Adaptive Distributed Network-Channel Coding For Cooperative Multiple Access Channel Jo˜ao Luiz Rebelatto, Bartolomeu F. Uchˆoa-Filho

Yonghui Li and Branka Vucetic

Communications Research Group Department of Electrical Engineering Federal University of Santa Catarina Florian´opolis, SC, 88040-900, Brazil Email: {jlrebelatto, uchoa}@eel.ufsc.br

Telecommunications Group Department of Electrical Engineering University of Sydney NSW 2006, Australia Email: {lyh, branka}@ee.usyd.edu.au

Abstract—In this work, we propose an adaptive distributed network-channel coding for a cooperative multiple access channel where M users cooperatively communicate with a common base station. The scheme is based on the recently proposed generalized dynamic-network codes (GDNC), in which the network code design that maximizes the diversity order was recognized as equivalent to the design of linear block codes over a nonbinary finite field under the Hamming metric. The aim here is to increase the system average code rate without reducing its diversity order, making use of a small quantity of feedback. The average rate and the diversity order are obtained analytically, and computer simulations are shown to agree with the analytical results. Index Terms—Cooperative communication, diversity order, linear block codes, nonbinary network coding.

I. I NTRODUCTION The concept of cooperation applied to multi-user wireless communications has been widely studied recently [1]– [7], due to its robustness against the channel fading. In a cooperative wireless communications system with multiple users transmitting independent information to a common base station (BS), besides broadcasting their own information, users help each other relaying their partner’s information. Through cooperation, each user’s information is transmitted by different users through independent channels, so the system diversity is increased [1], [2]. Network coding [8], [9], a method originally proposed to attain maximum information flow in a network, has recently been applied to cooperative wireless communications systems in order to improve their bit error rate (BER) performance [4]– [7], [10]. In a network coded system, relays are able to process information, performing linear combinations of their own information and information from their partners, with coefficients chosen from a finite field GF(q). In [4], it was shown that binary network coding (BNC)(with coefficients chosen from GF(2)) is not optimal for achieving full diversity in multiple user systems. The scheme proposed in [4], called dynamic-network codes (DNC), considers a fixed nonbinary network code. In the first time slot, each user broadcasts a single packet of its own to the BS as well as to the other users, which try to decode the packet. From the second time slot until the M th time slot, each user transmits to the BS M − 1 nonbinary linear combinations of the packets

that it could successfully decode. With DNC, by using an appropriately designed network code, the diversity order was shown to be higher than in BNC systems. The scheme is called “dynamic” in the sense that the network code is designed to perform well under the possible occurrence of errors in the inter-user channels. However, the design approach in [4] can become extremely complex as the number of users M grows. In [6], the problem of designing network codes subjected to link failures was recognized as equivalent to that of designing linear block codes over GF(q) for erasure correction. In particular, if the inter-user channels are error-free, it was shown that the diversity order equals the minimum Hamming distance of the block code, so the network transfer matrix should correspond to the generator matrix of an optimal block code under the Hamming metric. This equivalence between minimum distance and diversity order was proved in [7], where it was shown that if a generator matrix of a maximum distance separable (MDS) code is used as the network transfer matrix, the maximum diversity order is guaranteed. Thus, any class of MDS codes (such as the well-known class of Reed-Solomon codes [11]) could be adopt as the network code, avoiding the need of an exhaustive search in the network code design, as proposed in [4]. The scheme proposed in [6], called generalized dynamic-network codes (GDNC), is a generalization of the DNC proposed in [4]. In the GDNC scheme, a tradeoff between rate and diversity order can also be achieved, and both parameters can be made higher than in the DNC scheme. However, in both the DNC and the GDNC schemes, once the network code is designed, it remains fixed until the network configuration changes and a new code is required. This feature has a harmful effect on the average code rate, since users may still be transmitting parity packets after the BS having received sufficient information to recover all the data packets successfully. This waste of resources increases as the SNR increases. In this paper, we propose an adaptive network code design based on a small quantity of information fed back by the BS. The aim is to increase the average code rate (and consequently the system rate), without reducing the diversity order achieved by the original GDNC scheme. In the so-called FeedbackAssisted (FA) GDNC, the number of parity packets transmitted

