An Integer Programming Approach to UEP Coding for Multiuser Broadcast Channels

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by  Wook Jung

An Integer Programming Approach to UEP Coding
for Multiuser Broadcast Channels
Wook Jung Shih-Chun Chang
The Volgenau School of Engineering Department of Electrical and Computer Engineering
George Mason University, Fairfax, VA. 22030 George Mason University, Fairfax, VA. 22030
Email: [e-mail] Email: [e-mail]
4
1
0
r
2 Abstract—In this paper, an integer programming approach is 1 D1 mˆ1
introduced to construct Unequal Error Protection (UEP) codes
n
for multiuser broadcast channels. We show that the optimal
a codes can be constructed that satisfy the integer programming m=(m1,m2,...,mk)
J
bound. Based on the bound, we compute asymptotic code rate
1 and perform throughput analysis for the degraded broadcast c Broadcast r2
2 channel. • E Channel D2 mˆ2

. .
T] I. INTRODUCTION .. ..
I In this paper, we present an integer programmingapproach r
s. to construct unequal error protection (UEP) codes for mul- k Dk mˆk
c tiuserbroadcastchannels.Informationtheoreticfeaturesofthe
[
broadcast channel have been studied in [1], [2], [3] and the Fig.1. Broadcast communication system

1 designs of UEP codes have been well investigated in [4], [5],
v
[6], [7], [8], [9]. The UEP coding scheme corrects unequal
1
channel errors and recovers messages for all users effectively on the integer programming bound, we compute asymptotic
2
3 over the broadcast communication system. code rate and perform throughput analysis for the degraded
5 The multiuser broadcastcommunicationsystem is depicted broadcast channel.
. in Fig. 1. Due to the unequal error characteristics of the
1
0 broadcast channel, each receiver may have different level of II. UEPCODES FOR BROADCAST CHANNELS
4 error protection requirement denoted by
1 In [4], Masnick and Wolf have presented a UEP code
v: t=(t1,t2,...,tk) for 1≤i≤k (1) construction method. However, it has been claimed by the
authors that the optimal code construction beyond k = 5 is
i
X where ti is the error protection requirement for a receiver i. extremely difficult. It motivates us to develop an innovative
Let C be a (n, k) UEP code with a generator matrix G;
r approach based on integer programming to construct optimal
a
C ={mG|m ∈{0,1}, 1≤i≤k}. UEP codes beyond k =5.
i
Then a separation vector s = (s1,s2,...,sk) of the UEP A. Integer programming approach
code C is defined in [5] as follows
We begin our approach by specifying the separation vector
si =min{wH(mG) |mi =1}, 1≤i≤k (2) s=(s1,s2,...,sk) satisfies,
where wH(·) denotes the Hamming weight. Given si ≤sj for 1≤i<j ≤k. (4)
s ≥2t +1 for 1≤i≤k, (3)
i i Let A be a k×(2k −1) matrix consisting of all nonzero
a
the receiver i can correct up to t errors and can recover binary k-tuples as columns in increasing order.
i
themessagemˆ .Givenaspecificseparationvector,wedesign
UPE codes thait satisfy (2) based on an integer programming 000··· 01··· 11 a1
approach. 000··· 10··· 11  a2 
In the following sections, we introduce an integer pro- Aa =... ... ... ... ... ... ... ... ...= ...  (5)
   
gramming model to construct UEP codes and derive the 011··· 10··· 11 ak−1
   
performanceboundstoshowtheefficiencyofthecodes.Based 101··· 10··· 01  ak 
   
where ai = ai,1,ai,2,...,ai,2k−1 is a row vector. Let G Example 1: For k =3 and s=(3,5,7),
be a generator(cid:0)matrix of UEP code(cid:1)written as
0001111 a1
g1 Aa =0110011=a2,
g2 1010101 a3
G= . , (6)    
.
 .  and
 
