On the use of ordered statistics decoders for lowdensity paritycheck codes in space telecommand links
 Marco Baldi^{1, 2}Email authorView ORCID ID profile,
 Nicola Maturo^{1, 2},
 Enrico Paolini^{3} and
 Franco Chiaraluce^{1, 2}
https://doi.org/10.1186/s136380160769z
© The Author(s) 2016
Received: 16 April 2016
Accepted: 9 November 2016
Published: 29 November 2016
Abstract
The performance of short lowdensity paritycheck (LDPC) codes that will be included in the standard for nextgeneration space telecommanding is analyzed. The paper is focused on the use of a famous ordered statistics decoder known as most reliable basis (MRB) algorithm. Despite its complexity may appear prohibitive in space applications, this algorithm is shown to actually represent a realistic option for short LDPC codes, enabling significant gains over more conventional iterative algorithms. This is possible by a hybrid approach which combines the MRB decoder with an iterative decoding procedure in a sequential manner. The effect of quantization is also addressed, by considering two different quantization laws and comparing their performance. Finally, the impact of limited memory availability onboard of spacecrafts is analyzed and some solutions are proposed for efficient processing, towards a practical onboard decoder implementation.
Keywords
Lowdensity paritycheck codes Most reliable basis algorithm Ordered statistics decoding Space missions Telecommand links1 Introduction

convolutional codes,

Reed–Solomon codes,

parallel turbo codes,

lowdensity paritycheck (LDPC) codes.
A comprehensive description of all these families of codes can be found, for example, in [5].

