# Block Markov Superposition Transmission of RUN Codes

## Abstract

In this paper, we propose a simple procedure to construct (decodable) good codes with any given alphabet (of moderate size) for any given (rational) code rate to achieve any given target error performance (of interest) over additive white Gaussian noise (AWGN) channels. We start with constructing codes over groups for any given code rates. This can be done in an extremely simple way if we ignore the error performance requirement for the time being. Actually, this can be satisfied by repetition (R) codes and uncoded (UN) transmission along with time-sharing technique. The resulting codes are simply referred to as RUN codes for convenience. The encoding/decoding algorithms for RUN codes are almost trivial. In addition, the performance can be easily analyzed. It is not difficult to imagine that a RUN code usually performs far away from the corresponding Shannon limit. Fortunately, the performance can be improved as required by spatially coupling the RUN codes via block Markov superposition transmission (BMST), resulting in the BMST-RUN codes. Simulation results show that the BMST-RUN codes perform well (within one dB away from Shannon limits) for a wide range of code rates and outperform the BMST with bit-interleaved coded modulation (BMST-BICM) scheme.

## 1Introduction

Since the invention of turbo codes [1] and the rediscovery of low-density parity-check (LDPC) codes [2], many turbo/LDPC-like codes have been proposed in the past two decades. Among them, the convolutional LDPC codes [3], recast as spatially coupled LDPC (SC-LDPC) codes in [4], exhibit a threshold saturation phenomenon and were proved to have better performance than their block counterparts. In a certain sense, the terminology “spatial coupling” is more general, as can be interpreted as making connections among independent subgraphs, or equivalently, as introducing memory among successive independent transmissions. With this interpretation, braided block codes [5] and staircase codes [6], as the convolutional versions of (generalized) product codes, can be classified as spatially coupled codes. In [7], the spatially coupled version of turbo codes was proposed, whose belief propagation (BP) threshold is also better than that of the uncoupled ensemble.

Recently, block Markov superposition transmission (BMST) [8] was proposed, which can also be viewed as the spatial coupling of generator matrices of short codes. The original BMST codes are defined over the binary field . In [9], it has been pointed out that any code with fast encoding algorithms and soft-in soft-out (SISO) decoding algorithms can be taken as the basic code. For example, one can take the Hadamard transform (HT) coset codes as the basic codes, resulting in a class of multiple-rate codes with rates ranging from to , where is a positive integer [11]. Even more flexibly, one can use the repetition and/or single-parity-check (RSPC) codes as the basic codes to construct a class of multiple-rate codes with rates ranging from to , where is an integer [13]. It has been verified by simulation that the construction approach is applicable not only to binary phase-shift keying (BPSK) modulation but also to bit-interleaved coded modulation (BICM) [14], spatial modulation [15], continuous phase modulation (CPM) [16], and intensity modulation in visible light communications (VLC) [17].

In this paper, we propose a procedure to construct codes over groups, which extends the construction of BMST-RSPC codes [13] in the following two aspects. First, we allow uncoded symbols occurring in the basic codes. Hence the encoding/decoding algorithms for the basic codes become simpler. Second, we derive a performance union bound for the repetition codes with any given signal mapping, which is critical for designing good BMST codes without invoking simulations. We will not argue that the BMST construction can always deliver better codes than other existing constructions.^{1}*any* given signal set (of moderate size), *any* given (rational) code rate and *any* target error performance (of interest). We start with constructing group codes, referred to as RUN codes, with any given rate by time-sharing between repetition (R) codes and/or uncoded (UN) transmission. By transmitting the RUN codes in the BMST manner, we can have a class of good codes (called BMST-RUN codes). The performance of a BMST-RUN code is closely related to the encoding memory and can be predicted analytically in the high signal-to-noise ratio (SNR) region with the aid of the readily-derived union bound. Simulation results show that the BMST-RUN codes can approach the Shannon limits at any given target error rate (of interest) in a wide range of code rates over both additive white Gaussian noise (AWGN) channels and Rayleigh flat fading channels.

The pragmatic reader may question the necessity to construct codes over high-order signal constellations, since bandwidth efficiency can also be attained by BICM with binary codes. However, in addition to the flexility of the construction, the BMST-RUN codes have the following competitive advantages.

BMST-RUN codes can be easily designed to obtain shaping gain in at least two ways. One is designing codes directly over a well-shaped signal constellation, say, non-uniformly spaced constellation [18]. The other is implementing Gallager mapping for conventional signal constellations [19]. In both cases, neither optimization for bit-mapping (at the transmitter) nor iterations between decoding and demapping (at the receiver) are required.

