Academic literature on the topic 'Single-sender index coding problems'

This paper studies index coding with two senders. In this setup, source messages are distributed among the senders possibly with common messages. In addition, there are multiple receivers, with each receiver having some messages a priori, known as side-information, and requesting one unique message such that each message is requested by only one receiver. Index coding in this setup is called two-sender unicast index coding (TSUIC). The main goal is to find the shortest aggregate normalized codelength, which is expressed as the optimal broadcast rate. In this work, firstly, for a given TSUIC problem, we form three independent sub-problems each consisting of the only subset of the messages, based on whether the messages are available only in one of the senders or in both senders. Then, we express the optimal broadcast rate of the TSUIC problem as a function of the optimal broadcast rates of those independent sub-problems. In this way, we discover the structural characteristics of TSUIC. For the proofs of our results, we utilize confusion graphs and coding techniques used in single-sender index coding. To adapt the confusion graph technique in TSUIC, we introduce a new graph-coloring approach that is different from the normal graph coloring, which we call two-sender graph coloring, and propose a way of grouping the vertices to analyze the number of colors used. We further determine a class of TSUIC instances where a certain type of side-information can be removed without affecting their optimal broadcast rates. Finally, we generalize the results of a class of TSUIC problems to multiple senders.

SOX5encodes a conserved transcription factor implicated in cell-fate decisions of the neural lineage.SOX5haploinsufficiency induced by larger genomic deletions has been linked to a recognizable pediatric syndrome combining developmental delay with intellectual disability, mild dysmorphism, inadequate behavior, and variable additional features including motor disturbances. In contrast toSOX5-involving deletions, examples of pathogenicSOX5small coding variations are sparse in the literature and have been described only in singular cases with phenotypic abnormalities akin to those seen in theSOX5microdeletion syndrome. Here a novelSOX5loss-of-function point mutation, c.13C>T (p.Arg5X), is reported, identified in the course of exome sequencing applied to the diagnosis of an unexplained adult-onset motor disorder. Aged 43 years, our female index patient demonstrated abrupt onset of mixed generalized hyperkinesia, with dystonic and choreiform movements being the most salient features. The movement disorder was accompanied by behavioral problems such as anxiety and mood instability. The mutation was found to be inherited to the patient’s son who manifested abnormal behavior including diminished social functioning, paranoid ideation, and anxiety since adolescence. Our results expand the compendium ofSOX5damaging single-nucleotide variation mutations and suggest that SOX5haploinsufficiency might not be restrictively associated with childhood-onset syndromic disease.

<p><strong>Abstract.</strong> Atlas mapping is one of the priorities of the modern mapping industry. It is connected both with the unceasing popularity of traditional atlases in the printing version, and with the active introduction and use of interactive electronic and web atlases, as well as with the development of atlas information systems. Currently there are no generalizing methodological solutions for automating the atlas compilation processes, despite the importance and perspectives of atlas mapping. In addition, there is no software which ensures all requirements for the design of the atlas. The use of geographic information technologies in atlas mapping doesn’t solve all the problems involved in creating atlases. For example, GIS can be used for solution of individual processes, e.g. the development of spatial and thematic databases, the creation of thematic maps, the cartographic generalization, the project of graphical index of maps into atlas pages, etc. But GIS doesn’t provide for the creation of a finished cartographic work with an integrated structure and content. This fact undoubtedly leads to the high cost and laboriousness of creating atlases, to the long-term design and the presence of errors in their compilation.</p><p>The Department of Cartography of the Moscow State University of Geodesy and Cartography (MIIGAiK) carries out research work on the development of the theory and methods of automating atlas mapping processes. The research is being directed to the speed of creation and improvement of the quality of general geographical, thematic and complex atlases. As part of this research, methods and algorithms for the automated creation of atlases have been developed, providing solutions to the most time-consuming and rather complex processes of atlas mapping, among which are:</p><ol><li>The development of an optimal atlas structure;</li><li>Design of the mathematical basis of the atlas (selection of a scale series, cartographic projections, format and layout);</li><li>Formalization of the creation of geographic base maps for atlas;</li><li>The formation and visualization of reference information of the atlas;</li><li>Organization, storage and use of spatial-temporary data in electronic atlases.