the data rates (measured in bits/s) are usually very low compared with TM links,

TCs are originated and assembled on the ground.
The first above feature implies milder requirements for the errorcorrecting code. Similarly, because of the second feature, fewer limits and constraints are imposed on the available transmitted power. Recommendations for TC space links are issued traditionally by two organizations: the Consultative Committee for Space Data Systems (CCSDS) and the European Cooperation for Space Standardization (ECSS). The only errorcorrecting code so far included in these recommendations [6, 7] is a Bose–Chaudhuri–Hocquenghem (BCH) code. This linear block code is characterized by a codeword length n=63 bits, an information block length k=56 bits and is typically decoded O/B via harddecision decoding.
Reliable space telecommanding is of fundamental importance as the success of a mission may be compromised because of an error corrupting a TC message. This imposes strict constraints on the maximum tolerable error rates. In particular, the codeword error rate (CER) is defined as the ratio of the number of decoding failures to the total number of received codewords. Requirements are often specified in terms of average CER, and a typical value is CER ≤10^{−5}.
TC links are currently evolving to include new and more challenging mission profiles, characterized by much more demanding requirements. Compliance with these new requirements imposes the use of more advanced coding techniques than the BCH(63, 56) code. In space telecommanding, it is always of fundamental importance to ensure the integrity of very short emergency commands, with a very low data rate and limited latency. Such constraints impose the adoption of short codes.
The mentioned BCH code is short enough, but its performance is far from meeting the requirements of next generation missions. In the following analysis, we will refer to a classical binary phase shift keying (BPSK) modulation over an additive white Gaussian noise (AWGN) channel. This model is particularly suitable to describe deepspace communications. Using the standard BCH(63, 56) code with harddecision decoding, the signaltonoise ratio required to achieve CER ≈10^{−5} is E _{ b }/N _{0}≈9.1 dB, where E _{ b } is the energy per information bit and N _{0} the onesided noise power spectral density. This value of E _{ b }/N _{0} can be reduced, so improving performance, by applying softdecision decoding. For example, by using a decoder based on the BCJR algorithm [8], a gain of about 2.1 dB with respect to the harddecision decoder can be achieved. These E _{ b }/N _{0} values, however, remain too large for space missions of next generation. These missions shall be characterized by a significant increase of the supported data rate and/or maximum distance with respect to the present scenarios. Both these factors degrade the signaltonoise ratio and impose to achieve the target CER over a worse channel.
In order to achieve higher coding gains, new advanced coding schemes have been designed and evaluated [9]. After a long campaign of simulations and comparisons, two short binary LDPC codes have emerged as the most attractive candidates. Most of the steps necessary for their inclusion in the standard for space TC have already been completed, so that they will certainly be employed in nextgeneration missions.
LDPC codes can be decoded through classical softdecision iterative algorithms (IAs), such as the sumproduct algorithm (SPA) [10] or its simplified versions, e.g., minsum (MS) [11] or normalized minsum (NMS) [12]. As regards SPA, throughout this paper, we will refer to its loglikelihood ratio (LLR) implementation. By using LDPC codes with IAs, the coding gain is larger than with the BCH code, although the performance remains relatively far from the theoretical limits.
More substantial improvements can result from the adoption of algorithms based on ordered statistics decoding (OSD) [13]. In this paper, in particular, we consider the wellknown most reliable basis (MRB) algorithm [14]. MRB is a noniterative softdecision decoding algorithm able to approach the performance of a maximum likelihood (ML) decoder. The latter is optimal, in the sense of minimizing the decoding error probability. The main drawback of MRB is represented by its complexity, which makes its use problematic in TC links where decoding is performed O/B with very limited computational and memory resources. As mentioned, however, the length of the TC LDPC codes is small, namely, n=128 bits and n=512 bits, which makes MRB a potential candidate especially for the shorter code.
Complexity can be reduced by resorting to a hybrid solution, which combines MRB with IAs. More precisely, the hybrid approach consists of performing lowcomplexity decoding through an IA, at first, and invoking the MRB algorithm only when the IA is not able to find any valid codeword (detected error). The hybrid decoder has recently been used to decode also LDPC codes constructed on nonbinary finite fields [15], which represent another option for space TC links [16, 17]. Due to their higher decoding complexity, however, nonbinary LDPC codes are less attractive than their binary counterparts.
For the LDPC codes analyzed in this paper, the hybrid decoder outperforms the IA or the MRB algorithm when used individually. A qualitative explanation of this favorable behavior is as follows. The IA decoders here considered are not boundeddistance decoders: Therefore, they may be able to successfully decode softdecision sequences from the channel even if they have relatively large Euclidean distances from the BPSKmodulated transmitted codeword and, at the same time, they may fail to decode softdecision sequences at relatively small Euclidean distances from it. The MRB decoder instead works in a completely different way: Once having fixed its order i, it is able to correct all the error patterns involving i errors or less in the most reliable basis. Then, the two decoders complement one each other, this way improving the individual performances. On the other hand, it is evident that in case the IA is characterized by a high undetected codeword error rate (UCER), the MRB algorithm is not invoked in many times where it could correct and the performance of the plain MRB is better than that of the hybrid approach. This event does not occur for the considered codes that, by design, are characterized by values of UCER which are orders of magnitude lower than the CER. As an example, while the CER value at the working point must be at least as low as 10^{−5}, the UCER value at the same point must be at least as low as 10^{−9} [18]. Consequently, the MRB decoder is not invoked in a very few number of cases where it should be, and the advantage offered by the SPALLR to sporadically correct a (relatively) large number of errors dominates.
