Practical Implementation of Hybrid DLT Codes for Cooperative RelayCommunications †

Xilin Cheng∗, Rui Cao‡, and Liuqing Yang∗∗ Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523, USA

xlcheng@lamar.colostate.edu, lqyang@engr.colostate.edu‡LSI corporation, Milpitas, CA 95035, USA, ray.cao@lsi.com

Abstract—Forward error correction codes are commonlyadopted in dual-hop relay communications to ensure the link-layer communication reliability. Among all FEC codes, LubyTransform (LT) codes are favorable because of their low decodercomplexity and rate adaptability to channel dynamic fading. Toalleviate the high computation cost in the primitive LT-basedcooperative communications, hybrid decomposed LT (h-DLT)codes are proposed in our recent work. Theoretical analyses showthat reduced energy consumption and latency can be achievedfor h-DLT codes assisted cooperative relay communications. Inthis paper, the practical implementation of h-DLT codes incooperative relay communications with limited storage capabilityis investigated. As the direct application of the original h-DLTcodes to practical systems can induce high communication costs,we rst design a new type of h-DLT codes that enables energy-ef cient reliable cooperative communications. Then, the codeconstruction algorithms are provided and the correspondingcommunications protocol is devised. Finally, simulations areconducted to manifest the performance and bene ts of thecooperative relay communication system with the newly proposedh-DLT codes and the effects of multiple design factors, such asthe storage size at relays, the mode ratio, the number of relays,and etc.

I. INTRODUCTION

Cooperative relay communications have been extensive-ly studied in the literature to enhance the communicationreliability and extend range [1], [2]. This dual-hop relaycommunications have some unique requirements on the pro-tocol design. First, due to the independent channel fading onthe source-relay (S-R) and relay-destination (R-D) links, thecommunication reliability has to be guaranteed for both links[3]. Secondly, the end-to-end data delivery latency needs tobe reduced, especially for delay-sensitive applications, suchas underwater acoustic networks [4]. Thirdly, heterogeneousnode energy is a critical factor that limits the network lifetime.

In the literatures, fountain codes are proposed to improve thecommunication reliability and reduce the end-to-end latency ofdual-hop relay communication systems. In [3], [5], [6], inde-pendent fountain encoding is adopted at each hop to ensurethe dual-hop transmission reliability. As the relays need todecode and re-encode each received packet, high computationcost is incurred at relays. To address this issue, concatenatedencoding is adopted in [7], [8], [9], [10], where the relayssimply apply a second-layer encoding to the fountain-codedsource data without decoding. However, this can give rise tosigni cant decoding complexity at the destination. In orderto reduce computation complexity and cost while retainingthe communication reliability on both links, the concept of

† This work is in part supported by National Science Foundation undergrant #1129043 and Of ce of Naval Research under grant #N0014-07-1-0868.

decomposed fountain codes has been proposed. Typically, thedecomposed fountain codes consist of two-layer data encodingwhich can be performed collaboratively by the source andrelays. Analyses in [11] show that the asymptotic performanceof Decomposed Luby Transform (DLT) codes with two-layerrandom encoding is the same as that of the non-decomposedLT codes. The rst DLT code is proposed in [12], which isa special case of DLT codes with the second-layer encodingdegree xed to 2 or 4. In our recent work [13], generalizedhybrid DLT (h-DLT) codes are studied, which enable theexible computation cost allocation between two encoders and

readily cope with the node energy heterogeneity issue in dual-hop relay communications.

