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For a multiple input channel, one may define different capacity regions, according to the

The behavior of a multiple user channel is quite different from that of a point-to-point channel. For example, it is well known that the choice of the error criterion,

The difference of the two capacity regions could be intuitively explained as follows. For a channel with two inputs, two senders may not always well cooperate (although in most cases they may cooperate well). If the criterion of maximum probability of error is used, the worst case has to be counted. As a direct consequence, the capacity region in this case may be strictly smaller than the one under the criterion of average error probaility (where only the average case is considered). This also explains why for a broadcast channel (BC), which is a channel with one single input, the maximum-error-probability capacity region is always equal to the average-error-probability capacity region [

In network information theory, a lot of excellent work has been done on the criterion of average error probability (for example c.f. [

Another interesting issue is

Finally, let us take a look at the impact on the capacity regions made by different types of codes,

In general, for an MAC or IC, we can define the following 6 capacity regions:

average-error-probability capacity regions, without and with feedback;

maximum-error-probability capacity regions of random codes, without and with feedback;

maximum-error-probability capacity regions of deterministic codes, without and with feedback.

Similarly, when speaking of capacity region of a TWC, we must distinguish which type of codes and error criterion are used.

We observe that the relations of these capacity regions of MAC, IC, and TWC are similar, due to the fact that they all have multiple inputs. We refer to them as multiple input channels. The goal of the paper is to clarify the relation of the capacity regions of multiple input channels, but not to determine them. To simplify the discussion, we first assume that all channels have two inputs, and then extend them to the general case.

First, we show that

Consider the remaining 4 capacity regions. Recall that for a MAC which employs deterministic codes, feedback may enlarge the average- and maximum-error-probability capacity regions; and the maximum-error-probability capacity region may be strictly smaller than the average-error-probability capacity region. In particular, G. Dueck [

The choice of criterion of error makes no difference for random codes.

The maximum-error-probability capacity region for deterministic codes without feedback is contained by all the other capacity regions; and, the contained relation can be strict.

The average-error-probability capacity region with feedback contains all the other capacity regions; and, the containing relation can be strict.

The maximum-error-probability capacity region for deterministic codes with feedback may be strictly larger than, or strictly smaller than the average-error-probability capacity region without feedback.

For deterministic codes with feedback, if the channel is deterministic, the choice of criterion of error makes a difference; whereas if the channel is non-deterministic, it makes no difference.

When deterministic codes are employed, feedback may enlarge both the average- and maximum-error-probability capacity regions. However, the reasons may be different. To illustrate it, we provide a class of MAC, for which feedback may enlarge the maximum-error-probability capacity regions, but not the average-error-probability capacity regions. As

Moreover,

The rest of the paper is organized as follows. In the next section, we briefly describe our problems and review the contraction channel which was introduced in [

Consider a MAC, which has two inputs and one output. Two senders aim to send messages from their own message sets _{i}, i_{i}, i_{i}, i_{i}, i_{1}_{2}.

Consider an IC, which has two inputs and two outputs. Two senders aim to send their own messages from sets_{1} and_{2} to two receivers, respectively. Similarly, a code _{i}, i_{j}, j

For a MAC/IC, we say that feedback from the output ^{(}^{j}^{)} (for a MAC, the output index _{i},^{(}^{j}^{)} causally. I. e., the codeword can be written as

where

A TWC has two inputs and two outputs, and is used by two users for exchanging messages. The first user sends messages to the first input of the channel, and receives messages from the second output of the channel; and the second user sends messages to the second input, and receives messages from the first output. The symbol sent by a user at the time

To distinguish MAC, IC, and TWC from point-to-point channels and BC, we refer them as multiple input channels. Note that in general a MAC may have more than two inputs. In the following, we first consider a MAC with only two inputs; and results obtained will be extended to the general case later.

It is well known that, a discrete channel
_{1}_{2}, . . . , _{n}

where

For a multiple input channel, let _{e}_{1}_{2})) be the _{1}_{2}) ∈ _{1} _{2} (_{1} and _{2} are sent simultaneously through the channel). Then the

respectively, where _{i}_{i}|, i

We say that an encoder _{i}_{i}

In this paper, we assume that the random key is uniformly distributed. A random code may have two random encoders, or one random encoder and one deterministic encoder. If a random code has two random encoders, we assume that the two random keys _{1} and _{2} are independent. We shall see in the next section that the number of random encoders makes no difference in the capacity regions of an (ordinary) multiple input channel. However, we shall see in Subsection 6.2, that random codes with two random encoders, and random codes with one random encoder and one deterministic encoder may have different capacity regions for an arbitrarily varying MAC. Thus, we do not need to distinguish the number of random encoders, when speaking of the capacity region, until Subsection 6.2.