in the cooperation phase is properly chosen according to the feedback information. We obtain analytical expressions for the average rate and the diversity order. We also show computer simulations results that agree with the analytical results. The remainder of this paper is organized as follows. The next section presents the system model and some relevant previous works, including the BNC scheme [5], the DNC scheme [4], and the GDNC scheme [6], [7]. The motivation for this work is presented in Section III. Section IV presents the proposed FA-GDNC scheme. The analysis is also given in this section. Simulations results are presented in Section V. Finally, Section VI presents our conclusions and final comments.

where Po is the system overall outage probability. In this work, block fading means that fading coefficients are i.i.d. random variables for different packets but constant during the same packet. It is also assumed that the receivers have perfect channel state information (CSI), but the transmitters do not have any CSI1 . B. Dynamic-Network Codes In [4], it was shown that the use of nonbinary network coding is necessary to achieve a full diversity order, and the so-called DNC scheme was proposed. A simple 2-user DNC scheme is presented in Fig. 1. When both inter-user channels

II. P RELIMINARIES User 1

A. System Model We consider a network in which multiple users (M ≥ 2) have different information to send to a common base station (BS). One time slot (TS) is defined as the time period in which all the M users realize a single transmission (through orthogonal channels, either in time, frequency or code), that is, one TS corresponds to M transmissions. The baseband codeword received by user i at time t is given by yj,i,t = hj,i,t xj,i,t + nj,i,t ,

(1)

where j ∈ {1, · · · , M } represents the transmit user index and i ∈ {0, 1, · · · , M } the receive user index (0 corresponds to the BS). The index t denotes the time slot. xj,i,t and yj,i,t are the transmitted and the received packets, respectively. nj,i,t is the zero-mean additive white Gaussian noise with variance N0 /2 per dimension. The channel gain due to multipath is denoted by hj,i,t , and it is assumed to have independent identically distributed (i.i.d.) (across space and time) Rayleigh distribution with unit variance. Assuming xj,i,t ’s to be i.i.d. Gaussian random variables and considering all the channels with the same average signal-tonoise ratio (SNR), the mutual information Ij,i,t between xj,i,t and yj,i,t is Ij,i,t = log(1 + |hj,i,t |2 SNR).

(2)

Assuming powerful enough channel codes, xj,i,t can be correctly decoded if Ij,i,t > rj,i,t , where rj,i,t is the information rate from user j to user i in the time slot t. Considering that all the users have the same rate at all times, the index of r can be dropped. Thus, xj,i,t cannot be correctly decoded if |hj,i,t |2 < g,

(3)

r

−1 where g = 2SNR . The probability that such an event happens is called the outage probability. For Rayleigh fading, the outage probability is calculated as [2], [12]  Pe = Pr |hj,i,t |2 < g = 1 − e−g ≈ g, (4)

where the approximation holds for a high SNR region. Considering block fading, the diversity order D is defined as [12] − log Po D , lim , SNR→∞ log SNR

(5)

User 1

O RRRRRRI1 RRR BS RR) I1 I2 3  ffffffffff f I 2

User 2

RRR RRIR1 +I2 RRR BS RR) f3 f f f f fffIf1f+2I2 f

User 2

(a) Broadcast phase.

(b) Cooperative phase.