While ensuring excellent error rate performance, the hybrid decoder is characterized by a significantly lower average complexity than MRB decoding used alone. Moreover, such a performance is not seriously affected by practical issues like, for example, quantization, on condition that suitable quantization rules are applied.
Motivated by the above considerations, in this paper, a thorough analysis of the performance of the MRB algorithm, used individually or in hybrid form, is developed with reference to space TC applications. While a preliminary version of this analysis has been considered in [19], the study is here deepened by addressing all technical issues in a more complete way. Among them, we investigate the impact of limited memory availability (a typically very stringent constraint in O/B processing), with the aim to discuss also the problems arising in practical implementations. In fact, while the presence of enough O/B memory would help fast convergence of the decoding algorithm, memories are usually considered not reliable in hostile environments like the one characterizing space missions, so that their usage is generally minimized. As a consequence, “onthefly” computations are preferred to memory accesses. Therefore, we extend the analysis in [19] by addressing both the case of a limited amount of memory and the case of total absence of memory. We also study a parallel implementation for controlling other important variables, like the decoding latency.
The organization of the paper is as follows. In Section 2, the considered errorcorrecting codes are described. In Section 3, the various decoding procedures are presented and expressions for computing their complexities are provided. The performance of these schemes is then assessed in Section 4, also taking into account the quantization issues. Section 5 is devoted to the analysis of the impact of limited memory and to the latency evaluation. Concluding remarks are given in Section 6.
2 LDPC codes for space telecommand links
The two considered LDPC codes are described in detail in [16]. The first code has information block length k=64 bits and codeword length n=128 bits. The second code has information block length k=256 bits and codeword length n=512 bits. Hence, both codes have code rate R _{ c }=k/n=1/2. A short overview of their main features is reported in the following. In particular, in Section 2.1, their structure is described, by specifying their paritycheck matrices, while in Section 2.2, their weight spectrum properties are addressed.
2.1 Paritycheck and generator matrices
In the figure, I _{ M } and 0 _{ M } denote the M×M identity and zero matrices, respectively, and Φ is the first right circular shift of I _{ M }. This means that Φ has a nonzero entry at row i and column j if and only if j=(i+1) mod M. Moreover, Φ ^{2} represents the second right circular shift of I _{ M }, that is, Φ ^{2} has a nonzero entry at row i and column j if and only if j=(i+2) mod M, and so on. The ⊕ operator indicates elementwise modulo2 addition.
As an alternative representation, a k×n generator matrix G can be obtained from the paritycheck matrix H. The lengthn codeword c corresponding to a lengthk information block u can be then generated as c=u G. The matrices G for the two considered codes are reported in [16].
2.2 Weight distribution properties
The Hamming weight of a codeword is defined as the number of its nonzero elements. The performance of a linear block code under optimum ML decoding is governed by its weight spectrum, that is the number of codewords of Hamming weight w for all integer 0≤w≤n. Unfortunately, even when the weight spectrum is perfectly known (which is a very favorable condition, normally unverified for LDPC codes), it is not possible to find an exact analytical expression of the error rate achieved by the ML decoder, even for very simple communication channel models. Thus, it is a common practice resorting to analytical bounds, an example of which is represented by the union bound, that establishes an upper bound on the error rate of the considered code under ML decoding [20].
The first term of the sum, corresponding to w=d _{min}, is also known as the “error floor.” For sufficiently high values of E _{ b }/N _{0}, the error floor provides an excellent approximation of the performance of ML decoding. In this sense, it represents a first benchmark for any suboptimum decoding algorithm: the closer the error rate offered by the decoding algorithm to the error floor, the smaller the gap between the suboptimum decoder and the optimum ML decoder. As from (1), the computation of CER _{ UB } requires the knowledge of the complete weight spectrum of the code. It is known that for LDPC codes this may be a nontrivial task. For the considered codes, however, much work has been done to overcome this issue.
where the presence of the term A _{ w } x ^{ w } means that there are A _{ w } codewords with Hamming weight w. The multiplicities A _{14},A _{16} and A _{18} are exact [22]; this part of the weight spectrum has been obtained through computer searches using a carefully tuned “error impulse” technique [23]. The other multiplicities are lower bounds on the actual values and have been obtained by using the approach proposed in [24]. The overall estimate is anyway sufficiently stable and allows to draw a reliable union bound, as will be done in Section 4.
The multiplicities A _{40} and A _{42} appear to be exact, while A _{44} and A _{46} are approximate and, in general, there is not yet sufficient confidence on the reliability of the estimate, even as regards the value of d _{min}. This does not allow to draw a sufficiently reliable union bound for the LDPC(512, 256) code.
3 Decoding algorithms
As decoding algorithms we consider one OSD (the MRB algorithm) and three IAs (SPA, MS, and NMS). We investigate their performance when used alone or combined in the aforementioned hybrid decoding scheme.
3.1 MRB decoder
where σ ^{2} is the variance of the Gaussian thermal noise. The reliability of bit c _{ i } is then defined as L(c _{ i }).