In this work, we investigate the implementation of the h-DLT codes in cooperative relay communications from thepractical point of view. Due to the skewed degree distributionat the source, the direct application of h-DLT codes in [13]to the dual-hop relay communication system results in largeredundancy among the packets sent by the source, which caninduce the high communication cost. To address this issue,we design a new type of h-DLT codes for cooperative relaycommunications. With the newly proposed h-DLT codes, theS-R packet redundancy is largely reduced. For presentationclarity, we name the h-DLT codes in [13] as h-DLT I codesand the newly proposed h-DLT codes as h-DLT II codes in therest of the paper. To implement h-DLT II codes in dual-hoprelay communications, we rst investigate the decompositionalgorithm for the h-DLT II codes, and then a practical coop-erative protocol is designed based on h-DLT II codes. Timedivision multiple access (TDMA) is adopted as the multipleaccess method. In simulations, the bene ts of the h-DLT IIcodes assisted system are illustrated through comparisons withthe h-DLT I codes assisted system. In addition, the effects ofseveral system parameters are also revealed, which include thestorage size at relays, the mode ratio, and the relay number.

The rest of the paper is organized as follows. In Sec. II,the background of LT codes, DLT codes, and h-DLT codes isbrie y reviewed. In Sec. III, the h-DLT II codes constructionis presented. Then, the h-DLT II codes assisted cooperativerelay communications protocol is described in Sec. IV. Thesimulation results are provided in Sec. V. The summarizingremarks are given in Sec. VI.

II. BACKGROUND

A. LT codes

Fountain codes are rateless erasure codes with the propertythat unlimited coded data can be potentially generated fromthe source data, and from any subsets of the coded data with

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size equal to or slightly larger than the number of the sourcedata, the source data can be recovered [14].

LT codes [15] are the rst practical realization of fountaincodes. The corresponding encoding process of the source datacontaining K input packets consists of two steps:

1) First, the encoder randomly chooses integer d ∈ [1,K]as the degree of the coded packet according to a degreedistribution polynomial (DDP) μ(x) =

∑Ki=1 μix

i withμi representing the probability of choosing degree d = i.According to probability theory, DDP μ(x) has the fol-lowing properties: μ(1) = 1 and μi ≥ 0, i ∈ {1, · · · ,K}.

2) Secondly, d input packets are randomly selected from theK source packets and then XORed together to generateone LT packet.

LT decoding adopts the belief propagation (BP) technique torecover source packets from LT packets. With the encodingdegree and packet index information of each LT packet, abipartite graph is formed. The decoder starts by releasingpackets with degree one. Then, all edges connected to thedegree one packet(s) are removed. This is done recursivelyuntil no degree-one packet is left. If all K input packets arerecovered, the decoding is successful, otherwise, a failure isreported. To achieve high decoding success probability, a goodencoding degree distribution has to be designed. In [15], Lubydesigned the Robust Soliton distribution (RSD) as follows:

De nition 1 With two parameters δ ∈ [0, 1] and c ≥ 0, theRSD can be computed as:

μ(x) =ρ(x) + τ(x)

β, (1)

where β = ρ(1) + τ(1) is the normalizing constant, ρ(x) =x/K +

∑Ki=2 x

i/i(i − 1) is the DDP of the Ideal Solitondistribution, τ(x) =

∑K/R−1i=1 Rxi/iK+R ln(R/δ)xK/R/K ,

and R = c√K ln(K/δ).

One important property of RSD is that with K +O(√K ln2(K/δ)) RSD encoded packets, the BP decoder cansuccessfully recover all K source packets with the probabilityof at least 1− δ.

B. Decomposed LT (DLT) codes

Different from LT codes, DLT codes generate a packetusing two-layer random encoding. For each layer, the encodingprocess is in the same manner as LT codes, except with adifferent DDP. DLT encoding can be described as follows:

1) At the rst-layer encoder, the packets are encoded withDDP θ(x). The output packets are termed as DLT-1packets;

2) Then, the DLT-1 packets are input to the second-layerencoder with another DDP ω(x). The nal output packetsare called as DLT-2 packets.

