**2. Evaluation**

The most common causes of esophageal dysphagia are listed in Table 1.

**Table 1.** Common diseases of esophageal dysphagia and endoscopic management

The acceptance of endoscopy as a gold standard for testing for mucosal disease may bias evaluations of the sensitivity of other diagnostic modalities. Given that endoscopy was used as the gold standard for mucosal disease, the sensitivity of radiology could not have exceeded that of endoscopy [3].

If the patient history, barium swallow, or both suggest achalasia, manometry to confirm the diagnosis should generally precede the endoscopic evaluation, to better prepare for endo‐ scopic therapy.

### **2.1. GERD: Peptic esophageal stricture**

**2. Evaluation**

92 Seminars in Dysphagia

**Benign diseases** Yes

Esophageal web Yes

Eosinophilic esophagitis Yes

Caustic injury Yes

Radiation injury Yes

Drug-induced stricture Yes

Postendoscopic therapy stricture Yes

Diffuse esophageal spasm No

**Table 1.** Common diseases of esophageal dysphagia and endoscopic management

**Malignant diseases**

squamous cell)

**Motility disorders**

Esophageal carcinoma (adeno and

The most common causes of esophageal dysphagia are listed in Table 1.

Bougie Balloon

Peptic stricture Yes Transoral incisionless fundoplication

Schatzki ring Yes Electrocautery incision with a needle-knife

Anastomotic stricture Yes Biodegradable stent

Head and neck tumor No Self-expanding metal stents (SEMS)

Extrinsic compression No Self-expanding metal stents (SEMS)

Achalasia Yes Peroral endoscopic myotomy (POEM)

**Endoscopic dilation Other endoscopic treatments**

Radiofrequency ablation

Injection of corticosteroids or triamcinolone

papillotome

Yes Self-expanding metal stents (SEMS)

Injection of botulinum toxin

White light endoscopy is now the standard investigation procedure for identifying esophageal injury. An experienced endoscopist who takes the time to methodically inspect the esophagus is generally expected to succeed in diagnosing reflux esophagitis. There has been evidence, however, of interindividual variability in the endoscopic diagnosis of erosive reflux esopha‐ gitis and other lesions of the upper gastrointestinal tract. Krugmann reported detailed endoscopic findings for GERD [4]. Peptic esophageal stricture as a consequence of gastroeso‐ phageal reflux disease is the most frequent among benign esophageal strictures [6]. The typical case of peptic esophageal stricture is shown in Figure 1. There are multiple etiologies for benign esophageal stricture or stenosis, but the most frequent is the peptic stricture resulting from pathologic acid exposure in GERD. Dysphagia is a common symptom. When dysphagia is encountered, accurate diagnostic procedures (barium esophagogram, upper endoscopy with biopsies) have to be performed to exclude malignant causes first. Strictures can be divided into two categories anatomically: "simplex" or "complex." The former are short, focal, nonangu‐ lated, and wide enough to allow an endoscope to easily pass through. The latter are long and angulated, with a severely narrowed diameter. The strictures are also sometimes scored based on three parameters to optimize the therapeutic decisions: the stricture diameter, stricture length, and degree of difficulty of stricture dilation [7]. After this scoring, the stricture can be classified as type I (mild), type II (moderate), or type III (severe or critical). This classification is also useful for predicting the most appropriate therapeutic option.

**Figure 1.** Peptic structure

### **2.2. NERD (Nonerosive Reflux Disease)**

NERD is a subcategory of GERD characterized by troublesome reflux-related symptoms in the absence of esophageal mucosal erosions or breaks on conventional endoscopy, without recent acid-suppressive therapy [8]. Most patients with typical reflux symptoms show no evidence of erosive esophagitis on endoscopy. Upper gastrointestinal endoscopy is required to establish a diagnosis of NERD. Further investigation is required when alarming symptoms are present. Routine random biopsy is not currently recommended for the diagnosis of NERD. Additional diagnostic information is provided by ambulatory 24-hour intraesophageal pH-metry and impedance measurement with reflux-related symptom correlation.

#### **2.3. Schatzki ring and esophageal web**

Narrowing of the esophagus can be due to either a benign or malignant stricture formation, webs (mucosa or submucosa alone), or rings (mucosa, submucosa, and muscle). Esophageal rings and webs are both membranous structures in which a thin fold of tissue creates at least a partial obstruction of the esophageal lumen. Esophageal webs usually measure 2–3 mm wide. The obstruction is a smooth extension of normal esophageal tissue made up of mucosa or submucosa alone. Webs can be found anywhere along the esophagus, but classically they appear in the anterior postcricoid area of the upper esophagus. A web at this site constitutes the Paterson Brown–Kelly syndrome, otherwise known as Plummer–Vinson syndrome in the USA [9].

Esophageal rings are concentric, smooth, thin extensions of normal esophageal tissue, usually 3–5 mm thick. They consist of mucosa, submucosa, and muscle. Rings are often detected incidentally at barium studies or endoscopy. There is no sex difference in the incidence of rings overall, though multiple rings are usually found in young men. Rings are classified as types A, B, and C [9]. The A ring, an uncommon type located a few centimeters proximal to the esophagogastric squamocolumnar junction, is thought to be caused by normal physiologic smooth muscle contractions. The B ring (more commonly known as Schatzki ring) is actually a web, as it involves only mucosa and submucosa and tends to appear in the distal esophagus and as the proximal part of a hiatus hernia. The B ring is nonprogressive and usually presents in patients aged over 50 who experience intermittent dysphagia to solid food over periods spanning months or years. The C ring, another rare type, is found in the most distal portion of the esophagus. On X-ray, the C ring manifests as an indentation caused by diaphragmatic crural pressure. A and C rings are both unlikely to be readily seen on upper endoscopy. Hence, the B ring (Schatzki ring) is the most common esophageal ring found on either esophagogram or endoscopy. Schatzki rings rarely cause symptoms. Overall, esophageal rings with luminal narrowing significant enough to cause symptoms (13 mm or less) are seen in only about 0.5% of all esophagograms.

#### **2.4. Head and neck tumor and esophageal tumor**

A core cancer-specific symptom of head and neck tumors is difficulty in swallowing [10]. Most patients with tumors of the head and neck or esophagus present for medical attention because of dysphagia.

Among the many symptoms of esophageal cancer, dysphagia may have an especially adverse effect on quality of life (QOL) [11]. Dysphagia is the predominant symptom in more than 70% of patients with esophageal cancer. The optimum management of dysphagia caused by advanced primary EC has not yet been established, although continued progress toward this goal has been achieved in recent years.

Apart from the weight loss that may result, an inability to swallow comfortably or a tendency to regurgitate food may spoil meals shared with families and friends or induce patients to withdraw from social situations.

Granular cell tumors and solitary fibrous tumors of the cervical esophagus also cause dys‐ phagia [12, 13].

### **2.5. Eosinophilic esophagitis and infection**

**2.2. NERD (Nonerosive Reflux Disease)**

94 Seminars in Dysphagia

**2.3. Schatzki ring and esophageal web**

USA [9].

of all esophagograms.

of dysphagia.