For a random code _{r}_{e}_{r},_{1}_{2})) for a fixed message pair (_{1}_{2}) is a function of the random keys, and thus is a random variable. Therefore, it must be replaced by its expectation in the definitions of error probabilities. That is, the _{r}

respectively, where

The capacity regions of a multiple input channel are defined in a standard way. According to the presence of feedback, criterions of probabilities of error, and types of codes (random or deterministic), formally one might have 8 capacity regions.

As a matter of fact, every random code such that its average probability of error defined by (

For the criterion of the average probability of error, we have two capacity regions for a given channel _{f}

For the criterion of maximum probability of error, one may define capacity regions of random codes without and with feedback, denoted by _{r}_{r,f}_{d}_{d,f}

As a special kind of random codes, deterministic codes may not have larger capacity regions than random codes; and, feedback may not reduce capacity regions. Therefore, relationship of the six capacity regions are presented as follows.

Let C_{r} be a random code with a random encoder Φ_{1} and a deterministic encoder φ_{2} for a multiple input channel. Then {Φ_{1}(m_{1}) : m_{1} ∈ ℳ_{1}} is not necessary to be a set of independent random variables. However, one can obtain a random code
_{r}, such that_{1} ∈ _{1}

_{1} ∈ _{1}_{1}(_{1})

_{i}_{i}_{e}_{r},_{1}_{2}))_{1}_{2}) ∈ _{1} _{2}

_{1}(_{1})_{1} ∈ _{1}} _{e}_{r},_{1}_{2}))_{1} ∈ _{1}} _{2} ∈ _{2}

_{i}, i_{i}_{i}_{i}_{i}, i_{e}_{r},_{1}_{2}))_{1} ∈ _{1} _{2}(_{2}) = ^{(2)}.

In [

Dueck’s contraction channel is a discrete memoryless channel with a generic

Let

Let _{1} be a channel _{1} :
_{i}_{1} ∈
_{1}(_{1}_{1}_{2}) = 1 if and only if _{1}_{2}) = _{1}; and _{2} be the identity channel on {0_{2} ∈
_{2} ∈
_{2}(_{2}_{2}) = 1 if and only if _{2} = _{2}. Then the generic channel _{1}_{2}_{1}_{2}) = _{1}(_{1}_{1}_{2})_{2}(_{2}_{2}), for _{i}_{j}

Dueck [

_{1} (_{2}) and the second (first) output of the channel with the alphabet

Notice that a codeword ^{(2)}, which is sent to the second input of the contraction channel, always can be recovered from both outputs correctly with probability 1; and, the second output is a function of the first output of the channel. Therefore, one may assume that the receiver only accesses the first output of channel in scenario (2). Moreover, a code is decodable in scenario (1) if and only if it is decodable in scenario (2). Thus, its capacity regions in scenarios (1.w) and (2.w) are the same; and, its capacity regions in scenarios (1.f) and (2.f) are the same.

Let

where _{out}

In this section, we show that randomization at one encoder may enhance a code under the criterion of maximum error probability for a multiple input channel, without or with feedback, to achieve the average-error-probability capacity region. That is, for a multiple input channel _{r}_{r,f}_{f}

First, let us recall Chernoff bound which will be applied in this paper repeatedly.

_{1}_{2}_{L}_{l}_{l}

where the equality in (a) holds because _{1}_{2}_{L}_{l}_{l}^{x}

_{f}

_{r}_{f}_{r,f}

Since _{r}_{r,f}_{f}_{f}

Let _{f}_{1}_{2}) be a rate pair in _{f}_{i}, i

where _{i}, i

for _{2} ∈ _{2}; and,

and _{2}_{,}_{1} = _{2}\_{2}_{,}_{0}. Then it follows from (

This gives us
_{2}_{,}_{1}, we have a subcode _{0} of _{1}_{2}) ∈ _{1}_{2}_{,}_{0}, _{e}_{0}_{1}_{2})) = _{e}_{1}_{2})).