</li></ol><p> Figure 1 presents the proposed solutions for automating processes mentioned above. The list of solutions may be different depending on the type of atlas being created (general geographical, thematic, and complex atlas). So the following options can be when developing the structure of the atlas. The use of a unified system of classification and coding of thematic maps and taking into account the degree of knowledge (study) of the object and mapped phenomenon is being used only for a thematic or complex regional atlas. The definition of possible combinations of territories shown on a single atlas map is used only for thematic or general geographical atlas. For each type of atlas, the formation and use of the reference information system of the studied atlases can be used, as well as the identification and formalization of factors affecting the inclusion of a specific section or a separate map.</p><p>The results of the research will ensure the efficiency of creating atlases and increase their quality. Also its will help to meet the increasing consumer demand for atlas map products, especially in the form of electronic atlases and geo-portal solutions.</p>

Polymorphisms in several neurotransmitter-associated genes have been associated with variation in human personality traits. Among the more promising of such associations is that between the human dopamine receptor D4 gene ( Drd4 ) variants and novelty-seeking behaviour. However, genetic epistasis, genotype–environment interactions and confounding environmental factors all act to obscure genotype–personality relationships. Such problems can be addressed by measuring personality under standardized conditions and by selection experiments, with both approaches only feasible with non-human animals. Looking for similar Drd4 genotype–personality associations in a free-living bird, the great tit ( Parus major ), we detected 73 polymorphisms (66 SNPs, 7 indels) in the P. major Drd4 orthologue. Two of the P. major Drd4 gene polymorphisms were investigated for evidence of association with novelty-seeking behaviour: a coding region synonymous single nucleotide polymorphism (SNP830) and a 15 bp indel (ID15) located 5′ to the putative transcription initiation site. Frequencies of the three Drd4 SNP830 genotypes, but not the ID15 genotypes, differed significantly between two P. major lines selected over four generations for divergent levels of ‘early exploratory behaviour’ (EEB). Strong corroborating evidence for the significance of this finding comes from the analysis of free-living, unselected birds where we found a significant association between SNP830 genotypes and differing mean EEB levels. These findings suggest that an association between Drd4 gene polymorphisms and animal personality variation predates the divergence of the avian and mammalian lineages. Furthermore, this work heralds the possibility of following microevolutionary changes in frequencies of behaviourally relevant Drd4 polymorphisms within populations where natural selection acts differentially on different personality types.

Introduction: Increased prescribing of high potency opioids has been associated with increasing opioid addiction and linked to serious adverse outcomes including misuse, diversion, overdose and death. Problems related to opioids are a major Canadian public health concern yet few data are available on prescribing in most Canadian provinces. The objective of this study was to describe opioid prescribing in Ontario EDs and patient harms associated with this practice. Methods: We conducted a population-based cohort study among Ontario residents aged 15-64 years who were eligible for public drug coverage between April 2008 and March 2012. Using administrative databases, we identified patients with no opioid use in the past 12 months who received a prescription opioid from an emergency or family physician. Patients were followed for 2 years following their index prescription. The primary outcome was hospital admission for opioid toxicity and secondary outcome was dose-escalation exceeding 200 mg morphine equivalents (MEQ). Results: Of the 77,270 unique patients included, 33,492 (43.3%) and 43,778 (56.7%) prescriptions were issued by emergency physician (EP) and family physicians (FP), respectively. FP patients were older (45.9 vs 41.2 yr, MSD 0.35), had fewer ED visits (0.9 vs 2.3, MSD 0.46), and more FP visits (11.5 vs 8.7 MSD 0.31) in the year prior to their index visit. For combination products, EPs were more likely to prescribe oxycodone compared to FPs (37.2% vs 16.7%, Δ 20.5, 95% CI: 19.9, 21.2). For single agent products, EPs were more likely to prescribe hydromorphone compared to FPs (44.5% vs 21.7%, Δ 22.8, 95% CI: 20.4, 25.2). FPs were more likely to prescribe codeine either as a combination or single agent formulation. EP prescriptions led to significantly more hospital admissions for opioid toxicity (0.5% vs 0.3%, Δ 0.2, 95% CI: 0.1, 0.3), while FP prescriptions more often resulted in dose escalation beyond 200 mg MEQs (0.1% vs 0.7%, Δ 0.6, 95% CI: 0.4, 0.7). Conclusion: A large percentage of opioid-naïve patients receive an initial opiate prescription in the ED, where the use of high potency opioids is much more common, with 1/200 of these patients subsequently hospitalized for opioid toxicity. Creation of a physician accessible provincial registry would be useful to monitor opioid prescribing and dispensing, inform clinical practice, and identify patients at high-risk who may benefit from early interventions.