**2.4. Head and neck tumor and esophageal tumor**

NERD is a subcategory of GERD characterized by troublesome reflux-related symptoms in the absence of esophageal mucosal erosions or breaks on conventional endoscopy, without recent acid-suppressive therapy [8]. Most patients with typical reflux symptoms show no evidence of erosive esophagitis on endoscopy. Upper gastrointestinal endoscopy is required to establish a diagnosis of NERD. Further investigation is required when alarming symptoms are present. Routine random biopsy is not currently recommended for the diagnosis of NERD. Additional diagnostic information is provided by ambulatory 24-hour intraesophageal pH-metry and

Narrowing of the esophagus can be due to either a benign or malignant stricture formation, webs (mucosa or submucosa alone), or rings (mucosa, submucosa, and muscle). Esophageal rings and webs are both membranous structures in which a thin fold of tissue creates at least a partial obstruction of the esophageal lumen. Esophageal webs usually measure 2–3 mm wide. The obstruction is a smooth extension of normal esophageal tissue made up of mucosa or submucosa alone. Webs can be found anywhere along the esophagus, but classically they appear in the anterior postcricoid area of the upper esophagus. A web at this site constitutes the Paterson Brown–Kelly syndrome, otherwise known as Plummer–Vinson syndrome in the

Esophageal rings are concentric, smooth, thin extensions of normal esophageal tissue, usually 3–5 mm thick. They consist of mucosa, submucosa, and muscle. Rings are often detected incidentally at barium studies or endoscopy. There is no sex difference in the incidence of rings overall, though multiple rings are usually found in young men. Rings are classified as types A, B, and C [9]. The A ring, an uncommon type located a few centimeters proximal to the esophagogastric squamocolumnar junction, is thought to be caused by normal physiologic smooth muscle contractions. The B ring (more commonly known as Schatzki ring) is actually a web, as it involves only mucosa and submucosa and tends to appear in the distal esophagus and as the proximal part of a hiatus hernia. The B ring is nonprogressive and usually presents in patients aged over 50 who experience intermittent dysphagia to solid food over periods spanning months or years. The C ring, another rare type, is found in the most distal portion of the esophagus. On X-ray, the C ring manifests as an indentation caused by diaphragmatic crural pressure. A and C rings are both unlikely to be readily seen on upper endoscopy. Hence, the B ring (Schatzki ring) is the most common esophageal ring found on either esophagogram or endoscopy. Schatzki rings rarely cause symptoms. Overall, esophageal rings with luminal narrowing significant enough to cause symptoms (13 mm or less) are seen in only about 0.5%

A core cancer-specific symptom of head and neck tumors is difficulty in swallowing [10]. Most patients with tumors of the head and neck or esophagus present for medical attention because

impedance measurement with reflux-related symptom correlation.

The incidence of eosinophilic esophagitis (EoE) is registering an increase in adults. An allergic reaction to food is now established to play an important role in its etiology, and dietary interventions and biologic agents to block the inflammatory cascade are thought to hold promise as novel fields of clinical research. Biopsies should be obtained from the proximal and distal esophagus to evaluate for eosinophilic esophagitis in patients who present with dysphagia together with endoscopic findings suggestive or not suggestive of EoE, and also in patients without esophageal mechanical obstruction. Patients usually present with dysphagia, food impaction, and/or reflux-like symptoms, and biopsy of the esophagus typically shows more than 15 eosinophils per high-power field [14]. The dysphagia predominantly seen in EoE has been attributed to both organic and nonorganic (i.e., motility) disorders. Endoscopically, a normal-appearing esophagus is usually incompatible with a diagnosis of EE, although the findings can be subtle. EGD findings implicative of EoE include an attenuation of the subepi‐ thelial vascular pattern, linear furrowing (possibly extending along the whole length of the esophagus), surface exudates composed of eosinophils, or abscesses or strictures [15,16].

Endoscopic evaluation of the gastrointestinal tract remains a cornerstone of diagnosis, especially in patients with advanced immunodeficiency who are at risk for opportunistic infections (OIs) [17]. Infectious esophagitis may be caused by fungal, viral, bacterial, or even parasitic agents [18]. Acute onset of symptoms such as dysphagia and odynophagia is typical of this condition. Candida esophagitis most commonly appears in patients with hematologic malignancies or AIDS, or who use steroids for the treatment of disorders. Candida esophagitis is usually diagnosed when white mucosal plaque-like lesions are seen on esophagogastro‐ duodenoscopy. HSV esophagitis occurs most frequently in solid organ and bone marrow transplant recipients.

The diagnosis of herpes simplex virus esophagitis is usually based on endoscopic findings confirmed by histopathological examination. Well-circumscribed ulcerous lesions with a "volcano-like" appearance may appear in the mucosa of the distal esophagus, typically with diameters of less than 2 cm. Diffuse erosive esophagitis may also be present.

Cytomegalovirus esophagitis is observed in patients who have undergone transplanta‐ tion, are on long-term dialysis, are infected with HIV, or are receiving chronic steroid therapy. Esophagogastroduodenoscopy usually reveals large solitary ulcers or erosions in the distal esophagus.

#### **2.6. Esophagitis induced by caustic injury, radiation injury, or drugs**

A patient with a usual history of caustic substance ingestion and prolonged hospitalization for severe caustic damage was hospitalized again because of an increase in dysphagia and odynophagia [19]. The gold standard of safely assessing the depth, extent of caustic ingestion injury, and appropriate therapeutic regimen is EGD. The patients underwent EGD within 24 hours of admission and mucosal damage was graded using Zagar's modified endoscopic classification scheme [19].

Radiation therapy (RT), the primary modality for patients with tumors of the upper aerodigestive tract, allows larynx preservation [20]. Proximal esophageal strictures occur in 2–16% of patients after radiation therapy for cancers of the lung or head and neck. RT-induced laryngeal edema (due to inflammation and lymphatic disruption) is a common and expected side effect. Progressive edema and associated fibrosis detected by endoscope or barium swallowing can lead to long-term problems with phonation and swallowing. Aspiration pneumonia associated with dysphagia after intensive chemo radiation therapy has recently been reported at a growing frequency.

Drug-induced esophagitis mainly presents as chest pain, odynophagia, and dysphagia. In the agents known to induce esophagitis, antibiotics are the most common culprit. Other causative agents include nonsteroidal anti-inflammatory drugs, antihypertensive drugs, acetamino‐ phen, oral hypoglycemic agents (glimepiride), bisphosphonates (alendronate, ibandronate), ascorbic acid, and warfarin [21]. The most frequent endoscopic site of drug-induced esopha‐ gitis is the middle third of the esophagus. On endoscopy, drug-induced esophagitis manifests as ulcers of various forms such as kissing ulcers, erosions, or ulcers with bleeding, as patchy sections coated with drug materials or impacted pill fragments, or sometimes as strictures.