Next we construct a code with a random encoder and a deterministic encoder under the criterion of maximum error probability. To this end, we let ∑ be the permutation group on _{1}; and, _{k}, k^{2}_{1}

For a _{0}) as a code with the same message sets as _{0}, encoding functions _{1}_{,σ}_{1}(_{2}_{,σ}_{2}(·), and a suitable modification in decoder(s) (_{0} is
_{e}_{0})_{1}_{2})) = _{e}_{0}_{1})_{2})), for all (_{1}_{2}) ∈ _{1} _{2}_{,}_{0}. Thus by ^{2}}_{1}_{2}) ∈ _{1}_{2}_{,}_{0},

where (a) and (b) hold by _{2} ∈ _{2}_{,}_{0}. Consequently, by Chernoff bound, ^{2}_{1}_{2}) ∈ _{1}_{2}_{,}_{0},

Thus, by union bound, we have that with a probability at least
_{1}_{2}_{n}_{2}) has a realization (_{1}_{2}_{n}_{2}), such that for all (_{1}_{2}) ∈ _{1}_{2}_{,}_{0},

Now we are ready to construct a random code _{r}

We first prepare a code _{0}, and then find permutation groups (_{1}_{2}_{n}_{2}), satisfying (

The first sender uniformly at random generates a “key” _{1} from {1^{2}}, and then sends the outcome _{1} of _{1} in the first block, by using a code with _{1}_{,σ}_{k1} (_{1}) through the channel in the second block, if he/she wants to send the message _{1} and the outcome of the key is _{1}. At the same time, to send the message _{2}, the second sender sends _{2}_{,σk1}_{2})(= _{2}(_{2})) through the channel. That is, the two senders send their messages by using the code _{k}_{1} (_{0}) in the second block. Notice that the second sender does not need to know the outcome of the key, because his/her encoding function does not depend on the permutation _{k}_{1} . The ratio of the length of the first block in the whole length of the code can be arbitrarily small, when _{i}

At the receiver(s), first the key is decoded as _{k̂}_{0}).

Let _{0} be the event, of that an error occurs in the first block; and
_{1}_{2}) ∈ _{1}_{2}_{,}_{0},

where (a) holds since _{e}_{r}_{1}_{2})) ≤ 1 with probability 1; (b) holds since the average probability of error in the first block is no larger than

The part of the proof will be used in the next section and Section 6, for readers’ convenience, we summary them as the following lemma:

_{1}_{2}.

_{0} _{1} _{2}_{,}_{0}_{2}_{,}_{0} ⊂ _{2}_{e}_{2}) _{1}_{e}_{2}) _{2} ∈ _{2}_{,}_{0}.

_{k}, k^{2}_{0}

^{2} _{k}, k^{2} _{1}

Next, let us look into the reason

The proof of the theorem tells us that adding randomness with a vanishing rate at one single encoder is sufficient for a code fulfilling the criterion of maximum error probability to achieve the average-error-probability capacity region. That is, the cost for the randomization is very low. This suggests us to use random coding in the network communication, because in any sense a small maximum probability of error is better than a small average probability of error, although in most cases the latter is acceptable.

By Theorem 3.2, the 6 capacity regions in (

So far we have known that there is a MAC _{f}_{d}_{d,f}

By applying Cover-Leung inner bound [_{f}_{d,f}_{d,f}

In this section, we apply Lemma 3.3 to non-deterministic discrete memoryless and Gaussian multiple input channels with feedback.

An output of a multiple input channel is

_{0}, there exist two codewords ^{(}^{i}^{)}_{0}, and a subset

with probability at least 1

^{n}^{0(}^{α}^{+}^{o}^{(1))} parts with nearly equal probability, ^{(1)}^{(2)}) are the input to the channel.

Typically, we require that _{0} → ∞.

Obviously, for a non-deterministic discrete memoryless or a Gaussian multiple input channel, one may always choose a pair of codewords (^{(1)}^{(2)}) of length _{0}, such that the random output of the channel has positive entropy _{0}^{(1)} and ^{(2)} are sent to the channel. It is easy to see that, by taking the set of typical sequences as ^{n}^{0}^{α}^{n}^{0(}^{α}^{+}^{o}^{(1))} output typical sequences with nearly equal probabilities, if the output entropy is _{0}^{n}^{0(}^{α}^{+}^{o}^{(1))} parts, such that each part contains nearly equal number of typical sequences, and therefore has nearly equal probability, similarly for a Gaussian channel. That is, (*) follows.

