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Title

Practical Short-Length Coding Schemes for BinaryDistributed Hypothesis Testing

Author(s)Dupraz, Elsa; Adamou, Ismaila Salihou; Asvadi,
Reza; Matsumoto, Tad

Citation

2024 IEEE International Symposium on
Information Theory (IEEE ISIT 2024)

Issue Date

2024-07-15

Type

Conference Paper

Text version

author

URL

http://hdl.handle.net/10119/19325

RightsThis is the author's version of the work published at
arXiv, DOI: 10.48550/arXiv.2405.07697. This work
has been presented at International Symposium on
Information Theory (IEEE ISIT 2024), Athens,
Greece, July 7-12, 2024, and the paper is to be
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Description

2024 IEEE International Symposium on
Information Theory (IEEE ISIT 2024), Athens,
Greece, July 7-12, 2024

Japan Advanced Institute of Science and Technology

Practical Short-Length Coding Schemes for BinaryDistributed Hypothesis TestingElsa Dupraz1 , Ismaila Salihou Adamou1 , Reza Asvadi2 , and Tad Matsumoto1,3
1

2

IMT Atlantique, CNRS UMR 6285, Lab-STICC, Brest, France
Faculty of Electrical Engineering, Shahid Beheshti University, Tehran
3
JAIST and University of Oulu (Emeritus)

Abstract—This paper investigates practical coding schemes forDistributed Hypothesis Testing (DHT). While the literature
has extensively analyzed the information-theoretic performanceof DHT and established bounds on Type-II error exponents
through quantize and quantize-binning achievability schemes,
the practical implementation of DHT coding schemes has not
yet been investigated. Therefore, this paper introduces practicalimplementations of quantizers and quantize-binning schemesfor DHT, leveraging short-length binary linear block codes.
Furthermore, it provides exact analytical expressions for Type-I
and Type-II error probabilities associated with each proposed
coding scheme. Numerical results show the accuracy of the
proposed analytical error probability expressions, and enable tocompare the performance of the proposed schemes.

I. I NTRODUCTION
In modern communication systems, the communication objective has evolved beyond the mere reconstruction of original
information to encompass the application of Machine Learning
tasks such as decision making or classification upon received
data. This paper focuses on the specific case of Distributed
Hypothesis Testing (DHT) [1]. In the DHT setup, the decoder
aims to make a decision between two hypothesis H0 and H1 ,based on a coded version of the source X, and on someobserved side information Y . The performance of a DHT
scheme is typically evaluated from two error probabilities.
Type-I error refers to deciding H1 while H0 was true, andType-II error conversely involves selecting H0 when H1 istrue. To fully address DHT, it is essential to investigate bothinformation-theoretic performance limits and practical coding
schemes for this setup.
From an information-theoretic perspective, DHT was initiallyintroduced by Berger in [2]. Subsequently, Ahlswede andCsiszar established a lower bound on Type-II error-exponent
through an achievability scheme based on a quantizer [3].
Notably, this scheme was found to be optimal for testing
against independence [4]. Next, Shimokawa et al. improved
the lower bound presented in [3], extending its applicability
to general hypotheses, not restricted to testing against independence [5]. This enhancement was achieved by employing
a quantize-binning achievability scheme. The optimality ofthe quantize-binning scheme was then investigated on several
DHT problems. It was shown in [6] that this scheme is optimal
for a certain class of testing against conditional independence
problems, given that the tradeoff between quantization and binning is adressed properly. This analysis was further extended