BMST-RUN codes can be defined over signal sets of any size, such as 3-ary pulse amplitude modulation (3-PAM) and 5-PAM, which can be useful to transmit real samples directly [20].

The rest of this paper is organized as follows. In Section , we take a brief review of the BMST technique. In Section , we discuss constructing group codes with any given signal set and any given code rate. In Section , we propose the construction method of BMST-RUN codes and discuss the performance lower bound. In Section , we give simulation results and make a performance comparison between the BMST-RUN codes and the BMST-BICM scheme. In Section , we conclude this paper.

## 2Review of Binary BMST Codes

Binary BMST codes are convolutional codes with large constraint lengths [8]. Typically, a binary BMST code of memory consists of a short code (called the *basic code*) and at most interleavers [10]. Let be the basic code defined by a generator matrix over the binary field . Denote as blocks of data to be transmitted, where for . Then, the encoding output at time can be expressed as [10]

where is initialized to be for and are permutation matrices of order . For , the zero message sequence is input into the encoder for termination. Then, is mapped to a signal vector and transmitted over the channel, resulting in a received vector .

At the receiver, the decoder executes the sliding-window decoding (SWD) algorithm to recover the transmitted data [8]. Specifically, for an SWD algorithm with a decoding delay , the decoder takes as inputs to recover at time , which is similar to the window decoding (WD) of the SC-LDPC codes [21]. The structure of the BMST codes also admits a two-phase decoding (TPD) algorithm [10], which can be used to reduce the decoding delay and to predict the performance in the extremely low bit-error-rate (BER) region.

As discussed in [9], binary BMST codes have the following two attractive features.

Any code (linear or nonlinear) can be the basic code as long as it has fast encoding algorithms and SISO decoding algorithms.

Binary BMST codes have a simple genie-aided lower bound when transmitted over AWGN channels using BPSK modulation, which shows that the maximum extra coding gain can approach dB compared with the basic code. The tightness of this simple lower bound in the high SNR region under the SWD algorithm has been verified by both the simulation and the extrinsic information transfer (EXIT) chart analysis [24].

Based on the above two facts, a general procedure has been proposed for constructing capacity-approaching codes at any given target error rate [10]. Suppose that we want to construct a binary BMST code of rate at a target BER of . First, we find a rate- short code as the basic code. Then, we can determine the encoding memory by

where is the minimum SNR for the code to achieve the BER , is the Shannon limit corresponding to the rate , and stands for the minimum integer greater than or equal to . Finally, by generating interleavers uniformly at random, the BMST code is constructed. With this method, we have constructed a binary BMST code of memory using the Cartesian product of the R code , which has a predicted BER lower than within one dB away from the Shannon limit.

## 3RUN Codes over Groups

### 3.1System Model and Notations

Consider a symbol set and an -dimensional signal constellation of size . The symbol set can be treated as a group by defining the operation for . Let be a (fixed) one-to-one mapping . Let be a symbol to be transmitted. For the convenience of performance analysis, instead of transmitting directly, we transmit a signal , where is a sample of a uniformly distributed random variable over and assumed to be known at the receiver. The received signal , where denotes the component-wise addition over and is an -dimensional sample from a zero-mean white Gaussian noise process with variance per dimension. The SNR is defined as

where is the squared Euclidean norm of .

In this paper, for a discrete random variable over a finite set , we denote its *a priori message* and *extrinsic message* as and , respectively. A SISO decoding is a process that takes *a priori* messages as inputs and delivers extrinsic messages as outputs. We assume that the information messages are independent and uniformly distributed (i.u.d.) over .

### 3.2Repetition (R) Codes

Fig. shows the transmission of a message for times over AWGN channels.

#### Encoding

The encoder of an R code over takes as input a single symbol and delivers as output an -dimensional vector .

#### Mapping

The -th component of the codeword is mapped to the signal for , where is a random vector sampled from an i.u.d. process over .

#### Demapping

Let be the received signal vector corresponding to the codeword . The *a priori* messages input to the decoder are computed as

for .

#### Decoding

The SISO decoding algorithm computes the *a posteriori* messages

for making decisions and the extrinsic messages

for for iteratively decoding when coupled with other sub-systems.

#### Complexity

Both the encoding/mapping and the demapping/decoding have linear computational complexity per coded symbol.