Feedback in memoryless channel usually may play the following 3 roles:

(2)

(3)

The feedback will play the 3rd role in the following corollary. That is, by the property (*), we shall use a block of length _{0} to generate a random key of size

Note that

Therefore, if the topological interior of _{f}_{d,f}_{f}

_{f}

By (_{1}_{2}) ∈ _{f}_{1}_{2} is the message set of _{0} of _{1}_{2}_{n}_{2}) as described in Lemma 3.3. Next, we choose (^{(1)}^{(2)}) with the smallest possible _{0} in the property

Obviously,
_{0} no less than

We assume that the channel is a MAC. Our code consists of two blocks. In the first block, two senders send ^{(1)} and ^{(2)} (in the property (*)) respectively, no matter what messages they want to send. After the first block, both the first sender and the receiver learn the output _{0}. By assumption, we have that

In the case that _{0} does not occur, we assume that _{1} and _{2}_{k}_{0}) as described in Lemma 3.3. Recall (_{1}_{2})

which with (_{i}

Notice that the second sender does not need to know

Differently from an MAC, a TWC or IC has two outputs. Let _{i}

Corollary 4.1 tells us that, for a non-deterministic channel with feedback, the error criterion makes no difference on its capacity region. But, it is not always true in the general case, because we have learnt from the contraction channel [

Motivated by the following, we present two subclasses of MAC, as examples, in this section.

We have seen that one can apply feedback, not only for shifting the private messages to the common messages, but also for extracting common randomness, to help the transmission under the criterion of the maximum probability of error. On the other hand, the common randomness may not help the transmission under the criterion of the average probability of error, as random codes and deterministic codes have the same average-error-probability capacity region. So, it is expected that there exists a multiple input channel, for which feedback may enlarge the maximum-error-probability capacity region, but not the average-error-probability capacity region.

We wonder if there is a non-deterministic MAC, whose average- and maximum-error-probability capacity regions are different, since Dueck’s contraction channel is special for being deterministic.

First let us recall an inner bound on the average-error-probability capacity regions of discrete memoryless MAC

where
_{1}_{2}

for all (_{1}_{2}

It was shown by F. M. J. Willems [

We say that a MAC _{2} = _{1} ∈
_{1}_{2})

The maximum-error-probability capacity region of channels in

As
_{cl}_{d,f}_{f}

In fact, for all channels in

Since (ii) immediately follows from (i) and Corollary 4.1, it is sufficient for us to show (i).

To show (i), we only need to verify that, the average-error-probability capacity region of any channel in

where (_{1}_{2}_{X1X2Y}_{1}_{2}_{X1}(_{1})_{X2}(_{2})_{1}_{2}), for all (_{1}_{2}_{cl}_{cl}

To this end, we fix a _{1}_{2}) and the corresponding random output

due to the fact that by definition of
_{2} is uniquely determined by the random output

as _{1} and _{2} are independent. Then the bounds on _{2} and _{1} + _{2} in (

respectively. In addition, we notice that _{1} ≤ _{1}; _{2}) and _{2} ≤ _{2}) imply that _{1} + _{2} ≤ _{1}; _{2}) + _{2}). Thus for a

On the other hand, since for (_{1}_{2}_{2}; _{1}_{2}|_{1} + _{2} in (

That is, _{cl}_{1}_{2}). This completes our proof.

By Proposition 5.1, feedback may not enlarge the average-error-probability capacity region of a MAC in

Cover-Leung bound was obtained by the technique of

However, one can apply the 3rd role of feedback

In the previous subsection, we have that for a channel _{d,f}_{f}_{d}

Let _{*}. Let _{j}, j_{j}_{j}

for all _{1} ∈
_{2} ∈
_{1} ∈
_{2} = _{j}

Since the generalized contraction channels are in
_{1}_{2}) as given in (_{1} ∈

where (a) holds since _{1}; (b) holds since _{2} are independent. Next we let _{X2}(1) =

where _{1}; _{2} = 1) = _{1}; _{2} = 0) = _{*}; or, equivalently both _{X1|J} (·|0) and _{X1|J} (·|1) are optimal input of