Objective This update of a 2011 guideline developed by the American Academy of Otolaryngology–Head and Neck Surgery Foundation provides evidence-based recommendations on the pre-, intra-, and postoperative care and management of children 1 to 18 years of age under consideration for tonsillectomy. Tonsillectomy is defined as a surgical procedure performed with or without adenoidectomy that completely removes the tonsil, including its capsule, by dissecting the peritonsillar space between the tonsil capsule and the muscular wall. Tonsillectomy is one of the most common surgical procedures in the United States, with 289,000 ambulatory procedures performed annually in children <15 years of age based on the most recent published data. This guideline is intended for all clinicians in any setting who interact with children who may be candidates for tonsillectomy. Purpose The purpose of this multidisciplinary guideline is to identify quality improvement opportunities in managing children under consideration for tonsillectomy and to create explicit and actionable recommendations to implement these opportunities in clinical practice. Specifically, the goals are to educate clinicians, patients, and/or caregivers regarding the indications for tonsillectomy and the natural history of recurrent throat infections. Additional goals include the following: optimizing the perioperative management of children undergoing tonsillectomy, emphasizing the need for evaluation and intervention in special populations, improving the counseling and education of families who are considering tonsillectomy for their children, highlighting the management options for patients with modifying factors, and reducing inappropriate or unnecessary variations in care. Children aged 1 to 18 years under consideration for tonsillectomy are the target patient for the guideline. For this guideline update, the American Academy of Otolaryngology–Head and Neck Surgery Foundation selected a panel representing the fields of nursing, anesthesiology, consumers, family medicine, infectious disease, otolaryngology–head and neck surgery, pediatrics, and sleep medicine. Key Action Statements The guideline update group made strong recommendations for the following key action statements (KASs): (1) Clinicians should recommend watchful waiting for recurrent throat infection if there have been <7 episodes in the past year, <5 episodes per year in the past 2 years, or <3 episodes per year in the past 3 years. (2) Clinicians should administer a single intraoperative dose of intravenous dexamethasone to children undergoing tonsillectomy. (3) Clinicians should recommend ibuprofen, acetaminophen, or both for pain control after tonsillectomy. The guideline update group made recommendations for the following KASs: (1) Clinicians should assess the child with recurrent throat infection who does not meet criteria in KAS 2 for modifying factors that may nonetheless favor tonsillectomy, which may include but are not limited to multiple antibiotic allergies/intolerance, PFAPA (periodic fever, aphthous stomatitis, pharyngitis, and adenitis), or history of >1 peritonsillar abscess. (2) Clinicians should ask caregivers of children with obstructive sleep-disordered breathing and tonsillar hypertrophy about comorbid conditions that may improve after tonsillectomy, including growth retardation, poor school performance, enuresis, asthma, and behavioral problems. (3) Before performing tonsillectomy, the clinician should refer children with obstructive sleep-disordered breathing for polysomnography if they are <2 years of age or if they exhibit any of the following: obesity, Down syndrome, craniofacial abnormalities, neuromuscular disorders, sickle cell disease, or mucopolysaccharidoses. (4) The clinician should advocate for polysomnography prior to tonsillectomy for obstructive sleep-disordered breathing in children without any of the comorbidities listed in KAS 5 for whom the need for tonsillectomy is uncertain or when there is discordance between the physical examination and the reported severity of oSDB. (5) Clinicians should recommend tonsillectomy for children with obstructive sleep apnea documented by overnight polysomnography. (6) Clinicians should counsel patients and caregivers and explain that obstructive sleep-disordered breathing may persist or recur after tonsillectomy and may require further management. (7) The clinician should counsel patients and caregivers regarding the importance of managing posttonsillectomy pain as part of the perioperative education process and should reinforce this counseling at the time of surgery with reminders about the need to anticipate, reassess, and adequately treat pain after surgery. (8) Clinicians should arrange for overnight, inpatient monitoring of children after tonsillectomy if they are <3 years old or have severe obstructive sleep apnea (apnea-hypopnea index ≥10 obstructive events/hour, oxygen saturation nadir <80%, or both). (9) Clinicians should follow up with patients and/or caregivers after