Fig. 1. Two-user cooperative network with nonbinary network coding. (a) Each user broadcasts its own information and (b) each user transmits a linear combination over GF(4) of all the available information packets.

are not in outage, we can see that the BS is able to recover the two information packets I1 and I2 from any 2 out of the 4 received packets. An outage (for any of the two information packets) occurs only when at least 3 out of the 4 received packets are in outage. When a user cannot decode its partner’s information packet, which happens with probability Pe , it retransmits its own information. On receiving two copies of the same packet, the BS performs maximum ratio combining (MRC), and the outage probability was shown to be P2,MRC = Pe2 /2 [2]. For the 2-user DNC network presented in Fig. 1, it was shown in [4] that the outage probability, under the assumption of nonreciprocal inter-user channels (i.e., hj,i,t 6= hi,j,t ), is given by Po,DN C ≈ 4Pe3 . In an M -user DNC scheme, each user transmits M − 1 nonbinary linear combinations in the cooperative phase, and the diversity order was shown to be [4] DDNC = 2M − 1.

(6)

However, it is achieved with a low, fixed rate R = M/M 2 = 1/M . C. Generalized Dynamic-Network Codes In [6], the problem of designing network codes was recognized as equivalent to that of designing linear block codes over 1 The users do not have any CSI when transmitting in the broadcast phase. During the cooperative phase, the information fed back by the BS can be viewed as a partial CSI.

GF(q) for erasure correction. In particular, for perfect interuser channels, it was shown that the diversity order equals the minimum Hamming distance of the block code, so the network transfer matrix should correspond to the generator matrix of an optimal block code under the Hamming metric, and bounds from the classical coding theory, such as the Singleton bound [11] could be used as bounds for the diversity order. The DNC scheme was then extended by allowing each user to broadcast several (as opposed to just one) packets of its own in the broadcast phase, as well as possibly to transmit several nonbinary linear combinations (of all correctly decoded packets) in the cooperative phase. From the block coding perspective, the so-called generalized dynamic-network codes (GDNC) considers a longer codeword, with more parity symbols, which improves the Singleton bound. The GDNC scheme in its general form is illustrated in Fig. 2. Ij (t) represents the information (symbol or packet) transmitted by user j (j = 1, · · · , M ) in time slot t (t = 1, · · · , k1 ) of the broadcast phase (at left), and Pj (t′ ) corresponds to the parity (symbol or packet) transmitted by user j in time slot t′ (t′ = 1, · · · , k2 ) of the cooperative phase (at right). Following Fig. 2, each user first broadcasts k1 independent Broadcast phase user 1 .. . user M

I1 (1)

..

I1 (k1 )

...

. IM (1)

Cooperative phase ..

P1 (1)

. IM (k1 )

..

P1 (k2 )

.

... PM (1)

..

. PM (k2 )

Time

Fig. 2. GDNC protocol. Each user broadcasts k1 information packets and then transmits k2 parity-check packets composed of linear combinations over GF(q) of its own information and all the packets correctly decoded in the broadcast phase.

information packets2 . In the cooperative phase, each user transmits k2 parity-check packets consisting of nonbinary linear combinations of its k1 own information packets and the k1 (M −1) partners’ information packets (if decoded correctly). If a user cannot correctly decode an information packet from one of its partners, this information packet is replaced by an all-zero packet in the formation of the linear combination. Herein, as in [4], [6], we assume that the BS knows how each parity packet is formed. This may lead to some overhead, e.g., some extra information in packet headers. Yet, if Ij (t) is long enough, the overhead is negligible. Thus, the GDNC overall rate is given by: RGDNC =

k1 M k1 = . k1 M + k2 M k1 + k2

(7)

Applying (7) to the Singleton bound given by dmin = n − k + 1 [11], it was shown that the diversity order of the GDNC scheme is upper bounded by DGDNC ≤ k2 M +1, which cannot be achieved due to the errors in inter-user channels. 2 These packets can be sent in any order, since the channels are considered uncorrelated across both time and space.

1) Outage Probability and Diversity Order: In [6, Theorem 1] it was proven that if an appropriately designed network code is used as the network transfer matrix, the diversity order of the GDNC scheme when the inter-user channels are subjected to errors is DGDN C = M + k2 .