#### **2.7. Achalasia**

Achalasia is regarded as a disease exclusively involving the smooth muscle. About 70–80% of patients have absent or incomplete lower esophageal sphincter (LES) relaxation with wet swallows, while the remainder have complete but shortened relaxation [22]. The typical endoscopic findings are shown in Figure 2A,B. Patients with esophageal stasis are often unaware of this condition. Heartburn, though not infrequent, bears little relationship to the esophageal acid exposure in achalasia, regardless of whether heartburn is elicited by GER or esophageal stasis. Endoscopic examination can readily differentiate these disorders in most cases, although manometry may sometimes be required to make the distinction in equivocal cases. Readers may find interest in the AGA's recent technical review of the clinical uses of esophageal manometry and detailed descriptions of the procedure. Esophageal manometry is the gold standard test for esophageal motility disorders [23]. Esophageal manometry has been shown to be especially useful for definitively diagnosing achalasia or diffuse esophageal spasm and for detecting esophageal motor abnormalities associated with collagen-vascular disease.

(a) Fluid‐filled esophagus (b) Dilated esophagus

**Figure 2.** Achalasia

therapy. Esophagogastroduodenoscopy usually reveals large solitary ulcers or erosions in

A patient with a usual history of caustic substance ingestion and prolonged hospitalization for severe caustic damage was hospitalized again because of an increase in dysphagia and odynophagia [19]. The gold standard of safely assessing the depth, extent of caustic ingestion injury, and appropriate therapeutic regimen is EGD. The patients underwent EGD within 24 hours of admission and mucosal damage was graded using Zagar's modified endoscopic

Radiation therapy (RT), the primary modality for patients with tumors of the upper aerodigestive tract, allows larynx preservation [20]. Proximal esophageal strictures occur in 2–16% of patients after radiation therapy for cancers of the lung or head and neck. RT-induced laryngeal edema (due to inflammation and lymphatic disruption) is a common and expected side effect. Progressive edema and associated fibrosis detected by endoscope or barium swallowing can lead to long-term problems with phonation and swallowing. Aspiration pneumonia associated with dysphagia after intensive chemo radiation therapy has recently

Drug-induced esophagitis mainly presents as chest pain, odynophagia, and dysphagia. In the agents known to induce esophagitis, antibiotics are the most common culprit. Other causative agents include nonsteroidal anti-inflammatory drugs, antihypertensive drugs, acetamino‐ phen, oral hypoglycemic agents (glimepiride), bisphosphonates (alendronate, ibandronate), ascorbic acid, and warfarin [21]. The most frequent endoscopic site of drug-induced esopha‐ gitis is the middle third of the esophagus. On endoscopy, drug-induced esophagitis manifests as ulcers of various forms such as kissing ulcers, erosions, or ulcers with bleeding, as patchy sections coated with drug materials or impacted pill fragments, or sometimes as strictures.

Achalasia is regarded as a disease exclusively involving the smooth muscle. About 70–80% of patients have absent or incomplete lower esophageal sphincter (LES) relaxation with wet swallows, while the remainder have complete but shortened relaxation [22]. The typical endoscopic findings are shown in Figure 2A,B. Patients with esophageal stasis are often unaware of this condition. Heartburn, though not infrequent, bears little relationship to the esophageal acid exposure in achalasia, regardless of whether heartburn is elicited by GER or esophageal stasis. Endoscopic examination can readily differentiate these disorders in most cases, although manometry may sometimes be required to make the distinction in equivocal cases. Readers may find interest in the AGA's recent technical review of the clinical uses of esophageal manometry and detailed descriptions of the procedure. Esophageal manometry is the gold standard test for esophageal motility disorders [23]. Esophageal manometry has been shown to be especially useful for definitively diagnosing achalasia or diffuse esophageal spasm and for detecting esophageal motor abnormalities associated with collagen-vascular disease.

**2.6. Esophagitis induced by caustic injury, radiation injury, or drugs**

the distal esophagus.

96 Seminars in Dysphagia

classification scheme [19].

**2.7. Achalasia**

been reported at a growing frequency.

#### **2.8. Diffuse esophageal spasm**

Diffuse esophageal spasm (DES) is an uncommon disorder characterized by an impairment of ganglionic inhibition in the distal esophagus. Upper endoscopy should be performed as an initial evaluation of esophageal symptoms consistent with spastic disorders [24]. No specific endoscopic abnormality appears in most cases, but the endoscopist may notice disordered esophageal contractions. DES is defined by the presence of simultaneous contractions on conventional manometry. In higher resolution recordings by HRM with EPT, however, the propagation velocity varies greatly along the length of the esophagus and often progresses rapidly in some regions. Esophageal manometry, the putative gold standard in the diagnosis of achalasia, classically shows aperistalsis and failure of relaxation of the lower esophageal sphincter.

#### **2.9. Extrinsic compression**

Dysphagia is also caused by extrinsic compression of the esophagus associated with media‐ stinal diseases, tumors such as lung cancer and lymphoma, or infections such as tuberculosis or histoplasmosis [1,3].


*3.2. Decoding*

We use a maximum a posteriori (MAP) method to reconstruct a signal **x** in the NTGT.

$$
\hat{\mathbf{x}} = \arg\max P(\mathbf{x}|\mathbf{y}, \mathbf{A})\tag{6}
$$

The posteriori probability in (6) is as follows:

$$\begin{aligned} P(\mathbf{x}|\mathbf{y},\mathbf{A}) &= \frac{P(\mathbf{x},\mathbf{y},\mathbf{A})}{P(\mathbf{y},\mathbf{A})} \\ &\propto P(\mathbf{x},\mathbf{y},\mathbf{A}) \\ &= \sum\_{\mathbf{e}} P(\mathbf{x},\mathbf{y},\mathbf{A},\mathbf{e}) \\ &= \sum\_{\mathbf{e}} P(\mathbf{x})P(\mathbf{A})P(\mathbf{e})P(\mathbf{y}|\mathbf{x},\mathbf{A},\mathbf{e}) \end{aligned} \tag{7}$$

The last line of (7) is obtained using independent conditions, while the conditional probability *P*( **y**|**x**, **A**, **e**) is an indicator function that satisfies the following condition:

$$P(\mathbf{y}|\mathbf{x}, \mathbf{A}, \mathbf{e}) = \begin{cases} 1 & \text{if } \mathbf{y} = \mathbf{z} \oplus \mathbf{e}, \\ 0 & \text{if } \mathbf{y} \neq \mathbf{z} \oplus \mathbf{e}, \end{cases} \tag{8}$$

We define an error event if **xˆ** from (6) is not the same as the true realization of **x**. In other words, the probability of an error is expressed as *PE* = Pr{**xˆ** = **x**}.