tonsillectomy and document in the medical record the presence or absence of bleeding within 24 hours of surgery (primary bleeding) and bleeding occurring later than 24 hours after surgery (secondary bleeding). (10) Clinicians should determine their rate of primary and secondary posttonsillectomy bleeding at least annually. The guideline update group made a strong recommendation against 2 actions: (1) Clinicians should not administer or prescribe perioperative antibiotics to children undergoing tonsillectomy. (2) Clinicians must not administer or prescribe codeine, or any medication containing codeine, after tonsillectomy in children younger than 12 years. The policy level for the recommendation about documenting recurrent throat infection was an option: (1) Clinicians may recommend tonsillectomy for recurrent throat infection with a frequency of at least 7 episodes in the past year, at least 5 episodes per year for 2 years, or at least 3 episodes per year for 3 years with documentation in the medical record for each episode of sore throat and ≥1 of the following: temperature >38.3°C (101°F), cervical adenopathy, tonsillar exudate, or positive test for group A beta-hemolytic streptococcus. Differences from Prior Guideline (1) Incorporating new evidence profiles to include the role of patient preferences, confidence in the evidence, differences of opinion, quality improvement opportunities, and any exclusion to which the action statement does not apply. (2) There were 1 new clinical practice guideline, 26 new systematic reviews, and 13 new randomized controlled trials included in the current guideline update. (3) Inclusion of 2 consumer advocates on the guideline update group. (4) Changes to 5 KASs from the original guideline: KAS 1 (Watchful waiting for recurrent throat infection), KAS 3 (Tonsillectomy for recurrent infection with modifying factors), KAS 4 (Tonsillectomy for obstructive sleep-disordered breathing), KAS 9 (Perioperative pain counseling), and KAS 10 (Perioperative antibiotics). (5) Seven new KASs: KAS 5 (Indications for polysomnography), KAS 6 (Additional recommendations for polysomnography), KAS 7 (Tonsillectomy for obstructive sleep apnea), KAS 12 (Inpatient monitoring for children after tonsillectomy), KAS 13 (Postoperative ibuprofen and acetaminophen), KAS 14 (Postoperative codeine), and KAS 15a (Outcome assessment for bleeding). (6) Addition of an algorithm outlining KASs. (7) Enhanced emphasis on patient and/or caregiver education and shared decision making.

Objective This update of a 2011 guideline developed by the American Academy of Otolaryngology–Head and Neck Surgery Foundation provides evidence-based recommendations on the pre-, intra-, and postoperative care and management of children 1 to 18 years of age under consideration for tonsillectomy. Tonsillectomy is defined as a surgical procedure performed with or without adenoidectomy that completely removes the tonsil, including its capsule, by dissecting the peritonsillar space between the tonsil capsule and the muscular wall. Tonsillectomy is one of the most common surgical procedures in the United States, with 289,000 ambulatory procedures performed annually in children <15 years of age, based on the most recent published data. This guideline is intended for all clinicians in any setting who interact with children who may be candidates for tonsillectomy. Purpose The purpose of this multidisciplinary guideline is to identify quality improvement opportunities in managing children under consideration for tonsillectomy and to create explicit and actionable recommendations to implement these opportunities in clinical practice. Specifically, the goals are to educate clinicians, patients, and/or caregivers regarding the indications for tonsillectomy and the natural history of recurrent throat infections. Additional goals include the following: optimizing the perioperative management of children undergoing tonsillectomy, emphasizing the need for evaluation and intervention in special populations, improving the counseling and education of families who are considering tonsillectomy for their children, highlighting the management options for patients with modifying factors, and reducing inappropriate or unnecessary variations in care. Children aged 1 to 18 years under consideration for tonsillectomy are the target patient for the guideline. For this guideline update, the American Academy of Otolaryngology–Head and Neck Surgery Foundation selected a panel representing the fields of nursing, anesthesiology, consumers, family medicine, infectious disease, otolaryngology–head and neck surgery, pediatrics, and sleep medicine. Key Action Statements The guideline update group made strong recommendations for the following key action statements (KASs): (1) Clinicians should recommend watchful waiting for recurrent throat infection if there have been <7 episodes in the past year, <5 episodes per year in the past 2 years, or <3 episodes per year in the past 3 years. (2) Clinicians should administer a single