gk 0 ··· 0 0 ··· 0 1 ··· 1 g1
then each gi for 1≤i≤k can be represented as G=0 ··· 0 1 ··· 1 ··· 1 ··· 1=g2
1 ··· 1 0 ··· 0 1 ··· 1 g .
gi = ai,1,...,ai,1,ai,2,...,ai,2,··· ,ai,2k−1,...,ai,2k−1 ←x1 →←x2 →···←x7 →  3
(cid:16) (cid:17)
Then,
x1 x2 x2k−1
| {z } | {z } | {z }(7)
2k−1 A1= 0001111 =(a1),
where x is the code length n.
j=1 i (cid:0) (cid:1)
It follPows from (2) that, for a given non-decreasingsepara-
tion vector s=(s1,s2,...,sk), A2=(cid:18)00111101011010(cid:19)=(cid:18)aa22+a1(cid:19),
wH(g1)≥s1, (8a)
i−1 1010101 a3
wH(cid:16)gi+Xj=1mjgj(cid:17)≥si for 2≤i≤k (8b) A3=11011010011100=aa33++aa12 .
   
where mj ∈{0,1}. From (7), 1101001 a3+a1+a2
Therefore, the UEP code construction is to find x that
2k−1
w (g )= a x =a x⊤ for 1≤i≤k (9) minimizes the sum x1+x2+···+x7 satisfying
H i i,j j i
Xj=1 x4+x5+x6+x7 ≥ 3
where x=(x1,x2,...,x2k−1). Consequently, x2+x3+x6+x7 ≥ 5
i−1 i−1 x2+x3+x4+x5 ≥ 5
⊤
wH gi+ mjgj =ai+ mjajx . (10) x1+x3+x5+x7 ≥ 7
(cid:16) Xj=1 (cid:17)  Xj=1  x1+x3+x4+x6 ≥ 7
From (8), (9), and (10), let Ai contains all 2i−1 rows of x1+x2+x5+x6 ≥ 7
a for i=1 (11a) x1+x2+x4+x7 ≥7.
i
i−1 The optimal solution for this example shall be given in later
ai+ mjaj for 2≤i≤k, (11b) section.
Xj=1
B. Integer programming bound
then, we have the following inequalities
In [7], van Gils has derived a lower bound of a UEP code
Aix⊤ ≥b⊤i for 1≤i≤k (12) for a non-increasingseparation vector s′ =(s′1,s′2,...,s′k) as
whereA isa2i−1×(2k−1)matrixandb =(s ,s ,...,s )
i i i i i
isarowvectoroflength2i−1.Basedon(12),wecanformulate k s′
UEP code construction for a given non-decreasing separation ns′ ≥ (cid:24)2i−i1(cid:25). (13)
vector s=(s1,s2,...,sk) as an integer programmingmodel. Xi=1
We begin by introducing the bound of (13) for a purpose
UEP code construction with integer programming to give the following theorem.
Minimize ns =x1+x2+···+x2k−1 Theorem 1 (Integer programming bound): For a given k,
subject to let s=(s1,s2,...,sk) be a non-decreasingseparationvector,
⊤ ⊤ then the integer programming bound is given by
Ax ≥b
k
where A1 b⊤1 ns ≥ 2ks−ii . (14)
 A2  ⊤  b⊤2  Xi=1l m