979-8-3503-8284-6/24/$31.00 ©2024 IEEE

to encompass more generalized hypotheses, including those
involving independent and identically distributed (i.i.d.) binary
sources [7], non-binary sources [8], and non-i.i.d. sources [9].
The impact of discrete memoryless channels [10], MultipleAccess channels [11], or two-hop relay networks [12] on the
DHT performance was also analyzed. Multi-terminal source
coding for DHT was also considered in [13].
While the DHT information-theoretic performance is now wellknown across a wide range of source models and hypotheses,
the problem of designing practical coding schemes for thissetup has been by far less investigated. This paper takes a
step towards bridging this gap, by focusing on the scenario
where the sources X and Y are binary.
Previous achievability proofs suggest to consider either
quantizer-alone or quantize-binning coding schemes for DHT.
Some existing works have already introduced practical binary
quantizers [14], binning schemes [15], and quantize-binningschemes [16], all constructed with linear block codes. But theconstructions of [14]–[16] primarily focus on source reconstruction and typically involve very long source sequences,often exceeding 105 bits. However, DHT inherently deals
with short-length sequences, where just a few dozen bits may
suffice for making the correct decision. And the constructionof efficient short-length linear block codes is often known
to be a challenging problem [17]. Consequently, an important question arises regarding whether binary quantizers andquantize-binning schemes are efficient structures for practicalshort-length DHT.
This paper addresses practical coding schemes for DHT byproviding two main contributions. Firstly, it introduces andcompares three practical coding schemes. The first scheme,
named the truncation scheme, involves transmitting only the
first ℓ bits of the source sequence xn and serves as aperformance baseline. The second and third schemes arepractical quantizer-alone and quantize-binning schemes, bothconstructed with short-length binary linear block codes. Secondly, the paper provides exact analytical expressions for
Type-I and Type-II error probabilities of the three consideredcoding schemes. These new analytical tools should enable
the optimization and comparison of the proposed practicalschemes across a wide range of source and code parameters.
Numerical results validate the accuracy of the analytical error

2915

B. Error-exponent for binary DHT
We will now restate an existing lower bound on the error
exponent for the problem defined in Equation (1), from theliterature of information theory. Following the approach in [5],[6], we consider an encoding function:
Fig. 1. Distributed hypothesis testing scheme

fn : {0, 1}n → J1, 2v K,

probabilities by demonstrating their consistency with MonteCarlo simulations. Additionally, the paper compares the performance of the three schemes for sources of length n = 31
bits.
The outline of the paper is as follows. Section II describes
the binary DHT problem. Section III presents the truncation
scheme. Section IV introduces the quantizer scheme. Section V describes the quantize-binning scheme. Section VI
shows and discusses numerical results.

and a decision function gn : J1, 2v K × {0, 1}n → {0, 1}. We
consider a rate-limited setup in which v/n ≤ R.
For given functions (fn , gn ), we define Type-I error probability αn and Type-II error probability βn as in [7] as
αn = P(gn (fn (Xn ), Yn ) = 1|H0 ),

(3)

n

(4)

For a given value ϵ ∈ (0, 1) such that αn < ϵ, Type-II error
exponent θ is defined as [7]lim supn→∞

A. DHT for binary sources
We consider binary source vectors Xn and Yn both of lengthn. The encoder observes Xn , whereas the decoder observes
Yn , as illustrated in Fig. 1. The encoder sends a coded versionof Xn , while the decoder aims to make a decision between two
hypothesis H0 and H1 , based on both Yn and the received
coded data.
We assume that the n symbols of the sequences Xn and Yn
are i.i.d. and drawn according to random variables X andY , respectively. Furthermore, X and Y are jointly distributed
according to the model Y = X ⊕ E, where E is a binary
random variable independent of X, and P (X = 1) = 1/2.
Additionally, we denote p = P(E = 1) with 0 < p ≤ 1/2.
The two hypotheses are expressed as:
H0 : p = p0 ,(1)
H1 : p = p1 .
We assume, without loss of generality, that p0 < p1 . It is
worth noting that the probability distributions of both X andY are independent of the chosen hypothesis, given P (X =1) = 1/2. This model was investigated from an informationtheoretic perspective for instance in [5], [7], [8]. Furthermore,when p = 1/2, the problem (1) reduces to testing against
independence [3].

nβn = P(gn (fn (X ), Y ) = 0|H1 ).