intraoperative dose of intravenous dexamethasone to children undergoing tonsillectomy. (3) Clinicians should recommend ibuprofen, acetaminophen, or both for pain control after tonsillectomy. The guideline update group made recommendations for the following KASs: (1) Clinicians should assess the child with recurrent throat infection who does not meet criteria in KAS 2 for modifying factors that may nonetheless favor tonsillectomy, which may include but are not limited to multiple antibiotic allergies/intolerance, PFAPA (periodic fever, aphthous stomatitis, pharyngitis, and adenitis), or history of >1 peritonsillar abscess. (2) Clinicians should ask caregivers of children with obstructive sleep-disordered breathing and tonsillar hypertrophy about comorbid conditions that may improve after tonsillectomy, including growth retardation, poor school performance, enuresis, asthma, and behavioral problems. (3) Before performing tonsillectomy, the clinician should refer children with obstructive sleep-disordered breathing for polysomnography if they are <2 years of age or if they exhibit any of the following: obesity, Down syndrome, craniofacial abnormalities, neuromuscular disorders, sickle cell disease, or mucopolysaccharidoses. (4) The clinician should advocate for polysomnography prior to tonsillectomy for obstructive sleep-disordered breathing in children without any of the comorbidities listed in KAS 5 for whom the need for tonsillectomy is uncertain or when there is discordance between the physical examination and the reported severity of obstructive sleep-disordered breathing. (5) Clinicians should recommend tonsillectomy for children with obstructive sleep apnea documented by overnight polysomnography. (6) Clinicians should counsel patients and caregivers and explain that obstructive sleep-disordered breathing may persist or recur after tonsillectomy and may require further management. (7) The clinician should counsel patients and caregivers regarding the importance of managing posttonsillectomy pain as part of the perioperative education process and should reinforce this counseling at the time of surgery with reminders about the need to anticipate, reassess, and adequately treat pain after surgery. (8) Clinicians should arrange for overnight, inpatient monitoring of children after tonsillectomy if they are <3 years old or have severe obstructive sleep apnea (apnea-hypopnea index ≥10 obstructive events/hour, oxygen saturation nadir <80%, or both). (9) Clinicians should follow up with patients and/or caregivers after tonsillectomy and document in the medical record the presence or absence of bleeding within 24 hours of surgery (primary bleeding) and bleeding occurring later than 24 hours after surgery (secondary bleeding). (10) Clinicians should determine their rate of primary and secondary posttonsillectomy bleeding at least annually. The guideline update group made a strong recommendation against 2 actions: (1) Clinicians should not administer or prescribe perioperative antibiotics to children undergoing tonsillectomy. (2) Clinicians must not administer or prescribe codeine, or any medication containing codeine, after tonsillectomy in children younger than 12 years. The policy level for the recommendation about documenting recurrent throat infection was an option: (1) Clinicians may recommend tonsillectomy for recurrent throat infection with a frequency of at least 7 episodes in the past year, at least 5 episodes per year for 2 years, or at least 3 episodes per year for 3 years with documentation in the medical record for each episode of sore throat and ≥1 of the following: temperature >38.3°C (101°F), cervical adenopathy, tonsillar exudate, or positive test for group A beta-hemolytic streptococcus. Differences from Prior Guideline Incorporating new evidence profiles to include the role of patient preferences, confidence in the evidence, differences of opinion, quality improvement opportunities, and any exclusion to which the action statement does not apply. There were 1 new clinical practice guideline, 26 new systematic reviews, and 13 new randomized controlled trials included in the current guideline update. Inclusion of 2 consumer advocates on the guideline update group. Changes to 5 KASs from the original guideline: KAS 1 (Watchful waiting for recurrent throat infection), KAS 3 (Tonsillectomy for recurrent infection with modifying factors), KAS 4 (Tonsillectomy for obstructive sleep-disordered breathing), KAS 9 (Perioperative pain counseling), and KAS 10 (Perioperative antibiotics). Seven new KASs: KAS 5 (Indications for polysomnography), KAS 6 (Additional recommendations for polysomnography), KAS 7 (Tonsillectomy for obstructive sleep apnea), KAS 12 (Inpatient monitoring for children after tonsillectomy), KAS 13 (Postoperative ibuprofen and acetaminophen), KAS 14 (Postoperative codeine), and KAS 15a (Outcome assessment for bleeding). Addition of an algorithm outlining KASs. Enhanced emphasis on patient and/or caregiver education and shared decision making.