Upon receiving the signal from the AWGN channel and demodulating it into the sequence y, find the k most reliable received bits and collect them in a lengthk vector v ^{⋆}.

Perform GaussJordan elimination on the matrix G, with respect to the positions of the k most reliable bits, to obtain a systematic generator matrix G ^{⋆} corresponding to such bits.^{1}

Encode v ^{⋆} to obtain a candidate codeword c ^{⋆}=v ^{⋆} G ^{⋆}.

Consider all (or an appropriate subset of) test error patterns (TEPs). By definition, a TEP, noted by e, is a binary vector of length k and Hamming weight w≤i.

For each TEP e:

Calculate \(\tilde {\mathbf {v}} = \mathbf {v}^{\star }+\mathbf {e}\).

Encode \(\tilde {\mathbf {v}}\) to obtain a codeword \(\tilde {\mathbf {c}} = \tilde {\mathbf {v}}\, \mathbf {G}^{\star }\).

If the Euclidean distance between \(\tilde {\mathbf {x}} = \mathbf {1}  2\,\tilde {\mathbf {c}}\) and y is smaller than the one between x ^{⋆}=1−2 c ^{⋆} and y, then update the candidate codeword as \(\mathbf {c}^{\star } \leftarrow \tilde {\mathbf {c}}\).

At the end of the process, the algorithm delivers the codeword c ^{⋆} which minimizes the Euclidean distance from y, limited to the codeword set associated with the considered TEP set.
Hereafter, we denote by MRB(i) an instance of the algorithm with order i. Clearly, MRB(k) coincides with ML decoding, which is intractable in practice. Actually, the maximum number of TEPs to be tested, for each received sequence y, is equal to \(N_{\text {TEP}}^{\max } = \sum _{w=0}^{i} {k \choose w}\), and this number tends to become very large even for i≪k. So, in practice, only very small values of i can be considered.
 1.
The number of patterns to be tested is reduced by properly choosing a reliability threshold A such that, if the distance between the current candidate codeword and y is lower than A, then the decoding algorithm outputs the candidate codeword without testing the remaining TEPs.
 2.
The TEPs are commonly generated in ascending weight order. However, it is possible to precompute a list of most likely TEPs [25], that is, a list containing the TEPs ordered according to the probability to yield codewords at small distances from y, regardless of their weight w≤i. The threshold criterion at the previous point can then be applied to the ordered list.
One or both approaches allow to test an average number of patterns, denoted by N _{TEP} in the following, significantly smaller than \(N_{\text {TEP}}^{\max }\). As we will see in Section 3.3, this may have a significant impact on the decoding complexity.
There are also other tricks that may be used to reduce the MRB decoder implementation complexity. A very effective one is based on the observation that a large number of TEPs have overlapping supports, where the support of a TEP is defined as the set of its nonzero coordinates. Due to the code linearity, we can compute the XOR of two TEPs and encode it through G ^{⋆}. The resulting vector can then be added to the test codeword corresponding to the first TEP, in order to obtain the codeword corresponding to the second TEP. Since, for the small values of the order i we consider, the XOR of the two TEPs has a small Hamming weight, computing the corresponding codeword requires to sum a small number of rows of G ^{⋆}, thus reducing the computational burden. The procedure can be, obviously, iterated.
3.2 Hybrid decoder
As from Fig. 2, contrary to similar proposals appeared in previous literature, the MRB decoder is applied on the received sequence y and not on the output of the IA decoder. This permits us to circumvent the effects of the deteriorated soft values after a failure of the IA.
3.3 Decoding complexity
As mentioned in the previous sections, besides the error rate, another fundamental parameter for measuring the efficiency of a decoding algorithm is complexity. In case of the MRB algorithm, it is even the factor conditioning the applicability of the method.
A possible definition of complexity is in terms of the average number of binary operations required for decoding a single codeword. More practical issues, like chip area requirements, power consumption, number of lookup tables, number of flipflops, and memory occupation, should be considered as well. These factors, however, are strongly dependent on the hardware architecture and design choices, while we aim at performing a more general complexity assessment, for which the number of operations turns out to be a suitable metric.
In (4), we have taken into account that some of the operations are performed on real values; so, q is the number of quantization bits used to represent a real number. As we can see, the third term on the righthand side (r.h.s.) is proportional to the number of TEPs; therefore, using N _{TEP} instead of \(N_{\text {TEP}}^{\max }\), as permitted by the application of the speedup procedures described in Section 3.1, can yield a significant advantage.

To order n real values.

To perform GaussJordan elimination on the k×n original generator matrix G to obtain the generator matrix G ^{⋆} in systematic form, in which the information bits coincide with v ^{⋆}.

To perform the vectormatrix multiplication v ^{⋆} G ^{⋆}.