To show that the maximum-error-probability capacity region of a generalized contraction MAC is strictly smaller than its average-error-probability capacity region, we need an outer bound on the maximum-error-probability capacity region. G. Dueck [

The isoperimetric problem is a basic problem in combinatorics, which asks how large at least, the boundary of a subset with a given cardinality in a discrete metric space has to be. Its asymptotic version is known as “Blowing Up Lemma”, by people in Information Theory (e.g., c.f. [

For a subset
^{n}

where _{H}

^{n}. Then if

Let

without loss of generality, we may assume
_{1}_{2}_{n}^{n}^{n}^{n}^{n}_{H}^{(2)}) symbols at the coordinates ^{(2)} is sent to the second input of the channel. In other words,
^{(2)}. Thus by converse coding theorem for the point-to-point channels, we have that for all ∈

where _{H}_{H}^{(2)})^{(2)} ∈

Now we have to maximize the right hand side of (^{l}^{n}

which is maximized by

where |^{+} = max{0

Because for a fixed _{*} achieves the maximum value at

and
_{*} ≥ 0, and
_{*} = 0, the outer bound can be rewritten as

In particular, when _{*} = 1 and
_{*}

if _{*}

In this section, we extend our results to compound channels, arbitrarily varying channels, and channels with more than two inputs. For the sake of simplicity, we will focus on the discrete memoryless MAC. As one can see, it is not that difficult to extend the results in Subsections 6.1 and 6.3 to Gaussian MAC and other multiple input channels. However, it is not very easy to extend Theorem 3.2 to arbitrarily varying Gaussian channel defined in the sense of [

A _{1}_{2}_{n}

if the sequences
_{r}_{1} _{2}, we denote by _{e}_{1}_{2}); _{e}_{r}_{1}_{2}); _{j}_{j}, j_{r}_{1}_{2}) is sent and the state of the channel is

respectively; and, for a random code _{r}

respectively. In the following, we extend the results in Sections 3, 4 to compound channels.

First we extend Theorem 3.2 by showing that

for all compound channels
_{2} into two parts: _{2}_{,}_{0}(_{2}_{,}_{1}(

if and only if _{2} ∈ _{2}_{,}_{1}(_{2}_{,}_{0} = ∩_{s}_{∈}_{
}ℳ_{2}_{,}_{0}(_{0}.

By applying Lemma 3.3-(2) to each channel in the set of

Then by the union bound, with probability at least
_{1}_{2}_{n}_{2}), (_{1,} _{2, . . . ,} _{n}_{2}), such that for all (_{1}_{2})∈ _{1} _{2}_{,}_{0} and all

We omit the rest parts of the proof of of (

The extension of Corollary 4.1 to a compound MAC is straightforward. The proof follows from the argument in Section 4. Note that the definition of the non-deterministic compound channel slightly makes a difference here. If we define a non-deterministic compound MAC as a compound MAC, such that there is a pair of input letter (_{1}_{2}), for which no _{1}_{2}_{1}_{2}) for which no _{1}_{2}_{1}_{2}) takes a block. Surely, the sender and the receiver know from which block(s) they may extract randomness, because a “deterministic block” always produces the same output letter. The number of the blocks makes no difference, since their rates will vanish as the length of the code increases. Thus, the extension of Corollary 4.1 follows.

Similar to a compound MAC, an _{1}_{2}_{n}

if the sequences
_{1}_{2}_{n}_{r}_{1}_{2}, we denote by _{e}_{1}_{2}); _{e}_{r}_{1}_{2}); _{j}_{j}, j_{r}_{1}_{2}) is sent to the channel, and the channel is governed by a state sequence

respectively; and, for a random code _{r}

respectively.