II. P ROBLEM S TATEMENT
In what follows, J1, M K is the set of integers between 1 and M .
Random variables are represented by capital letters, such as X,
while their realizations are denoted by lower-case letters, likex. Vectors of length n, such as Xn , are in bold. In addition,
we use w(xn ) to denote the Hamming weight of the vector
xn , and d(xn , yn ) to denote the Hamming distance between
xn and yn . The binomial coefficient of the pair of integers(n, k) with k ≤ n is expressed as nk .

(2)

1
1
log2
≥ θ.
n
βn

(5)

A generic lower bound on the error exponent θ is provided
in [8]. Specifying this bound for the hypothesis testing problem defined in (1) leads toθ ≤ sup min R − [H2 (p0 ∗ δ) − H2 (δ)],(6)δ∈[0,1](p0 ∗ δ) log

1 − (p0 ∗ δ)
p0 ∗ δ
+ (1 − (p0 ∗ δ)) log
p1 ∗ δ
1 − (p1 ∗ δ).

Here, H2 is the binary entropy function, and ∗ is the binary
convolution operator defined as x ∗ y = (1 − x)y + (1 − y)x,
with 0 ≤ x, y ≤ 1.
C. Short-length nature of DHT
While the lower bound on the error exponent θ in (6) provides
a scaling law for Type-II error probability, it remains an
asymptotic result due to the limit as n tends to infinity indefinition (5). Nevertheless, it confirms the intuition that the
problem inherently involves short sequences. For instance,
consider parameters p0 = 0.05, p1 = 0.5, δ = 0.1, R = 0.4.
When evaluating the quantity e−nθ for these parameters withn = 100, the result is approximately 10−12 , and for n = 50,
it yields approximately 10−6 . This strongly suggests thatpractical schemes should focus on values of n less than
50. Hence, we now introduce three practical coding schemes
tailored for such short sequence lengths.
III. T RUNCATION S CHEME
A. Code construction
When considering DHT, there is no need to reconstruct all theexact values of the source bits xn . So a first straightforward
solution consists of sending the first ℓ symbols of the source
vector xn at coding rate R = ℓ/n. The decoder can thenperform a standard Neyman-Pearson (NP) test [18] on the pair(xℓ , yℓ ).

2916

Under a certain constraint αn < ϵ on Type-I error probability,
the NP lemma [18] states that the following test:
P1 (xℓ , yℓ ) < µP0 (xℓ , yℓ ),(7)minimizes Type-II error probability βn , where µ is a thresholdvalue chosen to satisfy the Type-I error constraint. In (7), P0
and P1 are the joint probability distributions of (xℓ , yℓ ) underhypothesis H0 and under hypothesis H1 , respectively. Given
that p0 < p1 , it is shown in [18] that the test described by (7)is equivalent to the condition:ℓX(xi ⊕ yi ) < λt ,

(8)

i=1

where λt ∈ N is an integer threshold value chosen so as tosatisfy the constraint αn < ϵ. Interestingly, this test depends
on the value of the parameter p0 solely through the choice ofthe value of λt .
B. Theoretical analysis
For this scheme, analytical expressions of Type-I and Type-II
error probabilities are given by [18]λt  
Xℓ j
p0 (1 − p0 )ℓ−j ,(9)αn(t) = 1 −j
j=0λt  
Xℓ j
p1 (1 − p1 )ℓ−j .
(10)βn(t) =
jj=0

algorithm since it may lead to an important loss in performanceon the considered codes. Instead, we will solve (11) exactly by
exhaustive search. Therefore, we consider any generator matrix
Gq , not necessarily obtained from an LDGM code.
The decoder first computes the quantized vector xnq = Gq zm
q .
n
n
n
Then, since P(zm
,
y
)
=
P(x
,
y
),
the
NP
test
(7)
reducesqqtonX(xq,i ⊕ yi ) < λq ,
(12)
i=1

where λq is an integer threshold. Here, compared to thetruncation scheme, the decision is taken from longer vectors
xnq and yn of dimensions n > m. On the other hand, the vector
xnq contains quantization errors compared to xn .
B. Theoretical analysis
We now provide exact analytical expressions of Type-I and
Type-II error probabilities for the quantization scheme. Con(q)(q)
sider the set of integers {Eγ }γ∈J0,d(q) K , where Eγ is themaxnnumber of words x of Hamming weight γ that belong to the(q)(q)
decision region C0 of xnq = 0n . In other words, xn ∈ C0nmmeans that the solution of (11) for x is 0 . We further denote
Pd(q)
(q)
(q)
max
N0 = γ=0Eγ .
Proposition 1. For the quantization scheme and for a threshold value λq , Type-I and Type-II error probabilities are givenby(q)