#### *3.3. Bounds for Group Testing Schemes*

Now consider the number of tests on successful decoding in the conventional GT models. The number of tests required to identify *K* defective samples out of all *N* samples for an adaptive GT algorithm with perfect reconstruction denotes as *m*(*N*, *K*). Moreover, for the case of a non-adaptive model, the number of tests is defined as *m*¯(*N*, *K*). The number of tests *N* required for individual testing is greater than *m*¯(*N*, *K*). Adaptive GT models require less or equal number of tests than those of non-adaptive GTs because they check the results of previous tests and perform the next tests, *m*(*N*, *K*) ≤ *m*¯(*N*, *K*). Even if the number of defective samples is one, at least one test must be performed, 1 ≤ *m*(*N*, *K*). Therefore, the number of tests has a wide range as follows:

$$1 \le m(N, K) \le m(N, K) \le N \tag{9}$$

From an information-theoretic bound, the minimum number of tests *M* for a GT framework with a sample space is obtained as [3],

$$M \ge \log\_2 |\mathcal{S}|\tag{10}$$

where S denotes the sample space. In addition, an information-theoretic performance is presented even for a GT framework with small error probability. It is expressed as an upper bound of the error probability for the number of tests required for successful decoding. This GT algorithm performs in such a way as the following bound on successful probability *Ps* for decoding of defective samples [25]:

$$P\_s \le \frac{M}{\log\_2 \binom{N}{K}}\tag{11}$$

In the past half century, many studies on GT models have been performed, and among them, well-known and important GT algorithms are introduced next. The first one to be considered is the binary splitting algorithm [3]. This algorithm solves the existing GT problems efficiently and is applicable to the adaptive GT models. So far, the reason this algorithm is used for GT problems is because of its simplicity and good performance. The number of tests required to reconstruct defective samples using the binary splitting algorithm is known through the following bounds:

$$M = \begin{cases} \begin{array}{c} N \\ (\log\_2 \sigma + 2)K + p - 1 \end{array} & \text{if } N \le 2K - 1, \end{cases} \tag{12}$$

where *σ* is the number of samples to be included for one test, and *p* is a uniquely determined nonnegative integer conditioning *p* < *K*.

Next, the definite defectives algorithm [26] is considered. This algorithm is suitable for non-adaptive GT models because an unknown input signal can be reconstructed using all of the test results at the same time through an iterative process. The feature of the definite defectives algorithm is attractive in that it can eliminate false negatives that may occur during the reconstruction process. As a result, the use of the definite defectives algorithm is more useful in applications where false negatives are sensitive or should not be present. For given *N* and *K*, the definite defective algorithm has the following lower bound for the number of tests *M* required for identifying defective samples if it is allowed an error rate of *σ*,

$$M \ge (1 - \sigma) \log\_2 \binom{N}{K} \tag{13}$$

This can be observed that (11) and (13) coincide with the same in the perfect reconstruction of defective samples.

#### **4. Necessary Condition for Complete Recovery**

## *4.1. Lower Bound*

In this section, we take into account a necessary condition for the number of tests required to identify defective samples in the NTGT model. We obtain the necessary condition using Fano's inequality theorem [27] presented in information theory. Fano's inequality is mainly exploited in channel coding theory, and describes the connection between error probability and entropy. In addition, in [28], the authors reviewed GT problems comprehensively and in-depth from an information theory perspective. The lower bound on the probability of an error is obtained by considering Fano's inequality theorem. From this lower bound, we are lead to the necessary condition for the number of tests to find all defective samples for the NTGT model. We first explain Fano's inequality theorem before deriving the necessary condition.

**Theorem 1** (Fano's inequality [27])**.** *Suppose there are random variables A and B of finite size. If the decoding function* Φ *that finds A by considering B is used, the following inequality holds:*

$$1 + P(\Phi(B) \neq A) \log\_2 |A| \ge H(A|B) \tag{14}$$

*where P*(Φ(*B*) = *A*) *is the probability of an error for the decoding function* Φ*, and the conditional entropy H*(*A*|*B*) *is defined as follows:*

$$H(A|B) = -\sum\_{a \in A} \sum\_{\beta \in B} P\_{AB}(a, \beta) \log P\_{A|B}\left(a|\beta\right) \tag{15}$$

*where PAB and PA*|*B are the joint probability and conditional probability, respectively.*

In the NTGT problem, we are able to obtain a lower bound on the probability of an error. This lower bound shows the minimum number of tests required to reconstruct an unknown signal, regardless of which decoding function is used. In this paper, our lower bound is a variant of the results obtained in [8]. Compared to [8], this work obtains the lower bound taking into account the measurement noise. However, the overall procedure of derivation is similar to each other because it uses Fano's inequality theorem.

**Theorem 2** (Lower bound)**.** *For any decoding function with the unknown sample signal defined in* (1) *and the measurement noise defined in* (4)*, a necessary condition for the probability of error PE to be less than an arbitrary small and positive value ρ for PE* < *ρ holds such that*

$$\frac{NH(\delta) - M + MH(\eta) - 1}{N} < \rho \tag{16}$$

*where H*(·) *is the entropy function.*

by

**Proof of Theorem 2.** Let **xˆ** be the estimated signal of **x** found using the decoding function. Considering the following process in terms of a Markov chain, we can say **x** → (**y**, **A**) → **xˆ**. Then, the following inequality is satisfied,

$$H(\mathbf{x}|\mathbf{y}, \mathbf{A}) \le H(\mathbf{x}|\mathbf{\hat{x}}) \tag{17}$$

Further, from Fano's inequality described in (14), the conditional entropy is bounded

$$H(\mathbf{x}|\mathbf{y}, \mathbf{A}) \le 1 + P\_E \log\_2 \left( 2^N - 1 \right) \tag{18}$$

Then, the probability of error is bounded in terms of the conditional entropy and the total number of samples *N*,

$$P\_E \ge \frac{H(\mathbf{x}|\mathbf{y}, \mathbf{A}) - 1}{N} \tag{19}$$

It needs to tackle the conditional entropy *H*(**x**|**y**, **A**). Let us divide and expand the following conditional entropy in more detail:

$$\begin{array}{rcl}H(\mathbf{x}|\mathbf{y},\mathbf{A})&=&H(\mathbf{x})-I(\mathbf{x};\mathbf{y},\mathbf{A})\\&=&H(\mathbf{x})-(I(\mathbf{x};\mathbf{A})+I(\mathbf{x};\mathbf{y}|\mathbf{A})) \\&\overset{(a)}{=}&H(\mathbf{x})-(H(\mathbf{y}|\mathbf{A})-H(\mathbf{y}|\mathbf{A},\mathbf{x})) \end{array} \tag{20}$$

where *I*(·) is mutual information, and equality (a) comes from the fact that **x** and **A** are independent of each other. Note that the smaller the term on the right side of (19), the lower the minimum value of the probability of error. This means that the conditional entropy, *H*(**x**|**y**, **A**), should be small as possible. As a result, on the last line of the right side in (20), the conditional entropy *H*(**y**|**A**) should be large; conversely, the conditional entropy *H*(**y**|**A**, **x** ) should be small.