The DLT decoder utilizes the same BP algorithm as the LTdecoder. Mathematically, the resultant DDP of the concate-nated random codes with θ(x) and ω(x) can be computed asμ̂(x) = ω(θ(x)). In order to achieve the decoding performancecomparable to the LT codes, the two encoding DDPs θ(x) andω(x) need to be delicately designed such that μ̂(x) resembles

the LT distribution μ(x). As shown in our previous work[13], the exact non-trivial decomposition of μ(x) with pos-itive coef cients does not exist, and approximate polynomialdecomposition algorithms can be developed by relaxing someDDP restrictions, e.g., θ(1) = 1 or ω(1) = 1. In [13], thealgorithm corresponding to the relaxed restriction θ(1) ≤ 1is developed. Similarly, we can develop the algorithm whichis corresponding to the relaxed restriction ω(1) ≤ 1 and amain component of the decomposition algorithm for h-DLT IIcodes, i.e., algorithm 2.Problem Statement 1 For a given DDP μ(x) with maximumdegree K , determine a nonnegative-coef cient polynomialθ(x) with θ(1) = 1, such that the following linear equationset has a nonnegative solution ω(x) with ω(1) ≤ 1,

Θ̃ω = μ̃, (2)

where ω = [ω1, ω2, · · · , ωDω ]T , μ̃ = [μ1, μ2, · · · , μDω ]

T , andΘ̃ de ned in [13] is a Dω ×Dω lower-triangular matrix offull-rank consisting of θi, i ∈ 1, · · · , Dθ.

For a smooth distribution μ(x), the decomposition algorithmcorresponding to the above problem statement is given as:

Algorithm 1: Degree Distribution Decomposition

Input: The target degree distribution μ(x)Result: The decomposed DDPs θ(x) and ω(x)Initialization: Set a small value for α ∈ [0, 1];while α < 1 do

for j = 1 to Dθ doif j = 1 then Choose a value for θ̂1 that satis es(8) in [13];else Compute the valid range [θ̂min,j , θ̂max,j ] forθ̂j according to Eq. (9) in [13];Calculate θ̂j = αθ̂min,j + (1− α)θ̂max,j ;

endCalculate θ(x) = θ̂(x)/θ̂(1) and Θ̃;Compute ω from Eq. (2);if ω ∈ [0, 1) and ω(1) ≤ 1 then output thecoef cients of θ(x) and ω(x) and break;else Increase α = α+ δα;

end

C. Hybrid DLT codes

The degree distribution decomposition algorithm providessatisfactory decomposition results for a smooth DDP, but mayfail to decompose non-smooth DDPs, e.g., RSD. Thus, h-DLTcodes are designed by extracting a decomposable part out ofthe distribution μ(x) for two-layer DLT encoding, while theremaining distribution is treated for one-layer LT encoding.The output degree distribution of the h-DLT codes can retainthat of the LT codes, and thus the decoding performance canbe improved over pure DLT codes.

The original h-DLT codes, namely h-DLT I, are proposedin [13] with the encoding process given as follows:

1) At the rst-layer encoder, a binary random numbergenerator is adopted to select an encoding mode. With

probability η, the encoder will choose the cooperativeDLT mode and generate a DLT-1 packet with encodingDDP θ1(x); with probability 1 − η, the encoder willoperate in the direct LT mode, and an LT packet isencoded with encoding DDP θ0(x). All coded packetsare labeled and sent to the second-layer encoder.

2) At the second-layer encoder, an encoding degree d ischosen with DDP ω(x). Then, d packets are randomlychosen from the inputs. If all selected packets are labeledas DLT-1, they are XORed together to generate an h-DLT packet which is the second-layer encoder output;otherwise, one LT packet is output as an h-DLT packet.

III. HYBRID DLT II CODES

In this section, we rst analyze the disadvantages of thedirect application of h-DLT I codes in practical relay commu-nication systems. Then, the h-DLT II codes are designed, andthe corresponding decomposition algorithm is developed.