According to our knowledge, most known works on AVMAC focused on the average-error-probability capacity regions. By elimination technique, J. H. Jahn [

Actually, one can define the 3rd kind of probabilities of error, which we call

Accordingly, we denote the semi-average error probability capacity regions of deterministic codes, without and with feedback by _{d}_{d,f}

The difference among the criterions of the average, semi-average, and maximum error probability regions is obvious. At first, we have that

Secondly, we observe that _{d}_{d}_{d}_{a}_{m}_{m}_{a}_{1}_{2})) :
_{1}_{2}) ∈

for all _{1} ∈
_{2} ∈
_{1}_{2}) ∈
_{1}_{2}) ∈
_{d}_{d}_{d}

respectively. Obviously we have that

We have discussed in Section 3 that, randomization at one single encoder is sufficient for a random code under the criterion of maximum probability of error to achieve the average-error-probability capacity region, and therefore the number of random encoders makes no difference for an (ordinary) MAC. Now, we shall see that it does make a difference for an AVMAC. Let _{r,d}_{r,r}

To show (

This can be done by simply modifying the proof of Theorem 3.2, similar to the extension to compound channels in last subsection. For a _{1}_{2}) ∈ _{d}

for all _{2} ∈ _{2} and _{r,d}_{r,d}_{1}_{2}) ∈ _{1} _{2}

By (_{0} in Lemma 3.3, which is needed by us. Thus, it is sufficient to show that, (by replacing _{0} in Lemma 3.3-(2) with _{1}_{2}_{n}_{2}), such that for all (_{1}_{2}) ∈ _{1}_{2}

since the rest part of the proof will directly follow from the proof of Theorem 3.2, similar to proof of extension to compound channel in the last subsection. Applying Lemma 3.3 to all _{1}_{2})∈ _{1}_{2}

That is, with a probability no less than

Next, we show the opposite relation

To do that, we have to prove that, given _{r,d}_{1} and a deterministic encoder _{2}, with the maximum probability of error
_{r,d}_{2} ∈ _{2} and _{e}_{r,d}_{1}_{2}); _{1} ∈ _{1}} are independent. Then, applying Chernoff bound, _{2} ∈ _{2}

Consequently, applying the union bound, we have

which can be arbitrarily close to 0, for a sufficiently large _{1} exponentially increases as _{r,d}

Now we proceed to the proof of (_{d}_{r,r}

Similarly to the proof of Theorem 3.2, for a deterministic code ^{(}^{i}^{)} on _{i}, i^{(1)}^{(2)})(_{i}_{i}_{i}_{i}_{i}, i^{(1)})^{−1}(_{1}(^{(2)})^{−1}(_{2}(_{i}, φ_{i}, i_{1}_{2}) are message sets, encoders, and decoder of

Next we randomly, independently and uniformly generate ^{2} random permutations
^{2} on _{i}_{1} = 1^{2}_{2} = 1^{2}, are ^{4} independent random variables with expectations

for all fixed (_{1}_{2}) ∈ _{1} _{2} and

As a direct consequence, by the union bound, with a probability at least
_{i}^{2}_{1}_{2})

Now we construct a two-block random code by using the code _{i}, i^{2}}, and send them in the first block independently, by using a code with average error probability at most
_{i}_{i}, i

Similarly to in Section 3, here we have the following corollary.

_{i}_{1}_{2}

_{i}_{1}_{2}

The results in Sections 3 and 4 can be easily extended to an MAC with _{i}, i

_{0} with message sets _{i}, i_{I,}_{0} ⊂ _{I}

for all _{I}_{I,}_{0}, where _{i}, i

^{(1)}^{(2)}^{(}^{I}^{−1)})(_{0})^{(}^{i}^{)} on _{i}, i_{i}^{(}^{i}^{)}(·)) for _{I}^{2} permutations
_{i}^{2} on _{i},_{i}^{2}

for all (_{1}_{2}_{I}_{−1}_{I}_{1}_{2} _{I}_{−1}_{I,}_{0}.

In [

We have seen that randomization at encoders serves as an efficient way to solve the problem caused by the bad combination of input codewords. As in many cases, a perfect cooperation in network communication is impossible, this suggests us to employ random codes.

So far we have known very little about the maximum-error-probability capacity regions. One reason is that in most cases a small average probability of error is acceptable. Perhaps, another reason is that, to determine the maximum-error-probability capacity regions is much harder than to determine the average-error-probability capacity regions. Nevertheless, from a theoretical point of view, to well understand the multiple user channels, exploring their maximum-error-probability capacity regions is necessary. Likely, the study on the maximum-error-probability capacity regions is closely related to the study on AVCs.

The author would like to thank Yanling Chen for many discussions and helpful suggestions in details for revising the paper. This work was partially supported by grant from the National Natural Science Foundation of China (Ref. No. 61271174).

The authors declare no conflict of interest.