(8)

It can be seen that the GDNC reduces to DNC when k1 = 1 and k2 = M − 1, with rate 1/M and diversity order 2M − 1. From (7) and (8), we can see that an appropriate choice of k1 and k2 can simultaneously improve rate and diversity order over the DNC scheme. 2) On the GDNC Network Code Design: Regarding the network code design, it was shown in [7] that a systematic matrix of a MDS code C with minimum Hamming distance dmin = M k2 + 1 being used as a transfer matrix of the GDNC scheme is a sufficient and necessary condition to guarantee the diversity order DGDNC = M + k2 . III. M OTIVATION

FOR THIS

W ORK

When the SNR increases, the probability that the BS correctly decodes all the information transmitted in the broadcast phase also becomes higher. In this case, the parity packets transmission is no longer necessary. Since the number of information packets transmitted during the broadcast in the GDNC scheme is equal to M k1 , this probability is then given by Mk1 Pr{None in outage} = (1 − Pe )Mk1 = P e , (9) where Pe is the single link outage probability given in (4) and ∆ P e = 1 − Pe . The increase of the probability in (9) with the SNR is exemplified in Table I, for a 2-user network with k1 = 2. We can see that, at SNR= 12dB, for example, unnecessary TABLE I P ROBABILITY OF NONE OF THE M k1 PACKETS BEING IN OUTAGE vs SNR, FOR M = 2 AND k1 = 2. SNR (dB)

0

4

8

12

Pr{None in outage}

0.19

0.52

0.77

0.90

parity-check packets are transmitted in 90% of the cases, compromising the transmission rate. This motivated us to modify the GDNC scheme accordingly, aiming to improve the system rate, while keeping the same diversity order. IV. F EEDBACK -A SSISTED G ENERALIZED DYNAMIC -N ETWORK C ODES In this work, we elaborate on the GDNC scheme, by assuming that the BS is able to send a very small amount of information back to the users regarding the success/failure in the decoding of the packets received during the broadcast phase. Each message fed back by the BS consists of only one bit, called outage (OUT) bit (or OUT flag). OUT = 0 means that the BS has correctly decoded the information packet

(or even a set of packets), and OUT = 1 means that the information packet (or part of a set of packets) has not been correctly decoded. In the GDNC scheme, each user broadcasts k1 information packets in the broadcast phase, resulting in M k1 packets broadcasted altogether per transmission round3. After receiving these packets (whether successfully or not), the BS informs the users through a feedback channel about the decoding process outcome. Herein, we consider that the BS’s feedback is limited to only one OUT bit per user after each broadcast phase. If the information packet length is long enough, this amount of feedback is negligible. Let OUTj (τ ) be the outage flag associated with user j in the transmission round τ . OUTj (τ ) = 0 if all the k1 information packets transmitted by user j in the round τ have been correctly decoded at the BS, and OUTj (τ ) = 1 otherwise. Thus, the users will collect a set of M feedback bits after each broadcast phase. In the cooperative phase of the GDNC scheme, according to Fig. 2, the number of parity-check packets transmitted per user is fixed and equal to k2 . Here, based on the received feedback, each user is allowed to adaptively choose the amount of paritycheck packets to be transmitted in the cooperative phase. Our goal is to increase the system rate while keeping the same diversity order as the original GDNC scheme. Let k2,j (τ ) be the number of parity-check packets transmitted by user j in the transmission round τ . We propose to chose k2,j (τ ) as  PM  0, if j=1 OUTj (τ ) = 0 k2,j (τ ) = k2 , if OUTj (τ ) 6= 0   ′ k2 , otherwise