A= . , b = . Proof: For the integer programmingproblem, we formu-
. .
 A.k   b.⊤k  late 2k−1 inequalities in m⊤atrix f⊤orm as
Ax ≥b .
TABLEI
Since the rows of A consist of 2k−1 linear combinationsof NUMERICALRESULTSFORs=(3,5,...,2k+1)
a for1≤i≤k,eachcolumnofAhas2k−1 ones[10,p.97].
i
Therefore, the sum of 2k−1 inequalities is Lowerbounds Integerprogramming Upperbounds
k (14) results,ns (18)
k
2k−1(x1+x2+···+x2k−1)≥ 2i−1si, 2 7 7 7
3 11 11 12
Xi=1
4 16 16 19
hence,
5 20 20 25
k
si 6 25 25 32
ns ≥ 2k−i. (15) 7 30 30 39
Xi=1
8 35 35 46
Since the separation vector s = (s1,s2,...,sk) is in non- 9 39 h40i 53
decreasing order, set s′i = sk−i+1 and then, transform the 10 44 h45i 60
bound of (13) into
11 49 h52i 67
k ′ k k 12 55 h58i 74
ns′ ≥Xi=1 (cid:24)2is−i1(cid:25)=Xi=1lsk2−i−i+11m=Xi=1l2ks−iim. (16) 1134 5694 hh6740ii 8818
For every i from 1 to k, 15 69 h76i 95
Note:h·iindicates sub-optimal length.
s s
i i
≥ . (17)
2k−i 2k−i
l m
Therefore,theactuallowerboundoftheintegerprogramming 100
is exactly the same as the bound of (16). Consequently 90
Upper bounds
80 Code construction
k s Lower bounds
ns ≥ 2k−ii . 70
Xi=1l m 60
Length 50
In the following, we also include an upper bound on the 40
length ns derived by Masnick and Wolf in [4], 30
20
ns ≤k+rU (18) 10
andrU isthesmallestnumberofcheckbitsrsuchthat2r >γ 02 4 6 8 10 12 14 16
Dimension k
where
2tk−1 n−1 k n−1−T k n−1−T Fig.2. Integerprogrammingresults withbounds.
i i
γ = − −
(cid:18) i (cid:19) (cid:18) 2t −2 (cid:19) (cid:18) 2t −1 (cid:19)
Xi=0 Xi=2 i Xi=2 i
(19)
and one of the optimal solutions occurs at
andT isthenumberofmessagebitsthatareprotectedagainst
i
i-bitormoreerrors.Numericalresultsforthelowerandupper x1 x2 x3 x4 x5 x6 x7
.
bounds are shown in Table I. 3 2 2 1 1 1 1
It means that i-th column of A appears x times in the
C. Integer programming results a i
optimal generator matrix. Therefore,
In Table I, we present the numerical results of integer
programming by using IBM ILOG CPLEX v12.5.1 [11] for 00000001111
s = (3,5,...,2k+1). From k = 2 to 8, ns are exactly the G=00011110011.
same as the lower bound, therefore the corresponding UEP 11100110101
 