The truncation scheme can be seen as a “no-coding”
setup [19]. In our numerical results, it will serve as a baseline
when evaluating the performance of the proposed coding
schemes.

αn(q) = 1 −

(q)

N0

Eγ(q) Γλ,j,γ pj0 (1 − p0 )n−j ,λ=0 γ=0 j=0(13)
βn(q) =IV. Q UANTIZATION S CHEME
In the literature of information theory, the seminal work of [3]proposed to build a DHT coding scheme from a quantizer
alone. We now introduce a practical short-length implementation of this scheme by using linear block codes.

λq dmax n
X
XX

1

1

λq d(q)
max

n
XXX

(q)
N0 λ=0 γ=0 j=0Eγ(q) Γλ,j,γ pj1 (1 − p1 )n−j ,(14)where for j = γ + λ − 2u and 0 ≤ u ≤ min(γ, λ) ≤ n,
 

γ
n−γ
Γλ,j,γ =
.
(15)
u λ−u

A. Code construction
In order to perform binary quantization, we consider thegenerator matrix Gq of size n × m of a linear block code [20].
Then, for a given source vector xn of length n, the encoder
produces a vector zm
q as [21]zmd(Gq zm , xn ).
q = arg minmz

Proof. Since by symmetry the quantizer error probability is
independent of the transmitted codeword [20], we consider
the all-zero codeword xnq = 0. We develop(11)

The codeword zm
q is transmitted to the decoder at a code rate
R = m/n.
In [14], [16], it is proposed to build efficient binary quantizers
using low density generator matrices (LDGM). LDGM codes
were considered so as to develop a low complexity messagepassing algorithm called Bias-Propagation to solve (11). However, the schemes introduced in [14], [16] consider very longcodes (more than 105 bits). Here, due to the short-length natureof the problem, we choose to discard the Bias-Propagation2917

αn(q) = 1 −λq
XP0 (w(Yn ) = λ)

(16)

λ=0
(q)

=1−

λq dmaxX
X Eγ(q)(q)
λ=0 γ=0 N0P0 (w(Yn ) = λ|w(Xn ) = γ)

(q)

=1−

λq dmaxnX
X Eγ(q) X(q)
j=0
λ=0 γ=0 N0λq d(q)max

P0 (d(Xn , Yn ) = j)

Γλ,j,γnjn  X X Eγ(q) Xn jΓλ,j,γ=1−
p0 (1 − p0 )n−j n .
(q)
j
j
j=0
λ=0 γ=0 N0(q)This gives (13). To obtain (14), we remark that βn =P
λqnλ=0 P1 (w(Y ) = λ) and follow the same steps as in (16),by replacing p0 by p1 .

min(d(qb)max ,λqb )X

PB (δ) =(q)(qb) dX
max

Eν


n
ν

ν=0

V. Q UANTIZE -B INNING S CHEME
We now propose a practical solution for the quantize-binningscheme which has been widely investigated in the literature of
information theory for DHT [5], [6], [8].
A. Code construction
In the quantize-binning scheme, we consider as before agenerator matrix Gq of size n × m. We also resort to theparity check matrix Hb of size k × m of another linear blockcode. After using Gq for binary quantization as described in
Section IV, the encoder uses the matrix Hb to computeuk = Hb zm
q .

where
n
(q) X

Eγ

(q)

γ=0 N0Γν,j,γ δ j (1 − δ)n−j ,

j=0

(22)