To do this, let us find the maximum and minimum values of the two conditional entropies, respectively.

$$\begin{array}{rcl}H(\mathbf{y}|\mathbf{A}) \leq H(\mathbf{y}) & = & H(\mathbf{z} \oplus \mathbf{e}) \\ & \leq & M \end{array} \tag{21}$$

where the first inequality is due to the definition of conditional entropy, and the last inequality comes from the fact that the result *yj* is either 0 or 1, *yj* values are independent of each other, and the maximum binary entropy is 1 in the case that Pr *yj* = 0 = Pr *yj* = 1 . Next, we take into account the other conditional entropy *H*(**y**|**A**, **x** ) which is minimized,

$$\begin{array}{rcl}H(\mathbf{y}|\mathbf{A},\mathbf{x})&=&H(\mathbf{z}\oplus\mathbf{e}|\mathbf{A},\mathbf{x})\\&=&H(\mathbf{e})\\&=&MH(\eta)\end{array}\tag{22}$$

where the second equality comes from how the randomness of **z** vanishes if **x** and **A** are known, the last equality being due to the independent events of **e**. Using (21) and (22), (20) can be rewritten as

$$H(\mathbf{x}|\mathbf{y}, \mathbf{A}) \le NH(\delta) - M + MH(\eta) \tag{23}$$

Finally, if (19) is changed to satisfy the condition *PE* < *ρ* where *ρ* is a small, positive value and *ρ* > 0, the following condition holds:

$$\frac{NH(\delta) - M + MH(\eta) - 1}{N} < \rho \tag{24}$$

This completes the proof of Theorem 2.

#### *4.2. Construction of Noisy Threshold Group Testing*

We now consider the result obtained from Theorem 2. First, Theorem 2 can be expressed as the ratio of the number of tests to the total number of samples as follows:

$$\frac{M}{N} > \frac{H(\delta) - \rho}{1 - H(\eta)}\tag{25}$$

It is advantageous to use the NTGT framework until the point *N* and *M* are equal. Otherwise, when *M* > *N*, individual testing becomes more effective than GT. This shows that NTGT can theoretically be used under the following noise conditions:

$$H(\eta) < 1 + \rho - H(\delta) \tag{26}$$

To design an NTGT framework, how to construct a group matrix is important. The key to this is shown in the proof of Theorem 2. Looking carefully at the conditions under which the inequality of conditional entropy holds in (21), the maximum conditional entropy *H*(**y**|**A**) is obtained when the following conditions are satisfied: Pr *yj* = 0 = Pr *yj* = 1 . This means that the NTGT system should be designed so that the output has an equal probability of being 0 or 1. Since **x** and **A** are independent of each other, the probability of an output of 0 is as follows:

$$\Pr\left(y\_{j} = 0\right) = \sum\_{t=0}^{T-1} \binom{N}{t} \left(\delta\gamma\right)^{t} \left(1 - \delta\gamma\right)^{N-t} = \frac{1}{2} \tag{27}$$

As shown in (27), it can be seen that there is a trade-off between *δ* and *γ*. In other words, to reconstruct a sparse signal, a high-density group matrix needs to be generated and used. Conversely, if the signal is not sparse, the group matrix should be designed with low density.

#### **5. Sufficient Condition for Average Performance**

## *5.1. Upper Bound*

Now we prove there is an upper bound on the probability of errors from the MAP decoding used in NTGT. We divide the proof of the upper bound into two parts: one considers the definition of the error event and the other part formulates the probability of errors.

We rewrite the a posteriori probability.

$$P(\mathbf{x}|\mathbf{y}, \mathbf{A}) \propto \sum\_{\mathbf{e}} P(\mathbf{x}) P(\mathbf{A}) P(\mathbf{e}) \mathbf{1}\_{\mathbf{y} = \mathbf{x} \oplus \mathbf{e}} \tag{28}$$

Note that both **A** and **y** are given and known. Using MAP decoding, we estimate with (28)

$$\mathfrak{X} = \arg\max\_{\mathbf{x}} \sum\_{\mathbf{e}} P(\mathbf{x}) P(\mathbf{A}) P(\mathbf{e}) \mathbf{1}\_{\mathbf{y} = \mathbf{z} \oplus \mathbf{e}} \tag{29}$$

An error event occurs if there is a feasible vector **x¯** = **x**, such that

$$\sum\_{\mathbf{v}} P(\bar{\mathbf{x}}) P(\mathbf{v}) \mathbf{1}\_{\mathbf{y} = \mathbf{w} \oplus \mathbf{v}} \ge \sum\_{\mathbf{e}} P(\mathbf{x}) P(\mathbf{e}) \mathbf{1}\_{\mathbf{y} = \mathbf{z} \oplus \mathbf{e}} \tag{30}$$

where **w** = ∑*Ajix*¯*i*≥*T* **Ax**¯ comes from (3), and **v** comes from a realization from (4). When given **y**, **A**, and **x**, we have one vector **e**, such that **e** = **z** ⊕ **y**. Then we can rewrite (30).

$$P(\bar{\mathbf{x}})P\_{\mathbf{V}}(\mathbf{y}\oplus\mathbf{w}) \ge P(\mathbf{x})P\_{\mathbf{e}}(\mathbf{y}\oplus\mathbf{z})\tag{31}$$

Therefore, an error event becomes equivalent to there existing a pair (**x**¯, **v**) such that

$$\begin{array}{c} \bar{\mathbf{x}} \neq \mathbf{x}, \\ \mathbf{y} = \mathbf{w} \oplus \mathbf{v} = \mathbf{z} \oplus \mathbf{e}, \\ P(\bar{\mathbf{x}}) P\_{\mathbf{v}}(\mathbf{y} \oplus \mathbf{w}) \ge P(\mathbf{x}) P\_{\mathbf{e}}(\mathbf{y} \oplus \mathbf{z}) \end{array} \tag{32}$$

So far, we have defined the error event and now we will derive an upper bound on the probability of error. When given **x** and **e**, we let *P*(I|**x**, **e** ) be the conditional error probability. We have an average error probability as follows:

$$P\_E = \sum\_{\mathbf{x}} \sum\_{\mathbf{e}} P(\mathbf{x}, \mathbf{e}) P(\mathcal{Z} | \mathbf{x}, \mathbf{e}) \tag{33}$$

We now introduce two typical sets that were defined in [27] (Ch.3.1). Let A *M* [**e**]*ε* and A *N* [**x**]*ε* be typical sets of **x** and **e** with respect to *P*(**x**) and *P*(**e**) as defined in (1) and (4). For any positive number *ε* and sufficiently large numbers of *N* and *M*, the two typical sets are defined as

$$\mathcal{A}\_{[\mathbf{x}]\varepsilon}^{N} = \left\{ \mathbf{x} \in \mathbf{2}^{N} : \left| -\frac{1}{N} \log P(\mathbf{x}) - H(\delta) \right| \le \varepsilon \right\} \tag{34}$$

and

$$\mathcal{A}\_{[\mathbf{e}]\varepsilon}^{M} = \left\{ \mathbf{e} \in \mathbf{2}^{M} : \left| -\frac{1}{M} \log P(\mathbf{e}) - H(\eta) \right| \le \varepsilon \right\} \tag{35}$$