#### Performance

Let denote the hard decision output. The performance is measured by the symbol-error-rate (SER) . Define , where denotes the subtraction under modulo- operation. Due to the existence of the random vector , the peformance is irrelevant to the transmitted symbol . We define

as the average Euclidian distance enumerating function (EDEF) corresponding to the error , where is a dummy variable. Then, the average EDEF for the R code over all possible messages and all possible vectors can be computed as

where denotes the average number of signal pairs with Euclidean distance , and . The performance under the mapping can be upper-bounded by the union bound as

where is the pair-wise error probability with .

From the above derivation, we can see that the performance bounds of the R codes are related to the mapping . In this paper, we consider as examples the BPSK, the signal set (denoted as -PAM), -PAM, -ary phase-shift keying (-PSK) modulation, -ary quadrature amplitude modulation (-QAM), or -PAM, which are depicted in Fig. along with mappings denoted by as specified in the figure. Fig. and Fig. show performance bounds for several R codes defined with the considered constellations. From the figures, we have the following observations.

The performance gap between the code and the uncoded transmission, when measured by the SNR instead of , is roughly dB.

Given a signal constellation, mappings that are universally good for all R codes may not exist. For example, as shown in Figure 4, is better than for rate () but becomes worse for rate ().

### 3.3Time-Sharing

With repetition codes over groups, we are able to implement code rates for any given integer . To implement other code rates, we turn to the time-sharing technique. To be precise, let be the target rate. There must exist a unique such that . Then, we can implement a code by time-sharing between the code and the code , which is equivalent to encoding information symbols with the code and the remaining symbols with the code , where is the time-sharing factor. Apparently, to construct codes with rate , we need time-sharing between the code and the uncoded transmission. For this reason, we call this class of codes as *RUN codes*, which consist of the R codes and codes obtained by time-sharing between the R codes and/or the uncoded transmission. We denote a RUN code of rate as . Replacing in Figure 1 the R codes with the RUN codes, we then have a coding system that can transmit messages with any given code rate over any given signal set.

#### Encoding

Let be the message sequence. The encoder of the code encodes the left-most symbols of into codewords of and the remaining symbols into codewords of .

#### Decoding

The decoding is equivalent to decoding separately codewords of and codewords of .

#### Complexity

Both the encoding/mapping and the demapping/decoding have the same complexity as the R codes.

#### Performance

The performance of the RUN code of rate is given by

which can be upper-bounded with the aid of (Equation 6). Performances and bounds of several RUN codes defined with BPSK modulation, -PAM, -PAM, -PSK modulation, or -QAM are shown in Figure 3 and Figure 4. We notice that the union bounds with BPSK modulation are the exact performances, while those with other signal sets are upper bounds to the performances. We also notice that the upper bounds become tight as the SER is lower than for all other signal sets. Not surprisingly, the performances of the RUN codes are far away from the corresponding Shannon limits (more than dB) at the SER lower than .

## 4BMST over Groups

### 4.1BMST Codes with RUN Codes As Basic Codes

We have constructed a class of codes called RUN codes with any given code rate over groups. However, the RUN codes perform far away from the Shannon limits, as evidenced by the examples in Fig. . To remedy this, we transmit the RUN codes in the BMST manner as inspired by the fact that, as pointed out in [9], any short code can be embedded into the BMST system to obtain extra coding gain in the low error-rate region. The resulted codes are referred to as BMST-RUN codes. More precisely, we use the -fold Cartesian product of the RUN code (denoted as ) as the basic code. Figure 5 shows the encoding structure of a BMST-RUN code with memory , where represents the basic encoder, , , represents symbol-wise interleavers, represents the superposition with modulo- addition, and represents the mapping . Let and be the information sequence and the corresponding codeword of the code at time , respectively. Then, the sub-codeword can be expressed as

where denotes the symbol-wise modulo- addition, for and is the interleaved version of by the -th interleaver for . Then, is mapped to the signal vector symbol-by-symbol and input to the channel. After every sub-blocks of information sequence, we terminate the encoding by inputting all-zero sequences to the encoder. The termination will cause a code rate loss. However, the rate loss can be negligible as is large enough.

### 4.2Choice of Encoding Memory

The critical parameter for BMST-RUN codes to approach the Shannon limits at a given target SER is the encoding memory , which can be determined by the genie-aided lower bound. Essentially the same as for the binary BMST codes [9], the genie-aided bound for a BMST-RUN code can be easily derived by assuming all but one sub-blocks are known at the receiver. With this assumption, the genie-aided system becomes an equivalent system that transmits the basic RUN codeword times. Hence the performance of the genie-aided system is the same as the RUN code obtained by time-sharing between the code and the code . As a result, the genie-aided bound under a mapping is given by

which can be approximated using the union bound in the high SNR region.

