l, containing π(i + 1), and being contained in K \ {π(m)} i m=1 .
Fig. 2: The illustration of the disjoint decomposition of k∈T A(k) in the case for K = 3, T = {1, 2, 3} and π = (1, 2, 3).The idea is that every time we pick a collection of sets descended from a set in the layer l, the number of unpicked sets in layer l − 1 is one, where layer l contains all subsets of K with cardinality l.Therefore, k∈[

IV. MAIN RESULTS
Theorem 1.Using the notations presented in Section III, a rate region for some pmf p(U (A)) and a function X = f (U (A)).
Proof.The achievable scheme is based on an exhaustive message splitting and a modified Marton's coding.More specifically, each message M k , for k ∈ K, is split into )) and will be decoded by receiver j if j ∈ S.Then, a modified Marton's coding is applied to send (2 K − 1) codewords (u n S : S ⊆ K, |S| = 1, . . ., K) that are jointly typical with each other.See more detailed proof in Section V.
To ensure arbitrary input pmfs p(U (A)) in the inner bound above, we need to establish a sufficient condition on the sizes of subcodebooks, which is related to the covering lemma.The multivariate covering lemma in [1] only consider sequences generated under simple dependence, e.g., given a sequence u n 0 ∼ P U0 .In our scheme there are (2 K − 1) RVs, represented by (U S : S ∈ P(K), S = ∅), and each symbol of codeword U n S is generated conditionally independent according to p(u S |u(A(S)\S)).At first glance it seems overwhelming to derived the sufficient condition such that all transmitted codewords are jointly typical.However, by dividing all subcodebooks into K levels and processing them recursively in a hierachical manner, we obtain the following new lemma: 2 nrS ], be pairwise conditionally independent sequences, each distributed according to S ∈ A l are mutually conditionally independent given (U n S (m S ) : S ∈ (A(S) \ S).Then there exists δ(ǫ l ) that tends to zero as ǫ l → 0 such that for all J l ⊆ A l and l ∈ [1 : (K − 1)].
Proof.See the proof in Appendix A.
Lemma 2. Consider the inequality condition in hierarchical covering lemma: then for all i ∈ [N ], we have With simple justification, the sum of lower bounds for S∈J l i r S , ∀i ∈ [N ] is not greater than the lower bound for S∈J l r S , which means combinations of split inequalities will not induce a smaller region bounded by the overall inequalities that simultaneously restrain all variables, Proof.It's equivalent to prove that for some pmf p(u 1 u 2 u 3 u 12 u 13 u 23 u 123 ) and a function x = f (u 1 u 2 u 3 u 12 u 13 u 23 u 123 ).

Remark 1 (Comparison with superposition coding and Marton's coding). Our achievable region generalizes that introduced by standard Marton's coding and superposition coding, i.e., both of them are special cases of our general-form intended coding.
Furthermore, the rate region in Theorem 1 contains the regions resulted from the two aforementioned coding schemes.
• Our coding degenerates into Marton's coding by setting U S = const, ∀S ∈ A 2:(K−1) , which indicates our rate region contains that derived by Marton's coding; • Since superposition coding is optimal for degraded/less noisy/more capable DM-BC, which is barely guaranteed in many cases, i.e., without knowing the concrete relation according to the Markov chain, our rate region contains that derived from superposition coding.In the future work, we will evaluate our rate region for some specific DM-BCs to show that our coding scheme strictly improves previously known inner bounds.