A. Motivations

In [13], the h-DLT I codes assisted cooperative communica-tions protocol is proposed. In the protocol, TDMA is adoptedas the multiple access method. As the packets generated atrelays are essentially the linear combination of the receivedpackets from the source, the decoding performance at thedestination is mainly determined by the decoding propertyof the encoded packets from the source. For h-DLT I codes,as seen from table I, the source has much smaller averageencoding cost than that of primitive LT codes, which meansmost of the encoded packets have low encoding degrees. Thus,the skewed DDP at the source can induce large redundancyamong the packets generated and the low decoding successprobability at the destination, which will degrade the systemperformance severely.

To increase the decoding success probability of the encodedpackets generated from the source, we design a new type ofh-DLT codes: h-DLT II. For the new codes, large encodingcost is allocated to the source to ensure the high decodingsuccess probability of the generated packets. In the following,the detailed code design is described.

B. h-DLT II codes

1) Encoding scheme: The encoding process of h-DLT IIcodes is given as follows.

1) Same as the rst layer encoding process of h-DLT I codes.2) At the second-layer encoder, one packet is selected from

the inputs. If the packet is an LT packet, it is directlyoutput as the h-DLT packet. Otherwise, an encodingdegree d is chosen with distribution ω(x). Then, d DLT-1 packets randomly selected from the inputs are XORedtogether to generate an h-DLT packet.

With this hybrid encoding scheme, the resultant degree dis-tribution of the output packets and the corresponding averageencoding degree for both encoders can be obtained.

2) Resultant degree distribution: For an h-DLT II code witha rst-layer encoding DDP θ(x) = ηθ1(x)+ (1− η)θ0(x) andsecond-layer DDP ω(x), the resultant degree distribution μ̂(x)is computed as:

μ̂(x) = ηω(θ1(x)) + (1− η)θ0(x). (3)

The average encoding degrees of the rst-layer encoder (C1)and the second-layer encoder (C2) are:

C1(η) = θ′(1) = ηθ

′1(1) + (1− η)θ

′0(1) (4)

C2(η) = (1− η) + ηω′(1) = η(ω

′(1)− 1) + 1. (5)

Differentiating Eq. (3) and setting x = 1, θ′0(1) can be

computed as:

θ′0(1) =

μ′(1)− ηω

′(1)θ

′1(1)

1− η. (6)

Substituting Eq. (6) into Eq. (4), we get:

C1(η) = θ′1(1)(1− ω

′(1))η + μ

′(1). (7)

As ω′(1) > 1, C1 in Eq. (7) is a linear decreasing function

of the encoding ratio η. Similarly, C2 in Eq. (5) is a linearincreasing function of the encoding ratio η.

Denote the resultant DDPs of the cooperative DLT modeand the direct LT mode as μ1(x) and μ0(x). From (3), itcan be obtained that μ1(x) = ηω(θ1(x)) and μ0(x) = (1 −η)θ0(x). De ne the mode ratio γ as the portion of the totaldistribution assigned to the cooperative DLT mode. Then, γ =μ1(1)/μ̂(1) = η. It’s notable that for h-DLT II codes, theencoding ratio η is also the mode ratio γ.

C. Hybrid distribution decomposition for h-DLT II codes

In order to facilitate the single layer BP decoding on theh-DLT packets, a target RSD μ(x) needs to be decomposedinto three distributions: θ1(x), θ0(x), and ω(x). Note that onlythe DLT mode DDP μ1(x) needs to be further decomposed.Hence, we can determine a proper decomposable μ1(x) for thecooperative DLT encoding and decompose it using algorithm1. Then, the remaining distribution μ(x) − μ1(x) is assignedto μ0(x) for the direct LT mode. This gives rise to the hybridRSD decomposition algorithm for h-DLT II codes.