Given a signal set of size with labeling , a rate and a target SER , we can construct a good BMST-RUN code using the following steps.

Construct the code over the modulo- group by finding such that and determining the time-sharing factor between the R code and the R code . To approach the Shannon limit and to avoid error propagation, we usually choose such that .

Find the Shannon limit under the signal set corresponding to the rate .

Generate interleavers of size uniformly at random.

### 4.3Decoding of BMST-RUN Codes

A BMST-RUN code can be decoded by an SWD algorithm with a decoding delay over its normal graph, which is similar to that of the binary BMST codes [9]. Figure 6 shows the unified (high-level) normal graph of a BMST-RUN code with and .

The normal graph can also be divided into *layers*, each of which consists of four types of nodes. These nodes represent similar constraints to those for binary BMST codes and have similar message processing as outlined below.

The process at the node is the SISO decoding of the RUN codes as described in Section 3.2.

The process at the node can be implemented in the same way as the message processing at a generic variable node of an LDPC code (binary or non-binary).

The process at the node can be implemented in the same way as the message processing at a generic check node of an LDPC code (binary or non-binary).

The process at the node is the same as the original one, which interleaves or deinterleaves the input messages.

Upon the arrival of the received vector (corresponding to the sub-block ) at time , the SWD algorithm takes as inputs the *a posterior* probabilities (APPs) corresponding to and uses the APPs corresponding to to recover , where the computation of APPs is similar to (). After is recovered, the decoder discards and slides one layer of the normal graph to the “right" to recover with received.

## 5Examples of BMST-RUN Codes

(dB) | BPSK |
|||||||

BPSK | ||||||||

BPSK | ||||||||

BPSK | ||||||||

BPSK | ||||||||

BPSK | ||||||||

BPSK | ||||||||

-PAM | ||||||||

-PAM | ||||||||

-PAM | ||||||||

-PAM | ||||||||

-PAM | ||||||||

-PAM | ||||||||

-PAM | ||||||||

-PAM | ||||||||

-PAM | ||||||||

-PAM | ||||||||

-PAM | ||||||||

-PAM | ||||||||

-PSK | ||||||||

-PSK | ||||||||

-PSK | ||||||||

-PSK | ||||||||

-QAM | ||||||||

uniformly spaced -PAM | ||||||||

non-uniformly spaced -PAM [18] | ||||||||

In this section, we present simulation results of several BMST-RUN codes over AWGN channels and Rayleigh flat fading channels, where code parameters are shown in Table ?. For all simulations, the encoder terminates every sub-blocks and the decoder executes the SWD algorithm with a maximum iteration number . Without specification, the decoding delay of the SWD algorithm is set to be .

### 5.1BMST-RUN Codes with One-Dimensional Signal Sets

Consider BMST-RUN codes of rates defined with BPSK modulation to approach the Shannon limits at the SER of . Figure 7 shows the required SNRs for the BMST-RUN codes to achieve the SER of . Also shown in Figure 7 is the channel capacity curve with i.u.d. inputs. It can be seen that the gaps between the required SNRs and the Shannon limits are within dB for all considered rates.

Consider BMST-RUN codes of rates defined with -PAM to approach the Shannon limits at the SER of . Figure 8 shows the SER performance curves for all codes together with their lower bounds and the corresponding Shannon limits. We can see that the performance curves match well with the corresponding lower bounds for all codes in the high SNR region. In addition, all codes have an SER lower than at the SNR within dB away from the corresponding Shannon limits, which is similar to the BPSK modulation case.

Consider a rate- BMST-RUN code of memory defined over two distinct -PAM constellations, where one consists of uniformly spaced signal points (under the mapping in Fig. ) and the other consists of non-uniformly spaced signal points (under the mapping in Fig. ) as designed in [18]. The SER performance curves with a decoding delay together with the lower bounds and the Shannon limits are shown in Fig. . From the figure, we can see that the BMST-RUN code has an SER lower than at the SNR about away from their respective Shannon limits for both uniformly spaced signal points and non-uniformly spaced signal points. In addition, the BMST-RUN code with non-uniformly spaced signal points performs about dB better than that with uniformly spaced signal points and also has a lower error floor.