 (q)
dmax
n
(q) n
X
X
Eγ X

(23)
PB̄ (δ) =Γi,j,w δ j (1 − δ)n−j 
(q)
N
γ=0
j=0
i=0
0


λqb
n X
(qb) (qb)
X
A
Γ
E
ν
i,ν,t
 .
 t n
×nt=1 ν=0νiProof. We consider the all-zero codeword xnq = 0. Under the
hypothesis H0 , we express(17)b0 , B) − P0 (Hb0 , B̄).
αn(qb) = 1 − P0 (H

(24)

k

The syndrom u is then transmitted to the decoder. In thiscase, the coding rate is given by R = k/n. At the decoder, inorder to apply the NP test (7), we first identify by exhaustive
search a vector ẑmq asẑmd(Gq zm , yn ) s.t. Hb zm = uk .
q = arg minmz

We then apply the following test:nX(x̂q,i ⊕ yi ) < λqb ,(18)

In this expression, B is the event that the correct sequencex̂q = xq was retrieved at the decoder, while B̄ is the eventthat an incorrect sequence x̂q ̸= xq was output by the decoder.
b0 is the event that hypothesis H0 was decided
In addition, H
b0 , B) andat the decoder. We further denote PB (p0 ) = P0 (HbPB̄ (p0 ) = P0 (H0 , B̄). We then express
PB (p0 ) =

(19)

(qb)
min(dmax,λqb )

where x̂nq = Gq ẑm
q , and λqb is an integer threshold.

X=

Next, according to [8], the binning allows us to leverage theside information vector yn so as to further reduce the coding
rate. However, it also introduces a binning error probability
which can impact Type-I and Type-II error probabilities.
While this problem was well investigated in the asymptotic
regime [8], we next discuss it for the considered short lengthcode construction.

(qb)

P0 (w(Yn ) = ν)

ν=0

Eν

n
ν

 .

(26)

Next, by following the same steps as in the proof of Proposition 1, we show that
d(q)
max
n

P0 (w(Y ) = ν) =n
X Eγ(q) X(q)
γ=0 N0j=0Γν,j,γ pj0 (1 − p0 )n−j , (27)which provides (22). We then write

B. Theoretical analysis(qb)
We now consider the decision region C0

for the all-zerocodeword of the quantize-binning scheme. Especially, a side(qb)
information vector yn belongs to C0 if the solution of (18)mm
for this vector is ẑq = 0 . We then define the set of integers
(qb)
(qb)
{Eν }ν∈J0,d(qb) K , where Eν is the number of words yn of

PB̄ (p0 ) =(20)

b0 , B̄|w(Yn ) = i), (28)P0 (w(Yn ) = i)P0 (Hwhere P0 (w(Yn ) = i) is given by (27). Next, we develop
b0 , B̄|w(Yn ) = i)
P0 (H

(qb)

Proposition 2. For the quantize-binning scheme and for a
threshold value λqb , Type-I and Type-II error probabilities are
given bynX
i=0

Hamming weight ν that belong to the decision region C0 .
(qb)
(qb)
We also define the set of integers {At }t∈J0,nK , where At
is the number of codewords xnq of Hamming weight t such thatnnmk
there exists zm
q that satisfies xq = Gq zq , and Hb zq = 0 . As
(qb)a result, the set {At }t∈J0,nK is the code weight distributionof the concatenated code.

αn(qb) = 1 − PB (p0 ) − PB̄ (p0 ),βn(qb) = PB (p1 ) + PB̄ (p1 ),b0 , B|w(Yn ) = ν) (25)P0 (w(Yn ) = ν)P0 (H

ν=0

i=1

max

n
X

=

λqb
n X
X

P0 (w(X̂nq ) = t, d(X̂nq , Yn ) = ν|w(Yn ) = i)t=1 ν=0

(29)
λqb
n X
X

(qb) (qb)Eν At Γi,ν,t


.
=
n
n
ν
i
t=1 ν=0(30)This provides the expression of PB̄ (p0 ) in (23).
(qb)

We obtain the expression of βn from the previous equations(qb)
by noticing that βn = PB (p1 ) + PB̄ (p1 ).