(10)

where k2 is the same as in the GDNC scheme and k2′ ∈ {0, 1, . . . , k2 }. If all the M k1 information packets have been correctly decoded (the sum of all the OUTs is equal to zero, corresponding to the first row of (10)), then no paritycheck is needed and k2,j (τ ) is set to zero for all j (which means that a new set of M k1 information packets may be transmitted next). If user j has at least one of its own k1 information packets incorrectly decoded by the BS, then it will set k2,j (τ ) = k2 parity packets. When all the k1 information packets broadcasted by user j are correctly decoded by the BS, but at least one out of the (M − 1)k1 information packets broadcasted by the other users is in outage, user j will collaborate with only k2′ ≤ k2 parity-check packets by setting k2,j (τ ) = k2′ . The value of k2′ is a system parameter and has to be optimized subjected to keep the diversity equal to M + k2 and increase the rate as much as possible. 3) Rate Analysis: Let T be the number of transmission rounds and Pτj denote the number of parity-check packets transmitted by user j in the round τ , where τ = 1, . . . , T . The total number of parity-check packets transmitted in 3 One transmission round is here defined as the composition of one broadcast phase plus one cooperative phase.

round τ is given by Pτ =

M X

Pτj ,

j=1

and its expected value can be obtained as follows   M X Pτ = E  Pτj  = M E [Pτ ]

(11a)

j=1

=M

X

Pr{Pτ = p} · p

(11b)

p

h Mk k1 1 = M Pe · 0 + (1 − P e ) · k2   i Mk1 k1 + 1 − Pe − (1 − P e ) · k2′    k1 (M−1)k1 = M k2 − M P e k2 − k2′ 1 − P e

(11c) (11d)

The average rate of FA-GDNC is then given by ROUT = =

M k1 M k1 + E [Pτ ] k1 k1 +k2 −P e

(12a)

k1    , (12b) (M−1)k1 k2 − k2′ 1 − P e

where E [Pτ ] is obtained from (11d). Since k2 ≥ k2′ and 0 ≤ P e ≤ 1, we can see from (12b) that ROUT ≥ RGDN C . 4) Outage Probability and Diversity Order: Denote by Dj,t ⊆ {1, . . . , M } the index set corresponding to the users that correctly decoded Ij (t), the information packet of user j in time slot t in the broadcast phase. For convenience, include index j itself to Dj,t . The number of users in Dj,t is denoted by |Dj,t |, and its probability is given by PDj,t = PeM−|Dj,t | (1 − Pe )|Dj,t |−1 , which corresponds to the probability of M − |Dj,t | out of M − 1 inter-user channels being in outage. We also define Qj,t ⊆ Dj,t \{j} as the index set corresponding to the users in Dj,t other than user j itself whose own information packets (all of them) have been correctly decoded by the BS. The size of Qj,t is denoted by |Qj,t |. The conditional probability of Qj,t given Dj,t can be approximated as PQj,t |Dj,t ≈ Pe|Dj,t |−|Qj,t |−1 , since the most probable scenario for Qj,t to happen is when exactly one packet of each of the |Dj,t | − |Qj,t | − 1 users in Dj,t \Qj,t is not received correctly by the BS. As j ∈ Dj,t and j ∈ / Qj,t , we must also have that |Dj,t | − |Qj,t | ≥ 1.

(13)

According to (10), the users belonging to Qj,t will contribute with k2′ parity-check packets in the cooperative phase, while the users belonging to the complement set Dj,t \Qj,t will contribute with k2 parity-check packets. Then, we have that the message Ij (t) is contained in (|Dj,t | − |Qj,t |)k2 + |Qj,t |k2′ + 1 packets (1 in the systematic part plus (|Dj,t | − |Qj,t |)k2 +

|Qj,t |k2′ as part of parities) transmitted to the BS through independent channels. For fixed Dj,t and Qj,t , it can be shown that the probability that the BS cannot recover Ij (t) is Po,j (Dj,t , Qj,t ) ≈