### 5.2BMST-RUN Codes with Two-Dimensional Signal Sets

Consider BMST-RUN codes of rates defined with -PSK modulation to approach the Shannon limits at the SER of . Figure 10 shows the SER performance curves for all codes together with their lower bounds and the corresponding Shannon limits.

Consider a BMST-RUN code of rate defined with -QAM (under the mapping in Figure 2) to approach the Shannon limit at the SER of , where an encoding memory is required. The SER performance curves with decoding delays and together with the lower bound and the Shannon limit are shown in Figure 11. Since a large fraction of information symbols () are uncoded in the basic code, a large decoding delay is required to approach the lower bound. With the decoding delay , the BMST-RUN code achieves the SER of at the SNR about dB away from the Shannon limit.

From the above two examples, we can see that BMST codes with two-dimensional signal constellations behave similarly as they do with one-dimensional signal constellations.

### 5.3Comparison with BMST-BICM

The examples in the previous subsections suggest that the proposed construction is effective for a wide range of code rates and signal sets. Also, the SWD algorithm is near-optimal in the high SNR region. Since binary BMST codes also have such behaviors and can be combined with different signal sets [14], we need clarify the advantage of BMST-RUN codes over groups. Some advantages have been mentioned in the Introduction. In this subsection, we will show that the BMST-RUN codes can perform better than the BMST-BICM scheme.

To make a fair comparison, we have the following settings.

For the BMST-BICM scheme, the basic codes are the RUN codes over , while for the BMST-RUN codes, the basic codes are the RUN codes over the modulo- group. Such setting ensures that both schemes have the same sub-block length in bits.

Both the BMST-RUN codes and the BMST-BICM scheme use the -PAM with the mapping in Figure 2.

For a specific code rate, the BMST-BICM scheme has the same encoding memory and the same decoding delay as the BMST-RUN code. The encoding memories are presented in Table ?, while the decoding delay is set to be for an encoding memory .

Since the performance of the BMST-BICM scheme can not be measured in SER, we compare the performance in BER. Figure 12 shows the BER performance curves for both the BMST-RUN codes (denoted as “RUN”) and the BMST-BICM scheme (denoted as “BICM”) together with the Shannon limits. Figure 13 shows the required SNRs to achieve the BER of for both the BMST-RUN codes and the BMST-BICM scheme together with capacity curve of -PAM under i.u.d. inputs. From these two figures, we have the following observations.

With the same encoding memory and decoding delay, the BMST-RUN codes achieve a lower BER than the BMST-BICM scheme for all considered code rates.

The BMST-RUN codes perform better than the BMST-BICM scheme in the lower code rate region and have a similar performance as the BMST-BICM scheme in the high code rate region.

### 5.4BMST-RUN Codes over Rayleigh Channels

It has been shown that BMST-RUN codes perform well over AWGN channels and are comparable to binary BMST codes with BICM. More interestingly and importantly, BMST construction is also applicable to other ergodic channels. Here, we give an example for fading channels as an evidence.

Consider BMST-RUN codes of rates defined with 4-PAM modulation (under the mapping in Fig. ) over Rayleigh flat fading channels. To approach the Shannon limits at the SER of , the required encoding memories for rates and are and , respectively. Fig. shows the required SNRs for the BMST-RUN codes to achieve the SER of . Also shown in Fig. is the channel capacity curve with i.u.d. inputs. It can be seen that the gaps between the required SNRs and the Shannon limits are about dB for all rates, which is similar to the case for AWGN channels.

## 6Conclusions

In this paper, by combining the block Markov superposition transmission (BMST) with the RUN codes over groups, we have proposed a simple scheme called BMST-RUN codes to approach the Shannon limits at any target symbol-error-rate (SER) with any given (rational) rate over any alphabet (of moderate size). We have also derived the genie-aided lower bound for the BMST-RUN codes. Simulation results have shown that the BMST-RUN codes have a similar behavior to the binary BMST codes and have good performance for a wide range of code rates over both AWGN channels and Rayleigh flat fading channels. Compared with the BMST with bit-interleaved coded modulation (BMST-BICM) scheme, the BMST-RUN codes are more flexible, which can be combined with signal sets of any size. In addition, with the same encoding memory, the BMST-RUN codes have a better performance than the BMST-BICM scheme under the same decoding latency.

## Acknowledgment

The authors wish to thank Mr. Kechao Huang and Mr. Jinshun Zhu for useful discussions.

### Footnotes

- Actually, compared with SC-LDPC codes, the BMST codes usually have a higher error floor. However, the existence of the high error floor is not a big issue since it can be lowered if necessary by increasing the encoding memory.

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