An instance of the multi-sender index coding problem consists of a set of senders collectively having all the messages demanded by a set of receivers. Each receiver knows a subset of messages a priori, called its side information, and receives all the transmissions from all the senders. Each sender avails the knowledge of the side information and the demands of all the receivers, and the knowledge of messages available with other senders, to broadcast coded messages called an index code. Transmissions of each sender are orthogonal in time with those of others. The objective is to minimize the total number of coded transmissions, such that each receiver can decode its demands. The multi-sender index coding problem is related to many problems of practical interest such as multi-source satellite communication, multi-source coded caching, coded cooperative data exchange and distributed computing among many others. The multi-sender index coding problem is NP-hard and the optimal codelength is known only for a few special classes of the problem. In a prior work (V. K. Gummadi, A. Choudhary, and P. Krishnan, \Index Coding : Rank-Invariant Extensions," arXiv preprint arXiv:1704.00687 [cs.IT], 3 Apr, 2017), some classes of single-sender index coding problems called rank invariant extensions were identified. Optimal scalar linear codes of any problem belonging to this class were obtained using those of a particular subproblem. In another line of research ( C. Thapa, L. Ong, S. J. Johnson, and M. Li, \Structural Characteristics of Two-Sender Index Coding," Entropy 21, no. 6, 615, 2019), optimal codes for some classes of the two-sender index coding problem were constructed using those of its subproblems. In this work, we identify some classes of single-sender index coding problems, and construct optimal scalar linear codes for the same using scalar linear codes of some subproblems. All the scalar linear codes need not be optimal for the constructed codes to be scalar linear optimal. This result generalizes the notion of rank invariant extensions and solves many classes of single-sender problems for optimal scalar linear codes. This result is also applied to construct optimal scalar linear codes for some classes of the two-sender index coding problem. Moreover, all unsolved problems belonging to a fundamental class of the two-sender problem are solved for optimal broadcast rates. Optimal linear codelength of error correcting index codes are established for some classes of single-sender problems and multisender problems using related parameters of some subproblems. Weakly secure index codes are constructed for some classes of the two-sender problem using those of its single-sender problems. Many results presented in this thesis can be extended to solve many multi-sender index coding problems. We first introduce the notion of joint extensions of any finite number of single sender groupcast index coding problems which generalizes the notion of rank invariant extensions. A class of joint extensions is identi fied, where optimal scalar linear codes are constructed using those of the subproblems. This result is used to construct optimal scalar linear codes for some classes of the two-sender groupcast index coding problem. Another class of joint extensions referred to as base induced joint extensions is identi fied, where a problem called the base problem dictates the relations among the subproblems in the jointly extended problem. We present an algorithm to construct scalar linear codes using those of the subproblems, for any base induced joint extension. We observe that the algorithm constructs optimal scalar linear codes for a particular class of base induced joint extensions. An index coding problem is said to be unicast if each message is demanded by only one receiver. We establish the optimal broadcast rate (total number of transmitted bits per message bit as message length tends to in finity) for all the unsolved problems belonging to a fundamental class of the two-sender unicast index coding problem introduced by Thapa et al. Related code constructions with finite length messages make use of optimal codes of the single-sender subproblems. The established optimal broadcast rates take into account both linear and nonlinear encoding schemes at the senders. The proof techniques employed are used to construct optimal linear codes for some classes of multi-sender unicast problems We then consider the problem of multi-sender groupcast index coding with error correction, where any receiver receives atmost a speci fied number of coded symbols erroneously. A parameter called the generalized independence number is established for base induced joint extensions and rank invariant extensions in terms of those of its subproblems. Using this result, optimal length of linear error correcting index codes are provided for some classes of the single-sender index coding problem. These results are used to construct optimal