To generate, on average, N _{TEP} TEPs and perform the relevant calculations.
where C_{IA} is the complexity of the IA preceding the (possible) MRB decoding attempt, while α represents the fraction of decoding instances in which MRB is invoked, i.e., the rate of the calls to MRB. This is because, in the hybrid approach, the MRB algorithm is invoked only when the IA fails. To be more precise, looking at the flowchart of the hybrid decoder in Fig. 2, it should be noted that the rate of calls to the MRB decoder equals the detected error rate of the IA. In fact, when an undetected error occurs for the IA, decoding terminates unsuccessfully without any call to MRB. So, in principle, α is different from the CER, as the latter captures both detected and undetected errors. However, since the undetected error rate of an IA is usually orders of magnitude smaller than the detected one (unless the minimum distance of the code is very poor—this is not the case for the codes considered in this paper), we can assume α≈CER, this way making a negligible error.
respectively, where I _{ave} is the average number of iterations and d _{ v } is the average column weight of the paritycheck matrix. The considered codes are both characterized by d _{ v }=4. Finally, it must be noted that, starting from the previous expressions, the corresponding decoding complexities per information bit can be obtained by dividing the r.h.s. of (4)–(8) by k.
4 Error rate versus complexity tradeoff evaluation
In this section, the CER/complexity tradeoff offered by each of the decoding algorithms considered in Section 3 is assessed, when used to decode the two LDPC codes described in Section 2. As previously stated, the modulation format is BPSK. Moreover, a maximum number of iterations I _{max}=100 has been used for each iterative algorithm. This value has been determined, from simulations, for the considered codes, as capable to obtain the best possible performance from the adopted IAs. This means that no significant extra gain is achievable by using a value of I _{max} larger than 100. An explicit example will be presented in Section 4.2. On the other hand, by using (6), (7), and (8), but with I _{max} in place of I _{ave}, we see that the complexity in the worst case (that is, when the decoder reaches the maximum number of iterations) grows linearly with I _{max}. As we will show in Section 5, the worst case complexity determines the worst case latency, the latter quantity being another design parameter that must be properly limited. As another argument to justify the limited gain induced by an increase in the maximum number of iterations, we can observe that in the region of medium/low error rates, I _{ave} is usually much lower than I _{max}, thus confirming that the maximum number of iterations is rarely reached (but it determines the maximum latency in the sense specified above).
4.1 LDPC ( 1 2 8 , 6 4 ) code
To confirm the goodness of the hybrid algorithm, we have realized a further experiment, by simulating the performance of a mixed (purely theoretical and impractical) decoder that feeds the received sequence to the inputs of both decoders but keeps only the output of the successful decoder. We have verified that the CER performance of this “ideal” decoder is practically coincident with that of the “real” hybrid decoder over the whole range of error rate values of interest.
The hybrid algorithm, which invokes MRB only upon a detected IA failure, has a complexity comparable with that of the plain MRB algorithm in the low E _{ b }/N _{0} regime (or, equivalently, in the high CER regime). By increasing E _{ b }/N _{0} (or, equivalently, by decreasing the CER), its complexity is gradually reduced, until it becomes comparable with that of the IA alone, in the high E _{ b }/N _{0} regime. At CER = 10^{−5}, the average number of operations required by the hybrid algorithm is still about one order of magnitude larger than that of the corresponding IA alone. Such a higher complexity, however, is compensated by the extragain ensured by the hybrid approach.
4.2 LDPC ( 5 1 2 , 2 5 6 ) code
As regards the MRB algorithm, the simulations have been performed with an order i=3, when the algorithm is used alone, as a larger order becomes too complex to manage for k=256 and n=512. This limited MRB order has a negative impact on the error rate performance, which turns out to be unsatisfactory. On the contrary, when the MRB algorithm is employed in the framework of the hybrid decoder, it is possible to increase its order up to i=4 with beneficial effects in terms of CER. Consequently, the performance of the hybrid decoder (here implemented with the SPA) reveals to be the best one, although the gain over the SPA or the NMS algorithms used alone is much less remarkable than for the short code—being now in the order of 0.15 dB.
5 Impact of limited memory

the adoption of a nonordered list (that occurs when the TEP list is generated onthefly, rather than storing it in a memory);