V. ACHIEVABLE CODING SCHEME FOR THEOREM 1
We first present our scheme for 3-receiver DM-BC as an illustration, and then extend it to general K-receiver DM-BC model for K ≥ 2.
To simplify notations, we denote C S (m S ) by C S (m) with cognition of the subscript of m from C S , similarly for U n S (m S , l S ) denoted by U n S (m, l) and its realization u n S (m S , l S ) denoted by u n S (m, l).
5) Analysis of the probability of error: Assume without loss of generality that is sent and let L 12 , L 13 , L 23 , L 1 , L 2 , L 3 be the index tuple of selected sequences We note that the subcodebook C S (m) consists of 2 n( RS −RS ) i.i.d.U n S (m, l) sequences, ∀S ∈ {1, 2, 3, 12, 13, 23}.By the hierarchical covering lemma (with r S = RS − R S ), we obtain a set of constraints for S ∈ {12, 13, 23} For receiver 2 and 3, the corresponding inequalities can be derived in a similar way.
From the lower bound of linear combinations of r S and the upper bound of linear combinations of RS about receiver 1, 2 and 3, eliminating R 123 , R 12 , R 13 , R 23 and R 1 , R 2 , R 3 by our elimination procedure, an example given in Figure .2, yields the characterization (12a).

B. Coding scheme for
Similarly to K = 3 case, we use the following notations and M S M 1,S , . . ., M K,S . 1) Codebook generation: Fix a pmf p(u(A)) and function x(u(A)) and let RS ≥ R S , ∀S ∈ A. Randomly and independently generate ).For all S ∈ A 1:(K−1) , construct a subcodebook C S (m) consisting of 2 n( RS −RS ) i.i.d.generated sequences u n S (m, j), (m S , j S ) ∈ To send the message tuple (m 1 , . . ., m K ) = (m S : S ∈ A), transmit x n (m S : S ∈ A).
3) Decoding.: Let ǫ > ǫ ′ .Decoder i, i ∈ K, declares that mi is sent if it is the unique message such that . 4) Analysis of the probability of error.: Assume without loss of generality that (M 1 , . . ., M K ) = (M S : ∀S ∈ A) = (1, . . ., 1) and let (J S : S ∈ A 1:(K−1) )) be the index tuple of the chosen sequences (U S : S ∈ A 1:(K−1) ) ∈ S∈A 1:(K−1) C S (1).Then decoder i makes an error only if one or more of the following events occur: Therefore, the probability of error for decoder i is upper bounded as ).To bound P (E 0 ), we utilize the hierarchical covering lemma proposed before.Noticing that sequences among the subcodebooks C S (m), S ∈ A l , l ∈ [1 : (K − 1)] are mutually conditionally independent, the usage of hierarchical covering lemma is straightforward with r S = RS − R S .Hence, P (E 0 ) tends to zero as n → ∞ if (10) holds for all l ∈ [1 : (K − 1)] and J l ∈ A l .
To bound P (E c 0 ∩ E i,1 ), by the conditional typicality lemma [1] it tends to zero as n → ∞. ), then by the conditional coding distribution, for all S ′ ∈ B i (S), we have U S ′ (M, j) / ∈ C S ′ (1).Thus, ∀l ∈ K, ∀J ⊆ A(i) by identifying S∈J B i (S), we obtain inequalities with the help of packing lemma: (17) 5) Eliminating ( Rk,S , r S : k ∈ K, S ∈ A): Due to massive numbers of Rk,S and r S , using standard Fourier-Motzkin elimination to obtain the achievable rate region is disastrous.We first present observations which help the elimination, and then find all valid constraints for k∈T R k for all T ⊆ K. Observation: Let S(i, J ) S∈J B i (S), and denote the right hand term of (17) by I S(i,J ) , thus (17) can be rewritten as below.
S∈S(i,J ) RS < I S(i,J ) .
(19b) b) If there exists i ′ and J ′ such that S(i, J ) ⊆ S(i ′ , J ′ ), then I S(i,J ) ≤ I S(i ′ ,J ) ′ . (20) From the rate-splitting procedure, we have R i = S∈A(i) R i,S , for all i ∈ K. Thus in order to find rate constraints for k∈T R k , we must find all valid constraints for k∈T S∈A(k) R k,S .By definitions in (16), we have (23) a) follows by the chain rule of entropy and (11); (b) holds because conditioning reduces entropy.Corollary 1. Define I 12 := min i∈{1,2} {I(U 12 U 123 ; Y i )}, I 13 := min i∈{1,3} {I(U 13 U 123 ; Y i )}, I 23 := min i∈{2,3}