linear error correcting index codes for some classes of the multi-sender groupcast index coding problem. We then consider the problem of index coding with weak security which consists of an adversary having some messages as side information. An index code is said to be weakly secure if the adversary can not obtain any information about any single message which it does not have using the index code and its side information. We show that the codes constructed for some classes of the two-sender groupcast index coding problem, using weakly secure codes of single-sender subproblems are also weakly secure. A subclass of the two-sender groupcast problem is identi fied where the constructed codes are optimal. Finally, we consider the problem of broadcasting with noisy side information introduced in S. Ghosh, and L. Natarajan, \Linear codes for broadcasting with noisy side information," arXiv preprint arXiv:1801.02868v1 [cs.IT], 9 Jan, 2018. Any wireless channel experiences outage due to fading. Some receivers erroneously decode their demanded messages during the fi rst round of transmissions from a sender. The sender can now avail the knowledge of the maximum number of erroneously decoded messages at any receiver (known as its error threshold), to reduce the number of retransmissions. The receivers make use of the erroneously decoded messages as side information (which is thus noisy) to decode their demands in the retransmission phase. We consider the problem with different error thresholds at different receivers and generalize a field-size independent encoding scheme given for the case with the same error threshold at every receiver. We show that this scheme saves at least two transmissions for a particular class of the problem with different error thresholds. We then extend the equivalence established between the problem of broadcasting with noisy side information and the index coding problem to the case with different error thresholds. We use this result to obtain optimal linear codes for a special class of the problem.

Research Doctorate - Doctor of Philosophy (PhD)
In index coding, a sender broadcasts messages through a noiseless broadcast channel to multiple receivers, each possessing a subset of the messages a priori, known as side-information. The sender knows the side-information available at each receiver, and uses that information to broadcasts coded messages, called an index code, such that all receivers can decode their requested messages using their side-information and the received index code. The aim is to find an index code with the minimum information bits per received message bits, called the optimal broadcast rate. The index-coding problem is applicable in many real-world contexts such as content-distribution networks (e.g., a network providing video-on-demand services), satellite communications, and distributed storage (e.g., data centers). In this dissertation, we investigate unicast-index-coding (UIC) problems, where each message is requested by only one receiver, each receiver requests only one message and each receiver has a subset of messages except its requested messages in side-information. These problems can be modeled by directed graphs. We apply graph-based approaches to the two types of the UIC problems based on the number of senders, namely single-sender unicast-index-coding (SSUIC) and two-sender unicast-index-coding (TSUIC) problems. In SSUIC, we propose a new scheme, called interlinked-cycle-cover (ICC) scheme, that exploits interlinked-cycle (IC) structures (a form of overlapping cycles that generalizes cliques and cycles) in directed graphs. This scheme provides an upper bound on the optimal broadcast rate for any SSUIC instance, and construct a linear index code. We prove that the ICC scheme is optimal over all linear and non-linear index codes for a class of infinitely many digraphs. We show that the ICC scheme can outperform existing schemes for some SSUIC instances with six receivers. Further, we extend the IC structures and the scheme. We find that the ICC scheme (including the extended scheme) provides the optimal broadcast rates for all message alphabet sizes for all SSUIC instances up to five receivers except eight problems. In TSUIC, firstly, we extend the existing SSUIC graph-based schemes, namely clique-cover, cycle-cover, and local-chromatic-number schemes to TSUIC. Then we investigate the TSUIC problems by its structural characterization. By considering three sub-problems of a TSUIC problem based on whether the receivers' requests are available at only one sender or both senders, we formulate its optimal broadcast rate as a function of the optimal broadcast rates of its three sub-problems. Furthermore, for this formulation, we extend the notion of confusion graphs and graph coloring to TSUIC. We characterize a class of infinitely many TSUIC instances where a certain type of side-information can be removed without affecting their optimal broadcast rates.