the adoption of an incomplete list (that occurs when only part of the list can be stored).
In the following, we consider the hybrid decoding algorithm, as it is more suitable (than plain MRB) in view of practical implementation. For the sake of simplicity, the numerical analysis is focused on the LDPC(128, 64) code under the assumption of using an MRB decoder of order i=4. Moreover, according to the suboptimal mechanisms described in Section 3.1, we use a stopping threshold A=24.5. Such a value of A has been heuristically optimized for the considered code.
In practice, ordering can be seen as a perturbation of the ascending weight rule that can cause some TEPs of weight j+z, with z integer greater than 0, but with a higher probability, to be processed before some TEPs of weight j. An explicative example is reported below.
Example 1
We see that patterns of weight 2, that is, with two nonzero coordinates, are intermingled with those of weight 1. For example, the pattern (63 64 0 0) is considered before the pattern (9 0 0 0). Similarly, the weight1 pattern (8 0 0 0) is considered after some weight2 patterns.
Related with the complexity issue, it is also interesting to have a preliminary evaluation of the latency due to the MRB decoder. Latency is a measure of the time required for decoding and its value is normally subjected to restrictions, which are expected to be particularly severe when the TC link is used in emergency conditions.
Average latency (in seconds) due to the MRB decoding process, by assuming f _{clock}=100 MHz
N _{Teu}  E _{ b }/N _{0}=1 dB  E _{ b }/N _{0}=2 dB  E _{ b }/N _{0}=3.5 dB 

1  4.05  2.21  2.45·10^{−1} 
10  4.05·10^{−1}  2.21·10^{−1}  2.48·10^{−2} 
100  4.09·10^{−2}  2.24·10^{−2}  2.77·10^{−3} 
1000  4.41·10^{−3}  2.54·10^{−3}  5.60·10^{−4} 
10,000  7.53·10^{−4}  5.81·10^{−4}  3.35·10^{−4} 
Worstcase latency (in seconds) due to the MRB decoding process, with and without ordered list, by assuming f _{clock}=100 MHz and a reduced number of TEPs
N _{Teu}  Ordered list  200,000 TEPs  Nonordered list  400,000 TEPs 

1  8.40  16.79 
10  8.40·10^{−1}  1.68 
100  8.43·10^{−2}  1.68·10^{−1} 
1000  8.73·10^{−3}  1.71·10^{−2} 
10,000  1.17·10^{−3}  2.01·10^{−3} 
For N _{Teu}=100, when the list is ordered the latency is at most about 84 ms while, when the list is nonordered, it is at most about 170 ms. By tolerating a maximum latency of this size, it is possible to generate the TEPs onthefly with a minimum impact on the performance.
In Table 2, the worstcase latency has been determined by considering the MRB algorithm only. In the hybrid approach, the latency due to the IA must be taken into account as well. For fixing the ideas, in the following we suppose to apply the SPA but, obviously, the analysis can be repeated for any other IA.
where L _{IA} represents the contribution due to the IA and L _{MRB} the contribution due to the MRB algorithm.
When SPA is used alone, (12) is satisfied by assuming the maximum admissible number of iterations, even larger than I _{max}=100, although in Section 4.2 we have shown this may yield a negligible improvement. For the hybrid algorithm, instead, in order to satisfy (12), besides I we can choose the value of N _{TEP} as well as the parallelization parameter N _{Teu}. A numerical example is reported next.
Example 2
The degrees of freedom offered by the choice of I and N _{TEP}, for a given value of N _{Teu}, can be used to optimize the error rate performance. In other words, we can search for the combination which allows obtaining the minimum value of E _{ b }/N _{0} for a given target CER. An example is reported next.
Example 3
Values of Eb/N0 required to achieve CER = 10^{−5} for different decoder configurations
N _{Teu}=1  N _{Teu}=100  SPA used alone  