TABLE II N ETWORK CODES OBTAINED FROM RS

(|D |−|Qj,t |)k2 +|Qj,t |k2′ +1 γPe j,t ,

where γ is the short for γ(k1 , k2 , Dj,t , Qj,t ), a positive integer representing the number (multiplicity) of outage patterns leading to that same probability. The overall outage probability is then given by: XX PDj,t PQj,t |Dj,t Po,j (Dj,t , Qj,t ) (14a) Po,j ≈

k1

k2

Rate

q

1

1

2/4

4

1

2

2/6

8

2

1

4/6

8

2

2

4/8

8

2

3

4/10

16

2

4

4/12

16

3

2

6/10

16

3

3

6/12

16

Dj,t Qj,t

≈

XX

M+(|Dj,t |−|Qj,t |)k2 +|Qj,t |(k2′ −1)

γPe

(14b)

Dj,t Qj,t

=γ

′

∗

PeM+k2 ,

k2

(14c) (14d)

where |∆|∗ corresponds to the |∆| = |Dj,t | − |Qj,t | value that results in the lowest exponent term in (14b), which is |∆|∗ = 1 according to (13), and γ ′ collects the multiplicities of all events Dj,t and Qj,t for which |Dj,t | − |Qj,t | = |∆|∗ . Regarding the parameter k2′ in (14b), we can see that its lowest value in order to achieve the same diversity order as the original GDNC scheme is k2′ = 1. When k2′ = 0, the diversity order is decreased by |Qj,t |. On the other hand, when k2′ ≥ 1, the diversity is not increased beyond M + k2 , since Qj,t with |Qj,t | = 0 (i.e., the empty set) is a possible event and it yields M + k2 as lowest possible exponent. Thus, we have proved that the diversity order of the FA-GDNC is still M + k2 . Let G = [I|P] be a generator matrix of a linear block code in its systematic form. Table II lists the parity-check matrix P of some MDS codes obtained from RS codes which can be used as the GDNC’s network transfer matrix [7], for M = 2 users. These matrices were obtained using SAGE [13]. Since the code rate is adaptive and may assume different values, different codes (with different rates) can be obtained from codes such as the ones presented in Table II by puncturing.

M = 2 USERS

Parity matrix P h i 3 2 2 3 h i 3 5 1 7 2 4 3 6 Obtained from the 4/8 code by puncturing the last 2 columns   3 7 3 6 5 7 7 4   2 4 6 1 5 5 3 2 Obtained from the 4/12 code by puncturing the last 2 columns   14 5 7 10 4 10 8 9  12 4 14 5 12 12 14 7    13 6 12 15 8 1 4 10 14 6 4 1 1 6 3 5 Obtained from the 6/12 code by puncturing the last 2 columns  11 2 4 6 14 12   1 11  2 4   6 13  4 12 11 13

13 2 12 12 10

10 10 11 2 14

14 5 8 6 10

10  9   12   6 4

1

0.9

Average Code Rate

≈ γ ′ PeM+|∆|

CODES FOR

0.8 GDNC Approach 1, simulation Approach 1, analytical Approach 2, simulation Approach 2, analytical

0.7

0.6

0.5

V. S IMULATION R ESULTS In this section, we present some simulation results in order to support the the analysis presented in Section IV. Throughout the simulations, we consider that the information rate r in each link is equal to 0.5 bit per channel use4 . Fig. 3 shows the average rate versus SNR for a 2-user network adopting the GDNC scheme in [7] with k1 = 2 and k2 = 2, FA-GDNC with k1 = 2, k2 = 2 and k2′ = 2 (referred herein as Approach 1) and FA-GDNC with k1 = 2, k2 = 2 and k2′ = 1 (referred as Approach 2). The GDNC scheme’s rate does not depend on the SNR range and, for the given parameters, is equal to 0.5 (the same holds for the BNC [5] and DNC [4] schemes). We can see that, when k2′ = 2, the rate is substantially increased. However, by setting k2′ = 1, it is increased even further. We can also see that the simulation results match the analytical ones with great precision. 4 It should be noted that the choice of the information rate r is irrelevant for the purpose of obtaining the diversity order.