codes are optimal. The optimal results are listed in Table II.
Letˆs=(sˆ1,sˆ2,...,sˆk)denotetheseparationvectorobtained
For9≤k ≤15,insteadoffindingoptimalsolutions,welimit from the corresponding G. If
our search for sub-optimal solutions due to time constraint.
The results and both lower and upper bounds are plotted in sˆi ≥si for 1≤i≤k,
Fig. 2.
then G satisfies (2). In this example,
Example 2: Theoptimalsolutionobtainedfromtheinteger
ˆs=(4,6,7) ≥ s=(3,5,7).
programming for s=(3,5,7) is
There are many combinations of x for 1 ≤ i ≤ 2k −1
7 i
ns = xi =11, that are summed to the optimal value of ns (e.g., there are 3
Xi=1 solutions of xi to ns = 7 when k = 2, 7 solutions of xi to
TABLEII
OPTIMALRESULTSOFINTEGERPROGRAMMING
k ns non-zeroxi for1≤i≤2k−1 ˆs
x1x2x3
2 7 (3,5)
4 2 1
x1x2x3x4x5x6x7
3 11 (4,6,7)
3 2 2 1 1 1 1
x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15
4 16 (8,8,8,9)
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19
5 20 (4,8,8,10,11)
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x1 x2 x3 x4 x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
6 25 (4,6,8,10,12,13)
x21x32x33x34x35
1 1 1 1 1
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10x11x12x13x14x15x16x17x18x19x20
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
7 30 (3,5,8,9,12,14,15)
x21x22x23x32x33x34x35x64x65x104
1 1 1 1 1 1 1 1 1 1
x1 x2 x3 x4 x5 x6 x7 x8 x9x10x11 x12 x13 x14 x15x16x17x18x19x20
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
8 35 (3,5,7,11,11,13,15,17)
x21x22x23x32x33x34x35x64x65x66x126x127x128x129x184
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
ns = 11 when k =3, 2063 solutions of xi to ns = 16 when and
k =4, and so on).
x+1
∆j =ax 2x−1+···+1 +ax−1 2x−2+···+1
D. Asymptotic code rates Xj=2 (cid:0) (cid:1) (cid:0) (cid:1)
+···+a2(2+1)+a1
For s = (3,5,...,2k+1), we present the corresponding =ax(2x−1)+ax−1 2x−1−1 +···+a0(1−1)
asymptotic code rates based on the lower bounds as follows.
x (cid:0) (cid:1)
Theorem 2: Let RUEP be the rate of UEP code whose =k− ai (24)
separation vector is s=(3,5,...,2k+1). Then Xi=0
Therefore,
RUEP ≈0.2 when k ≫1. (20)
1
η =3k+1, (25a)
j
Proof: Let ns satisfy the bound of (14), then Xj=0
x+1 x x+1
k k−1
2i+1 2(k−j)+1 ηj =k− ai+ δj, (25b)
ns = (cid:24) 2k−i (cid:25)= (cid:24) 2j (cid:25). (21) Xj=2 Xi=0 Xj=2
Xi=1 Xj=0
and
k−1
Letηj , (2k−2j+1)/2j and k =ax2x+ax−12x−1+ ηj =k−⌊log2k⌋−2. (25c)
···+a0 wh(cid:6)ere x=⌊log2k⌋, th(cid:7)en j=Xx+2
Consequently,
2k+1 , j =0
 k , j =1 k−1
ηj =ax2x−j+1+···+aj−120+δj ,2≤j ≤x+1 ns = ηj ≈ 5k when k ≫1. (26)
1 , x+2≤j ≤k−1 Xj=0
whereδj ∈{0,1} [12, p.47,51]. (22) Thereforethecoderate isconvergedtoRUEP ≈ 5kk =0.2.
Let ∆ ,η −δ for 2≤j ≤x+1, then The actual rates of UEP codes based on R = k are
j j j ns
illustrated in Fig. 3 for k ≤ 215, which shows asymptotic
∆j =ax2x−j+1+ax−12x−j+···+aj−120, (23) convergenceto 0.2 as proven in Theorem 2.
S T1 T2 Tk−1 Tk
0.3 1−α1 1−α2 1−αk
0 • • • • •
α α α
1 2 k
0.28
α 1 α 2 α k
1 • • • • •
0.26 1−α1 1−α2 1−αk
Rate0.24 Fig.4. CascadedBSCsfordegradedbroadcast channel [3].
0.22
0.3
0.2 0.29
code rate, k/n
s
2^102^11 2^15 0.28
Dimension k throughput, R
T
0.27
Fig.3. UEPcoderatesfor2≤k≤215. 0.26
Rate0.25
0.24
E. Throughput of degraded broadcast channels
0.23
Theproposedbroadcastchannelmodelcanbeconsideredas
0.22
a degraded broadcast channel that has k component channels
[3]. Let S be a source input and T be outputs of the 0.21
i
componentchannels where S,Ti ∈{0,1} for 1≤i≤k, then 0.20 20 40 60 80 100 120
Dimension k
theerrorcharacteristicsofthechannelcanbedescribedwithk
cascaded binary symmetric channels (BSCs) with parameters
Fig. 5. Code rate and throughput of the degraded broadcast channel for
α ∈[0,1] as shown in Fig. 4.
i 2≤k≤27.
Letp denotethebiterrorprobabilityoftheeachcomponent
i
channel, then
we have demonstrated the achievable asymptotic code rates
pi =pi−1(1−αi)+(1−pi−1)αi, 1≤i≤k (27)
in comparison to the throughput of the degraded broadcast
where p0 = 0 [3]. Let ns satisfy the bound in (14) for a channel.
given separation vector s = (3,5,...,2k+1), then mi can Even though we limit to send 1 bit to each receiver, the
besuccessfullydeliveredtoreceiveriwhenthecorresponding integer programming approach can be extended to send a
component channel introduces less than or equal to i errors. multi-bit message or a packet to each receiver, we refer this
Therefore the average rate of successful transmission of a bit to UEP network coding in a follow-up paper.
to receiver i, denoting by θ , can be given as
i
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i
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