E _{ b }/N _{0} [dB]  4.75  4.09  5.10 
I  170  20  244 
N _{TEP}  70  21,125   
From the table, we see that the usage of the hybrid algorithm is advantageous with respect to the SPA used alone for both the considered cases, with a gain in the order of 0.35 dB when N _{Teu}=1 and more than 1 dB when N _{Teu}=100.
6 Conclusions
We have investigated the advantages resulting from the application of the MRB algorithm, used alone or in hybrid form, to the new short LDPC codes for TC space links. We have discussed the impact of quantization, showing that rather small numbers of quantization bits are necessary to obtain performance very close to that of the ideal, unquantized case.
Special attention has been devoted to the evaluation of the complexity, expressed as the average number of binary operations required per each decoded codeword. Closed form expressions have been adopted for such a purpose. Our investigation has revealed that, as opposed to the common belief, the hybrid algorithm is a realistic and appealing option, and that it allows achieving, for both codes, the best known performance with an acceptable complexity.
An optimal implementation of the MRB algorithm would require the availability of an ordered TEP list and a rather large memory, which may be unavailable O/B. Therefore, we have also investigated the implications of using a nonordered list and/or an incomplete list. We have shown that an “onthefly” implementation is even possible with limited loss and estimated the required decoding latency.
Although the proposed analysis refers to a very specific application, that is, space TC links, many findings are usual in a wider sense, e.g., for the application of short codes to machinetomachine communication or for ultrareliable communication with short codes.
7 Endnote
^{1} Since G is a full rank matrix by definition, its k rows being linearly independent vectors of length n>k, the GaussJordan elimination always succeeds. However, in case the k columns of G corresponding to the initial k most reliable positions are not linearly independent, it may be necessary to replace some of the most reliable bits in v ^{⋆} with some other bit outside the initial set.
8 Appendix
being T the clipping threshold (i.e., the maximum value admitted) and d the quantization step. Both T and d are related to the number of quantization bits, q. The main advantage of the logarithmic rule, against the uniform one, is in the fact that the quantization levels are denser for small input values. Since the LLR values close to 0 are responsible for maximum decoder uncertainty, it is evident that reducing the quantization error on these values allows to reduce the decoding errors. Further details can be found in [27].
Declarations
Acknowledgements
The authors wish to thank Dr. Kenneth Andrews for helpful discussion on the weight spectrum estimation. They also wish to thank Prof. Ioannis Chatzigeorgiou for having suggested the test with the ideal decoder.
This work was supported in part by the European Space Agency (ESA/ ESTEC) under the contract 4000111690/14/NL/FE “Next Generation of Uplink Coding Techniques – NEXCODE”. The final goal of the NEXCODE Project is the hardware and software implementation of LDPC decoding schemes for TC.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 JL Massey, in Advanced Methods for Satellite and Deep Space Communications (Ed. J. Hagenauer), Lecture Notes in Control and Information Science, 182, ed. by J Hagenauer. Deepspace communications and coding: a marriage made in heaven (SpringerHeidelberg and New York, 1992), pp. 1–17.View ArticleGoogle Scholar
 GP Calzolari, M Chiani, F Chiaraluce, R Garello, E Paolini, Channel coding for future space missions: new requirements and trends. Proc. IEEE. 95(11), 2157–2170 (2007).View ArticleGoogle Scholar
 T de Cola, E Paolini, G Liva, GP Calzolari, Reliability options for data communications in the future deepspace missions. Proc. IEEE. 99(11), 2056–2074 (2011).View ArticleGoogle Scholar
 F Chiaraluce, in Proc. 22nd Conference on Software, Telecommunications and Computer Networks (SoftCOM 2014). Error correcting codes in telecommand and telemetry for European Space Agency missions: an overview and new perspectives (Split, 2014).Google Scholar