0.4

3

4

5

6

7 8 SNR (dB)

9

10

11

12

Fig. 3. System overall rate versus SNR (dB) for a 2-user network adopting the GDNC scheme with k1 = 2 and k2 = 2, Approach 1 with k1 = 2 and k2 ∈ {0, 2}, and Approach 2 with k1 = 2 and adaptive k2,j ∈ {0, 1, 2}.

Regarding the outage probability and diversity order, Fig. 4 presents the frame error rate (FER) versus SNR for the 2user network adopting the same schemes considered for Fig. 3 as well as the BNC and the DNC schemes (DNC over GF(4)). The generator matrix for the GDNC and the proposed schemes, with k1 = k2 = 2 and over GF(8), is obtained from Table II. We assume that there exists a channel code with which it is possible to recover the transmitted packet if |hj,i,t |2 ≥ g. If |hj,i,t |2 < g, an outage is declared. As expected, FA-GDNC presents the same diversity order as

−1

1

10

Simul BNC Outage BNC Simul DNC Outage DNC Simul App.1 GDNC Outage App.1 GDNC Simul App.2 GDNC

FER

0.9

Average Code Rate

−2

10

GDNC Approach 1, simulation Approach 1, analytical Approach 2, simulation Approach 2, analytical Approach 2, asymptotic

−3

10

0.8

0.7

0.6

−4

10

0.5

3

4

5

6

7 8 SNR (dB)

9

10

11

12

Fig. 4. FER versus SNR (dB) for a 2-user system, adopting the BNC scheme, the DNC scheme (over GF(4)), Approach 1 with k1 = 2 and k2 = {0, 2}, and Approach 2 with k1 = 2 and k2,j = {0, 1, 2}, both over GF(8) and with transfer matrix obtained from Table II. Recalling that the GDNC scheme (with k1 = k2 = 2 and in GF(8)) has the same performance as Approach 1.

the GDNC scheme, while outperforming the BNC and DNC schemes. The SNR gap between the analytical and simulated curves are due to the Gaussian input assumption made in (2). However, we can see that the simulated diversity order (curve slope) matches the analytical one. Since the code when setting k2′ = 1 contains fewer parity-check packets if compared to the one with k2′ = 2, a worse performance in terms of array coding was already expected. However, again, we can see that both schemes have the same curve slope. When the number of users increases, the asymptotic behaviors of the FA-GDNC scheme with k2′ = 2 and k2′ = 1 are different. This is illustrated in Fig. 5, where the average rate is plotted against the number of users for a network adopting the GDNC scheme with k1 = 2 and k2 = 2, FA-GDNC with k1 = 2, k2 = 2 and k2′ = 2 and 1. We can see that, while the rate when k2′ = 2 tends to the same rate as the GDNC scheme, the rate of the FA-GDNC scheme when k2′ = 1 tends to a higher value, confirming our expectation. VI. C ONCLUSIONS

AND

F INAL C OMMENTS

In this work, we elaborate on the generalized dynamicnetwork codes (GDNC) introduced in [6], [7] by proposing an adaptive network coding design approach, named feedbackassisted generalized dynamic-network codes (FA-GDNC). In those works, the network code design was recognized as equivalent to a classical code design, and concepts from classical coding theory were used to increase the system diversity order. Herein, the aim was to increase the system average rate by appropriately choosing the number of parity packets transmitted to the base station in an adaptive way. This choice is made based on a small quantity of information fed back from the BS, regarding to the success/failure in the decoding process of the packets received in the broadcast phase. Rate and diversity analyses were carried out, and

0.4

5

10

15

20 25 30 Number of users

35

40

45

50

Fig. 5. System overall rate versus number of users M for a network with k1 = 2 and SNR = 10 dB, considering the GDNC scheme with k2 = 2, the Approach 1 with k2 ∈ {0, 2} and the Approach 2 with adaptive k2,j ∈ {0, 1, 2}.

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