Algorithm 2: Hybrid RSD Decomposition

Input: The target degree RSD distribution μ(x) and thedesired mode ratio γd

Result: The decomposed DDPs θ(x) and ω(x)Initialization: Construct a smooth distributionμ̃(x) = (ρ(x) + τ̃ (x))/β with τ̃ (x) =

∑k/Ri=1 (R/ik)xi;

(1) Compute μ1(x) = γdμ̃(x)/μ̃(1);(2) Decompose μ1(x) into θ1(x) and ω̃(x) usingAlgorithm 1;(3) Calculate θ0(x) = (μ(x) − ω̃(θ1(x)))/(1 − γd);(4) Compute the decomposed distributionθ(x) = γdθ1(x) + (1− γd)θ0(x) and ω(x) = ω̃(x)/ω̃(1).

One advantage of the hybrid RSD decomposition algorithmfor h-DLT II codes is that μ(x) can be decomposed only if

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Overhead( )

Suc

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ful d

ecod

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abili

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Primitive LTh DLT II ( = 1/4)h DLT II ( = 1/2)h DLT II ( = 3/4)

Fig. 1. The average recovery ratio with respect to different overheadfor the primitive LT code and the h-DLT II codes

μ̃(x) is decomposable. The other advantage is that only oneiteration is needed. In contrast, the hybrid RSD decompositionalgorithm for h-DLT I codes needs several iteration with eachiteration performing the degree distribution decomposition,until the exact mode ratio is reached.

D. h-DLT II codes performance

For a target RSD μ(x) with parameters k = 1000, c = 0.08and δ = 0.05, the corresponding h-DLT II distributions θ(x)and ω(x) are computed using algorithm 2, and the resultantdistribution μ̂(x) can be calculated according to Eq. (3). TheKL divergence shows that Dh-DLT II = 0 for all γ value, whichmeans the hybrid RSD decomposition algorithm for h-DLT IIcodes generates exactly the same distribution as the RSD. InFig. 1, we simulate the probability of successful decoding withrespect to different overheads for the primitive LT code and h-DLT II codes. It can be observed that h-DLT II codes achievesimilar decoding performance as the primitive LT code.

The average encoding degrees at both encoders are comput-ed using Eqs. (7) and (5). Table I gives the average encodingdegrees of the h-DLT II codes, the h-DLT I codes and theconcatenated LT code. It can be seen that the average encodingdegrees of h-DLT II codes for both layers are smaller thanthat of the concatenated LT code, demonstrating the lowercomputation property of h-DLT II codes. By comparing withh-DLT I codes, h-DLT II codes have the higher C1, but smallerC2. The encoding cost ratio of h-DLT II codes is around 4,while the encoding cost ratio of h-DLT I codes is around1, which means that h-DLT I codes distribute the encodingburden over the two encoders evenly, while h-DLT II codesput more encoding burden on the rst-layer encoder.

IV. HYBRID DLT II CODES ASSISTED COOPERATIVE

COMMUNICATIONS PROTOCOL

Consider a cooperative communication system with onesource node, N relays, and one destination node. Given themode ratio γ and the RSD distribution, we can obtain the

TABLE IAVERAGE ENCODING DEGREE

Code Type C1 C2 Ratio (C1/C2)h-DLT II (γ = 0.8) 8.93 2.77 3.22h-DLT II (γ = 0.6) 9.61 2.33 4.12h-DLT II (γ = 0.4) 10.29 1.89 5.44h-DLT I (γ = 0.8) 2.85 3.47 0.82h-DLT I (γ = 0.6) 2.94 3.06 0.96h-DLT I (γ = 0.4) 3.58 2 1.79Concatenated LT 11.64 11.64 1

encoding DDPs θ1(x), θ0(x), and ω(x) through the hybridRSD decomposition algorithm 2. The h-DLT II codes assistedcooperative communications protocol consists of three parts:source encoding and broadcast, relay encoding and forward-ing, and destination decoding. Without loss of generality,TDMA is utilized in this protocol, i.e., the source and therelays transmit packets in order at different time slots.