(21)

2918

Fig. 2. ROC curve for the BCH code (31, 16, 7) user as a quantizer, comparedto the truncation scheme.

Fig. 3. ROC curve for the quantize-binning scheme built from the BCH code
(31, 16, 7) for quantization combined with the Reed-Muller code (16, 5, 8)
for binning

VI. N UMERICAL R ESULTS
This section provides Monte-Carlo simulation results for the
proposed code constructions, and compares them with thetheoretical Type-I and Type-II error probabilities.
A. Truncation versus quantization
First of all, we evaluate the performance of the quantization
scheme introduced in Section IV and compare it against thetruncation scheme described in Section III. We set parameters
p1 = 0.5 (testing against independence), and p0 = 0.1
or p0 = 0.07. For the quantization, we consider the BCH(31, 16)-code with minimum distance dmin = 7. As a result,after applying the quantizer, m = 16 bits are sent to thedecoder. Therefore, we consider for comparison the truncation
scheme with ℓ = 16. For these two schemes, Fig. 2 shows thereceiver operating characteristic (ROC) curves which provide
the Type-II error with respect to the Type-I error, obtained for
different threshold values λt , λq ∈ J1, mK. Note that the ROC
curves are usually considered for the evaluation of hypothesis
tests. In Fig. 2, the plain curves come from Monte-Carlosimulations averaged other 10000 trials, while the dashedcurves come from the error probability expressions provided
in Sections III-B and IV-B.
For the considered two values of p0 , we observe that thequantizer scheme performs better than the truncation scheme.
This is due to the fact that with the quantizer, the decision
is made on n = 31 bits instead of m = 16 bits, although
the quantized vector xnq contains errors compared to theoriginal xn . In addition, we observe that the theoretical TypeI and Type-II error probabilities are closely consistent with
the Monte Carlo results. This is because the error probability
expressions take into account the considered code through the(q)terms Eγ . As a result, the theoretical expressions are foundto be relevant tools for the DHT code design.
B. Truncation versus quantize-binning
We now evaluate the performance of the quantize-binningscheme proposed in Section V compared to the truncation
scheme. For the quantize-binning scheme, we consider theBCH (31, 16) code for the quantizer, and the Reed-Muller(16, 5) code with dmin = 8 for the binning part. As a result,only k = 11 coded bits are sent to the decoder. Therefore, forcomparison, we consider the truncation scheme with ℓ = 11.
Fig. 3 shows the ROC curves for the considered two schemes,
both for p1 = 0.35 and p0 = 0.01 or p0 = 0.03. In this caseagain, plain curves come from Monte-Carlo simulations averaged over 10000 trials, and dashed curves come from the error
probability expressions obtained in Sections III-B and V-B.
We observe that the quantize-binning scheme performs muchbetter than the truncation scheme, since it makes decision on
n = 31 bits instead of ℓ = 11 bits for the truncation scheme,
despite the fact that both quantizer and binning can introduce
errors. We also observe that the theoretical Type-I and TypeII error probabilities are closely consistent with the practical
performance.
Finally, it is worth noting that the performance of the quantizebinning scheme strongly depends on the considered pair ofcodes used for quantization and binning. In this regard, thederived theoretical error probability expressions can serve asa useful tool for the design of efficient pairs of codes. Also
we did not include the entropy-check condition of [8] in ourquantize-and-binning scheme, since this did not bring any
further improvement in terms of practical Type-I and Type-II
error probabilities compared to the truncated scheme.
VII. C ONCLUSION
In this paper, we have introduced two practical coding schemes
for binary DHT, one built with a binary quantizer, and the otherbuilt with a quantize-binning scheme. Both schemes were
designed from short linear block codes. For each consideredscheme, we also derived theoretical expressions of Type-I and
Type-II error probabilities. Simulation results demonstrated
the superiority of the proposed schemes compared to thebaseline truncation scheme, and also showed the accuracy ofthe proposed theoretical expressions. From a practical pointof view, future works will include an interleaver design toimprove the performance of the concatenated construction, as
well as complexity reduction of the decoders so as to allow
for larger code length to be considered.

2919

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