From the Shannon–McMillan–Breiman theorem [27] (Ch.16.8), we obtain the following two bounds:

$$P\left(\left|-\frac{1}{N}\log P(\mathbf{x}) - H(\delta)\right| \le \varepsilon\right) \ge 1 - \varepsilon \tag{36}$$

and

$$P\left(\left|-\frac{1}{M}\log P(\mathbf{e}) - H(\eta)\right| \le \varepsilon\right) \ge 1 - \varepsilon \tag{37}$$

Now we define the space of the pair(**x**, **e**) with respect to the two typical sets. Let U and U*c* be the sets for the pair (**x**, **e**) such that

$$\mathcal{U} = \left\{ \mathbf{x} \in \mathfrak{L}^{N}, \mathbf{e} \in \mathfrak{L}^{M} : \left( \mathbf{x} \in \mathcal{A}\_{[\mathbf{x}]\varepsilon}^{N} \bigcap \mathbf{e} \in \mathcal{A}\_{[\mathbf{e}]\varepsilon}^{M} \right) \right\} \tag{38}$$

and

$$\mathcal{M}^{\mathbb{C}} = \left\{ \mathbf{x} \in \mathbf{2}^{N}, \mathbf{e} \in \mathbf{2}^{M} : \left( \mathbf{x} \notin \mathcal{A}\_{[\mathbf{x}] \boldsymbol{\varepsilon}}^{N} \bigcup \mathbf{e} \notin \mathcal{A}\_{[\mathbf{e}] \boldsymbol{\varepsilon}}^{M} \right) \right\} \tag{39}$$

where U is the joint typical set for the pair (**x**, **e**), since **x** and **e** are independent.

**Theorem 3** (Upper bound)**.** *In NTGT one, a distribution of defective samples defined in* (1) *and noise probability defined in* (4)*, for any small ε, the ratio of the number of tests M to the total number of samples N is upper-bounded:*

$$\frac{M}{N} > \frac{H(\delta) + \varepsilon}{1 - H(\eta) - \varepsilon} \tag{40}$$

**Proof of Theorem 3.** The probability of error is bounded as

$$\begin{split} P\_{\mathcal{E}} &= \sum\_{(\mathbf{x},\mathbf{e}) \in \mathcal{U}} P(\mathbf{x}) P(\mathbf{e}) P(\mathcal{Z}|\mathbf{x}, \mathbf{e}) + \sum\_{(\mathbf{x},\mathbf{e}) \in \mathcal{U}^{c}} P(\mathbf{x}) P(\mathbf{e}) P(\mathcal{Z}|\mathbf{x}, \mathbf{e}) \\ &\stackrel{(a)}{\leq} \sum\_{(\mathbf{x},\mathbf{e}) \in \mathcal{U}} P(\mathbf{x}) P(\mathbf{e}) P(\mathcal{Z}|\mathbf{x}, \mathbf{e}) + \sum\_{(\mathbf{x},\mathbf{e}) \in \mathcal{U}^{c}} P(\mathbf{x}) P(\mathbf{e}) \\ &\stackrel{(b)}{\leq} \sum\_{(\mathbf{x},\mathbf{e}) \in \mathcal{U}} P(\mathbf{x}) P(\mathbf{e}) P(\mathcal{Z}|\mathbf{x}, \mathbf{e}) + 2\varepsilon \end{split} \tag{41}$$

where (a) is due to *P*(I|**x**, **e** ) ≤ 1, and (b) comes from the following,

$$\begin{split} \sum\_{(\mathbf{x},\mathbf{e}) \in \mathcal{U}^{\xi}} P(\mathbf{x})P(\mathbf{e}) &= 1 - \sum\_{(\mathbf{x},\mathbf{e}) \in \mathcal{U}} P(\mathbf{x})P(\mathbf{e}) \\ &= 1 - \sum\_{\mathbf{x} \notin \mathcal{A}^{N}\_{[\mathbf{x}] \times \varepsilon}} P(\mathbf{x}) \sum\_{\mathbf{e} \notin \mathcal{A}^{M}\_{[\mathbf{e}] \times \varepsilon}} P(\mathbf{e}) \\ &\leq 1 - (1 - \varepsilon)(1 - \varepsilon) \\ &\leq 2\varepsilon \end{split} \tag{42}$$

This is because **A** is randomly generated as defined in (2); then we can define the following event as

$$\mathcal{E}\left(\mathbf{x}, \mathbf{e}; \bar{\mathbf{x}}, \mathbf{v}\right) = \left\{ \left(\mathbf{x}, \mathbf{e}; \bar{\mathbf{x}}, \mathbf{v}\right) : \mathbf{z} \oplus \mathbf{e} = \mathbf{w} \oplus \mathbf{v} \right\} \tag{43}$$

The conditional error probability *P*(I|**x**, **e** ) is the probability of the union of all the events in (43) with respect to all pairs (**x**¯, **v**) that satisfy (32). Thus, the conditional error probability in (33) can be rewritten as

$$P(\mathcal{Z}|\mathbf{x}, \mathbf{e}) = \Pr\left\{ \bigcup\_{\mathbf{x}, \mathbf{v}: P(\mathbf{x})P(\mathbf{v}) \ge P(\mathbf{x})P(\mathbf{e})} \mathcal{E}(\mathbf{x}, \mathbf{e}; \mathbf{\bar{x}}, \mathbf{v}) \right\} \tag{44}$$

Using the union bound in (41), we have the following bound:

$$\begin{split} \boldsymbol{P}\_{\mathrm{E}} &\leq \sum\_{(\mathbf{x},\mathbf{e}) \in \mathcal{U}} \mathrm{P}(\mathbf{x}) \, P(\mathbf{e}) \sum\_{(\mathbf{x},\mathbf{v}) : P(\mathbf{x}) \, P(\mathbf{v}) \geq P(\mathbf{x})} \mathrm{P}\left(\mathcal{E}\left(\mathbf{x},\mathbf{e};\mathbf{\bar{x}},\mathbf{v}\right)\right) + 2\varepsilon \\ &= \sum\_{(\mathbf{x},\mathbf{e}) \in \mathcal{U}} \mathrm{P}(\mathbf{x}) P(\mathbf{e}) \sum\_{(\mathbf{x},\mathbf{v})} P\left(\mathcal{E}\left(\mathbf{x},\mathbf{e};\mathbf{\bar{x}},\mathbf{v}\right)\right) \Phi(\mathbf{x},\mathbf{\bar{x}},\mathbf{e},\mathbf{v}) + 2\varepsilon \end{split} \tag{45}$$

where Φ(**x**, **x**¯, **e**, **v**) is the indicator function, such that *P*(**x**¯)*P*(**v**) ≥ *P*(**x**)*P*(**e**).