 TK Moon, Error Correction Coding: Mathematical Methods and Algorithms (Wiley, Hoboken, 2005).View ArticleMATHGoogle Scholar
 CCSDS, TC synchronization and channel coding. Blue Book. CCSDS 231.0B2 (2010).Google Scholar
 ECSS, Space data links—telecommand protocols, synchronization and channel coding. ECSSEST5004C (2008).Google Scholar
 LR Bahl, J Cocke, F Jelinek, J Raviv, Optimal decoding of linear codes for minimizing symbol error rate. IEEE Trans. Inf. Theory. IT20(2), 284–287 (1974).MathSciNetView ArticleMATHGoogle Scholar
 M Baldi, F Chiaraluce, R Garello, N Maturo, I Aguilar Sanchez, S Cioni, Analysis and performance evaluation of new coding options for space telecommand links—Part I: AWGN channels. Int. J. Sat. Commun. Netw. 33(6), 509–525 (2015).View ArticleGoogle Scholar
 J Hagenauer, E Offer, L Papke, Iterative decoding of binary block and convolutional codes. IEEE Trans. Inf. Theory. 42(2), 429–445 (1996).View ArticleMATHGoogle Scholar
 M Fossorier, M Mihaljevic, H Imai, Reduced complexity iterative decoding of lowdensity parity check codes based on belief propagation. IEEE Trans. Commun. 47(5), 673–680 (1999).View ArticleGoogle Scholar
 J Chen, MP Fossorier, Near optimum universal belief propagation based decoding of lowdensity parity check codes. IEEE Trans. Commun. 50(3), 406–414 (2002).View ArticleGoogle Scholar
 M Fossorier, S Lin, Softdecision decoding of linear block codes based on ordered statistics. IEEE Trans. Inf. Theory. 41(5), 1379–1396 (1995).View ArticleMATHGoogle Scholar
 Y Wu, CN Hadjicostis, Softdecision decoding using ordered recodings on the most reliable basis. IEEE Trans. Inf. Theory. 53(2), 829–836 (2007).MathSciNetView ArticleMATHGoogle Scholar
 M Baldi, F Chiaraluce, N Maturo, G Liva, E Paolini, A hybrid decoding scheme for short nonbinary LDPC codes. IEEE Commun. Lett. 18(12), 2093–2096 (2014).View ArticleGoogle Scholar
 CCSDS, Short block length LDPC codes for TC synchronization and channel coding. Orange Book CCSDS 231.1–O–1 (2015).Google Scholar
 M Baldi, M Bianchi, F Chiaraluce, R Garello, I Aguilar Sanchez, S Cioni, in Proc. IEEE 78th Vehicular Technology Conference (VTC Fall 2013). Advanced channel coding for space mission telecommand links (Las Vegas, 2013).Google Scholar
 M Baldi, N Maturo, G Ricciutelli, F Chiaraluce, in IEEE 21st IEEE Symposium on Computer and Communications (ISCC 2016). On the error detection capability of combined LDPC and CRC codes for space telecommand transmissions (Messina, 2016), pp. 1105–1112.Google Scholar
 M Baldi, N Maturo, F Chiaraluce, E Paolini, in Proc. 6th International Conference on Information and Communication Systems (ICICS 2015). On the applicability of the most reliable basis algorithm for LDPC decoding in telecommand links (Amman, 2015).Google Scholar
 I Sason, S Shamai, Performance analysis of linear codes under maximumlikelihood decoding: a tutorial. Found. Trends Commun. Inf. Theory. 3(1–2), 1–225 (2006).View ArticleMATHGoogle Scholar
 GC Clark, JB Cain, Error Correction Coding for Digital Communications (Plenum Press, New York, 1981).View ArticleMATHGoogle Scholar
 K Andrews, Weight enumerators for LDPC codes. CCSDS 2015 spring meetings presentation, Pasadena (2015).Google Scholar
 D Declercq, M Fossorier, in Proc. IEEE International Symposium on Information Theory (ISIT 2008). Improved impulse method to evaluate the low weight profile of sparse binary linear codes (Toronto, 2008), pp. 1963–1967.Google Scholar
 XY Hu, MPC Fossorier, E Eleftheriou, in Proc. 2004 IEEE International Conference on Communications (ICC 2004), 2. On the computation of the minimum distance of lowdensity paritycheck codes (Paris, 2004), pp. 767–771.Google Scholar
 A Kabat, F Guilloud, R Pyndiah, in Proc. IEEE Global Telecommunications Conference (GLOBECOM ’07). New approach to order statistics decoding of long linear block codes (Washington, 2007), pp. 1467–1471.Google Scholar
 M Fossorier, Iterative reliabilitybased decoding of lowdensity parity check codes. IEEE J. Select. Areas Commun. 19(5), 908–917 (2001).View ArticleGoogle Scholar
 M Baldi, F Chiaraluce, G Cancellieri, Finiteprecision analysis of demappers and decoders for LDPCcoded MQAM systems. IEEE Trans. Broadcast. 55(2), 239–250 (2009).View ArticleGoogle Scholar