1) Source encoding and broadcast: With K data packetsto be transmitted, the source generates coded packetsusing the rst-layer DDP θ(x) = γθ1(x) + (1− γ)θ0(x).An mode type ID bit is attached to each coded packetto indicate its encoding mode: LT (ID = 0) or DLT-1(ID = 1). The coded packets are continually generatedand broadcast to the relays at the source’s transmissiontime slots, until an acknowledgement (ACK) is receivedfrom the relays.

2) Relay encoding and forwarding: After receiving thepacket from the source, each relay undergoes an errordetection process, e.g., cyclic redundancy check (CRC) atthe physical layer. If CRC succeeds, the packet is furtherprocessed and the packet ID is retrieved: the receivedpacket with ID = 0 is treated as the h-DLT packet withID = 0 without storage, while the received packet withID = 1 is stored in the memory, and an h-DLT packetwith ID = 1 is generated from storage. The generationof the h-DLT packet with ID = 1 consists of two steps:First, an encoding degree d is randomly chosen accordingto distribution ω(x); secondly, d DLT-1 packets fromstorage are XORed together to generate an h-DLT packet.If CRC fails, the packet is dropped, and an h-DLT packetwith ID = 1 is generated from storage. Each relay keepsforwarding h-DLT packets attached with their mode typeID bit to the destination in its own time slots, until anACK is received from the destination. The received ACKis also forwarded to the source by the relay.

3) Destination decoding: Depending on the mode type IDof the received h-DLT packet, the destination choosesthe corresponding reception technique. For the h-DLTpacket with ID = 0, joint decoding is performed atthe physical layer, as relays may forward the samepacket when the source broadcasts the LT packet. Anycooperative transmission schemes, e.g., maximum ratiocombination (MRC), can be adopted to enhance thecommunication reliability; for each h-DLT packet withID = 1, independent decoding is conducted. After CRCchecks, all correctly received packets are forwarded tothe BP decoder to recover the source packets, and all

packets with error are deleted. If all K source packetsare decoded, an ACK is sent back to the relays.

Practically, the relays have the limited storage capability,and the storage space will be full when the number of storedpackets exceeds the storage size. In such a situation, one pack-et has to be discarded in the storage space to accommodate thenewly arrived packet. There are mainly two kinds of storageschemes. For the storage scheme I, the newly arrived packetalways replaces the packet which stays longest in the storagespace, while for the storage scheme II, the packet which ismostly used is replaced. To implement the storage schemeII, each packet in the storage space is assigned a counter. Ifa packet is selected for the XOR operation, its counter valueincreases by one. The newly arrived packet always replaces thepacket which has the largest counter value, and the counter ofthe new packet is initiated to be zero.

V. SIMULATION RESULTS

In this section, we simulate the h-DLT II codes assistedcooperative communications protocol under different channelerasure rates and collect the average number of transmissionsper packet for full recovery of all data at the destination,i.e., the ratio of the total number of the packets generatedby the source and the relays to the number of the source data.The results are compared with those of the h-DLT I codesbased schemes to illustrate the bene ts of h-DLT II codes. Inaddition, the effects of different design parameters are shown.

A. Codes bene ts and storage schemes

The communication cost is evaluated by the average numberof transmissions per packet. With the mode ratio γ = 0.5and the storage size of 50, we simulate the communicationcost of cooperative relay communications protocols based onthe h-DLT I and II codes and with different storage schemes.The results are plotted in Fig. 2. From the gure, it can beobserved that the h-DLT II codes assisted scheme requiresless communication cost compared with the h-DLT I codesassisted scheme. As explained in Sec. III-A, their performancedifference comes from the redundancy among the encodedpackets at the source. For h-DLT I codes, due to the very smallaverage encoding degrees for the generated packets from thesource, large redundancy exists among the received packets atthe relays and the destination, which renders the worse systemperformance.