$$\Phi(\mathbf{x}, \bar{\mathbf{x}}, \mathbf{e}, \mathbf{v}) = \begin{cases} 1 & \text{if } P(\bar{\mathbf{x}})P(\mathbf{v}) \ge P(\mathbf{x})P(\mathbf{e})\\ 0 & \text{if } P(\bar{\mathbf{x}})P(\mathbf{v}) < P(\mathbf{x})P(\mathbf{e}) \end{cases} \tag{46}$$

The indicator function is bounded [29] (Ch. 5.6) for 0 < *s* ≤ 1.

$$\Phi(\mathbf{x}, \vec{\mathbf{x}}, \mathbf{e}, \mathbf{v}) \le \left(\frac{P(\vec{\mathbf{x}})P(\mathbf{v})}{P(\mathbf{x})P(\mathbf{e})}\right)^{s} \tag{47}$$

For *s* = 1 in (47), we have the following bound:

$$P\_{\rm E} \leq \sum\_{(\mathbf{x},\mathbf{e}) \in \mathcal{U}} \sum\_{(\mathbf{x},\mathbf{v})} P(\mathbf{x}) P(\mathbf{v}) P\left(\mathcal{E}\left(\mathbf{x},\mathbf{e};\mathbf{\bar{x}},\mathbf{v}\right)\right) + 2\varepsilon \tag{48}$$

From the definition in (43), note that the probability *P*E(**x**, **e**; **x¯**, **v**) is

$$P\left(\mathcal{E}\left(\mathbf{x}, \mathbf{e}; \overline{\mathbf{x}}, \mathbf{v}\right)\right) = \Pr\left(\mathbf{w} \oplus \mathbf{v} = \mathbf{z} \oplus \mathbf{e}\right) \tag{49}$$

where

$$\begin{split} \boldsymbol{P}\_{\mathcal{E}} & \leq \sum\_{(\mathbf{x}, \mathbf{e}) \in \mathcal{U}} \sum\_{(\mathbf{x}, \mathbf{v})} P(\hat{\mathbf{x}}) P(\mathbf{v}) P(\mathcal{E} \left( \mathbf{x}, \mathbf{e}; \hat{\mathbf{x}}, \mathbf{v} \right)) + 2 \varepsilon \\ &= \sum\_{(\mathbf{x}, \mathbf{e}) \in \mathcal{U}, \|\mathbf{e}\| \in \mathbf{v}\|\_{0} - d\_{2}} \sum\_{\|\mathbf{x}\|\_{0} - d\_{1}} \sum\_{\|\mathbf{e}\| \in \mathbf{v}\|\_{0} - d\_{2}} P(\hat{\mathbf{x}}) P(\mathbf{v}) P\left( \mathbf{z} \oplus \mathbf{w} = \mathbf{e} \oplus \mathbf{v} \, \middle| \, \|\mathbf{x}\|\_{0} = d\_{1}, \, \|\mathbf{e} \oplus \mathbf{v}\|\_{0} = d\_{2} \right) \end{split} \tag{50}$$

In (50), we find the following probability depending on the number of nonzero elements *d*1 and *d*2:

$$P\left(\mathbf{z}\oplus\mathbf{w}=\mathbf{e}\oplus\mathbf{v}\,\middle|\,\|\mathbf{\dot{x}}\|\_{0}=d\_{1},\,\|\mathbf{e}\oplus\mathbf{v}\|\_{0}=d\_{2}\right)=\prod\_{j=1}^{M}P\left(z\_{j}\oplus w\_{j}=e\_{j}\oplus v\_{j}\middle|\,\|\mathbf{\dot{x}}\|\_{0}=d\_{1},\,\|\mathbf{e}\oplus\mathbf{v}\|\_{0}=d\_{2}\right)$$

$$=P\left(z\_{j}\oplus w\_{j}=1\middle|\,\|\mathbf{\dot{x}}\|\_{0}=d\_{1},\,\|\mathbf{e}\oplus\mathbf{v}\|\_{0}=d\_{2}\right)^{d\_{2}}\tag{51}$$

$$\times P\left(z\_{j}\oplus w\_{j}=0\middle|\,\|\mathbf{\dot{x}}\|\_{0}=d\_{1},\,\|\mathbf{e}\oplus\mathbf{v}\|\_{0}=d\_{2}\right)^{M-d\_{2}}$$

$$=(1-P\_{0})^{d\_{2}}P\_{0}^{M-d\_{2}}$$

where each row is independent. Given this, we define the following probability:

$$P\_0 \stackrel{\Delta}{=} \Pr(z\_j \oplus w\_j = 0 || \| \mathbf{\bar{x}} ||\_0 = d\_1) \tag{52}$$

We can divide *P*0 in (52) into two parts. If *d*1 < *T*,

$$\begin{split} \mathbf{P}\_0 &= \Pr\left(z\_j = 0\right) \Pr\left(w\_j = 0\right) + \Pr\left(z\_j = 1\right) \Pr\left(w\_j = 1\right) \\ &= \Pr\left(z\_j = 0\right) \end{split} \tag{53}$$

Otherwise,

$$\begin{split} P\_0 &= \text{Pr}\{z\_j = 0\} \left(\sum\_{t=0}^{T-1} \binom{d\_1}{t} \gamma^t (1-\gamma)^{(d\_1-t)}\right) + \text{Pr}\{z\_j = 1\} \left(\sum\_{t=T}^{d\_1} \binom{d\_1}{t} \gamma^t (1-\gamma)^{(d\_1-t)}\right) \\ &= P\_{z,0}(\delta,\gamma) P\_{w,0}(d\_1,\gamma) + (1-P\_{z,0}(\delta,\gamma))(1-P\_{w,0}(d\_1,\gamma)) \end{split} \tag{54}$$

where

$$\begin{aligned} P\_{z,0}(\delta,\gamma) \stackrel{\Delta}{=} \Pr(z\_{\bar{j}}=0) &= \sum\_{t=0}^{T-1} \binom{N}{t} (\delta\gamma)^t (1-\delta\gamma)^{N-t}, \\ P\_{w,0}(d\_1,\gamma) \stackrel{\Delta}{=} \Pr(w\_{\bar{j}}=0) &= \sum\_{t=0}^{T-1} \binom{d\_1}{t} \gamma^t (1-\gamma)^{d\_1-t} \end{aligned} \tag{55}$$