In addition, the storage scheme II shows better performancethan the storage scheme I, especially for the h-DLT I codes.In the storage scheme I, a coming packet always replaces theoldest packet in the storage space. Because of the randomencoding at the relay, the replaced packet may not be chosenfor encoding into any forwarded packets, and some packetsremaining in the memory may have been selected for encodingmultiple times. These can cause either information loss orcollision at the destination. On the other hand, by replacingthe most used packet, the storage scheme II resolves the issue.Thus it provides better performance.

B. Affecting factors

1) Effect of storage space: The storage space is an impor-tant resource for relays. In Fig, 3, the communication cost

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h−DLT I, Storage Scheme Ih−DLT I, Storage Scheme IIh−DLT II, Storage Scheme Ih−DLT II, Storage Scheme II

Fig. 2. The communication cost for the h-DLT I and h-DLT IIschemes with storage size 50

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Fig. 3. The communication cost of the h-DLT II codes basedcooperative protocol with different storage sizes

of the h-DLT II codes assisted cooperative protocol is plottedwith different storage sizes. The storage scheme II is utilized,the mode ratio is chosen as γ = 0.5, and the packet erasureprobability is 0. From the gure, it can be observed that thecommunication cost remains nearly constant with the storagesize, which means that the storage size have little in uenceon the communication cost. Thus, to save the storage space atrelays, the storage size should be chosen to be small. On theother hand, the storage size cannot be less than the maximumencoding degree of ω(x). Thus, the best storage size for relaysis Dω.

2) Effect of mode ratio: In Fig. 4, the effect of the moderatio η is investigated. It’s notable that η = 0 corresponds tothe primitive LT-based communications protocol with no relayencoding. From the gure, it can be observed that at the lowpacket erasure probability, smaller η has lower communicationcost, while larger η provides better performance at the high

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Fig. 4. The communication cost of the h-DLT II based cooperativeprotocol with different mode ratio η. Storage scheme II is adopted,and the storage size is 50.

packet erasure probability. This implies that as channel erasurerate increases, more DLT encoding at the relay is bene cial,which con rms the design objective of the h-DLT codes.

3) Effect of relay number: In cooperative communications,multiple relays can be implemented to improve the communi-cation reliability. The same setup can be adopted in h-DLT IIcodes assisted schemes. For the LT mode, the relays forwardthe same LT packets to the destination, and MRC is utilizedat the destination; for the DLT mode, each relay transmits adifferent encoded packet to the destination, which providesmore reliability against channel erasures. With γ = 0.5 andthe storage scheme II, we simulate the performance of the h-DLT II codes assisted systems with different number of relaysin Fig. 5. It can be observed that for small packet erasure prob-ability, the single-relay system provides the best performance,while multi-relays systems have better performance for thehigh packet erasure probability. This con rms that when thechannel condition gets worse, the reliability can be enhancedby increasing the relay number.

VI. CONCLUSIONS

In this paper, we investigated the practical implementationof h-DLT codes in cooperative relay communications withlimited storage space at relays. Due to the high communicationcost caused by large redundancy among the received packetsat relays for h-DLT I codes, we designed the h-DLT II codeswhich reduce packet redundancy at relays a lot. The encodingand distribution decomposition schemes for the newly pro-posed h-DLT codes were presented, and results show that theh-DLT II codes achieve similar performance as the primitiveLT codes. Then, we proposed an h-DLT II codes assistedcooperative communication scheme. Simulations show thatthe h-DLT II codes assisted scheme outperforms the h-DLT Icodes assisted scheme with much smaller communication cost.In addition, affecting factor analyses indicate that h-DLT IIcodes assisted cooperative systems are insensitive to the relay

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h−DLT II, 1 relayh−DLT II, 2 relaysh−DLT II, 3 relays

Fig. 5. The communication cost of the h-DLT II based cooperativeprotocol with different relay numbers

storage size, and higher mode ratio and larger relay numberare bene cial for the cases with larger channel erasure rates.

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