The maximum for *P*0 by looking at *Pz*,0(*δ*, *γ*) = 1/2 and *Pw*,0(*d*1, *γ*) = 1/2 from the fact that *P*0 in (54) is concave with respect to *Pz*,0(*δ*, *γ*) and *Pw*,0(*d*1, *γ*). Therefore, its bound is

$$P\_0 \le \frac{1}{2} \tag{56}$$

Using (51) and (56), (50) can be bounded as follows:

$$\begin{split} &P\_{\mathbb{E}} \leq 2^{-M} \sum\_{d\_1=0, \mathbf{x} \neq \mathbf{x}} \sum\_{(\mathbf{x}, \mathbf{e}) \in \mathcal{U}, \mathbf{x} \parallel \mathbf{e} \parallel d\_0 = d\_1} P(\bar{\mathbf{x}}) \left( \sum\_{\mathbf{v}} P(\mathbf{v}) \right) + 2\varepsilon \\ &\leq 2^{-M} \sum\_{d\_1=0, \mathbf{x} \neq \mathbf{x}} \sum\_{(\mathbf{x}, \mathbf{e}) \in \mathcal{U}, \mathbf{x} \parallel \mathbf{e} \parallel d\_0 = d\_1} P(\bar{\mathbf{x}}) + 2\varepsilon \\ &\leq 2^{-M} \sum\_{\mathbf{x} \in \mathcal{A}^{M}\_{[\mathbf{x}]}} \sum\_{\mathbf{e} \in \mathcal{A}^{M}\_{[\mathbf{e}]}} \sum\_{d\_1=0, \mathbf{x} \neq \mathbf{x}} P(\bar{\mathbf{x}}) + 2\varepsilon \\ &= 2^{-M} \left| \mathcal{A}^{N}\_{[\mathbf{x}] \varepsilon} \right| \cdot \left| \mathcal{A}^{M}\_{[\mathbf{e}] \varepsilon} \right| \sum\_{d\_1=0, \mathbf{x} \neq \mathbf{x}} P(\bar{\mathbf{x}}) + 2\varepsilon \\ &\leq 2^{-M} \left| \mathcal{A}^{N}\_{[\mathbf{x}] \varepsilon} \right| \cdot \left| \mathcal{A}^{M}\_{[\mathbf{e}] \varepsilon} \right| + 2\varepsilon \\ &\leq 2^{-M} 2^{N(H(\delta) + \varepsilon)} 2^{M(H(\eta) + \varepsilon)} + 2\varepsilon \\ &= 2^{N(H(\delta) + \varepsilon) + M(H(\eta) + \varepsilon) - M} + 2\varepsilon \end{split} \tag{57}$$

As the probability of error is less than 1, the exponent term on the right side of (57) is bounded by

$$N(H(\delta) + \varepsilon) + M(H(\eta) + \varepsilon) - M < 0\tag{58}$$

Then, the ratio of *M* to *N* is

$$\frac{M}{N} > \frac{H(\delta) + \varepsilon}{1 - H(\eta) - \varepsilon} \tag{59}$$

This completes the proof of Theorem 3.

#### *5.2. Discussion for Necessary and Sufficient Conditions*

In this section, we discuss the results obtained from Theorems 2 and 3. The result from Theorem 2 allows us to solve the lower bound in the NTGT problem using Fano's inequality. The minimum number of tests required to recover all defective samples with *δ* probability out of *N* samples is also obtained. In other words, Theorem 2 is a necessary condition for any probability of error to be smaller than *ρ*. Conversely, Theorem 3 leads to the upper bound on the probability of an error using the MAP decoding method. This condition refers to the upper bound on performance and is the sufficient condition to allow us to reconstruct defective samples.

We show that the results of Theorems 2 and 3 coincide with each other. Finding and presenting the necessary and sufficient conditions for the number of tests required in the NTGT problem is significant for TGT. In addition, as shown in (27) above, a system design method for NTGT was proposed so that the probability that a test result is 0 and the probability that it is 1 are the same depending on threshold *T*.


*2.9. Scanning Electron Microscopy (SEM)*

The adhesion and biofilm formation over the cotton fabric were assessed by SEM. The tubes containing MPB and specimens of untreated and treated cotton fabrics were inoculated with a suspension of *P. aeruginosa.* After incubation for 24 h, the specimens were washed with phosphate-buffered saline, dried and coated with gold with Jeol JFC-1200 fine coater (Jeol Ltd., Tokyo, Japan) and then investigated by Jeol JSM-5510 SEM (Jeol Ltd., Tokyo, Japan).


*3.2. Survey-Based Data Compilation*

A questionnaire was designed and shared with the health counselors and advisors (from the National Institute of Health), medical practitioners (from Shifa International Hospital, Pakistan Institute of Medical Sciences, and Aga Khan University Hospital), professors, and associate professors (from Quaid-i-Azam University, King Edward College, and Forman Christian College University) from the fields of Sociology, Medicine, and Psychology to obtain their expert feedback for pre-examination purposes. These expert participants played a dual role in the assessment of the questionnaire. Firstly, they commented on the contents of the questions to improve their clarity and quality. It established the content validity of the questionnaire. Secondly, they filled in the questionnaire for pilot testing to verify the functionality of the questionnaire. It established the face validity of the questionnaire [95]. The profiles of the participatory role-playing individuals are given in Appendix A (Table A1).

A questionnaire in English was combined with Urdu translation format removing any language barriers and producing informed feedback. This online survey was conducted in Pakistan during May–June 2020. In the face of the ongoing pandemic outbreak, the questionnaire was floated in numerous Facebook (Facebook Inc., Menlo Park, CA, USA) and WhatsApp (WhatsApp Inc., Menlo Park, CA, USA) groups among the social circles of friends, friends' friends, colleagues, colleagues' friends, and scholars and students from universities across universities. Ethical considerations were included by stating the research aims and scope in the questionnaire's introductory paragraph to ensure the respondents' informed consent. Furthermore, the confidentiality and anonymity of respondents were also guaranteed during the questionnaire conduction. Following Kamenidou et al. [96], the questionnaire conduction process was based on mixed non-probability sampling, which involved convenience, snowball, and criteria sampling procedures. The recruitment criterion was mainly based on the age of the respondents. Respondents below 18 years of age were advised not to fill in the questionnaire. Also, the individuals reluctant to provide their consent were excluded. (i.e., exclusion criteria). Moreover, the respondents needed to be residents of Pakistan. Further, since the questionnaire was conducted online, respondents on social media (Facebook and WhatsApp) were the only population available to generate the data sample (i.e., inclusion criteria). The respondents were from heterogeneous backgrounds in terms of occupation, qualification, and household income, among other traits. It considerably led the findings to be generalized for the population belonging to heterogeneous backgrounds. The survey was conducted from a total of 931 respondents. After initial scrutiny, 828 questionnaires were found completely and appropriately filled in by the respondents. Those questionnaires were declared valid for analysis purposes. The designed questions are presented in Appendix B (Table 2).


**8.6. External neutron sources for reproduction coefficient increase in thermal reactor**

External neutron sources can be included in neutron balance if it is linked somehow with reactor work. There are schemes, where neutrons, which are formed in reactions between accelerated protons and nuclei of heavy metals, are emitted to the reactor's core. This scheme has significant lack of reactor work reliability because of use less reliable source – electric nuclear installation. When autonomic electric nuclear installation is used only for 233U formation by neutron absorptions in thorium, reactor work almost is not depending on work reliability of electric nuclear installation. Work [21] demonstrates that raw uranium usage can be increased up to 100 % with use of comparatively small energy expenses (less than 2,5 % of produced energy).

Note, that modern thermal reactors, which use less than 1% or raw uranium, about 1% of produced energy spent for fuel enrichment.

Possible characteristics of joint work of fission reactor and neutron source based on fusion reactor are shown [22].