In PAC learning, error tolerance refers to the ability of an algorithm to learn when the examples received have been corrupted in some way. In fact, this is a very common and important issue since in many applications it is not possible to access noise-free data. Noise can interfere with the learning process at different levels: the algorithm may receive data that have been occasionally mislabeled, or the inputs may have some false information, or the classification of the examples may have been maliciously adulterated. == Notation and the Valiant learning model == In the following, let X {\displaystyle X} be our n {\displaystyle n} -dimensional input space. Let H {\displaystyle {\mathcal {H}}} be a class of functions that we wish to use in order to learn a { 0 , 1 } {\displaystyle \{0,1\}} -valued target function f {\displaystyle f} defined over X {\displaystyle X} . Let D {\displaystyle {\mathcal {D}}} be the distribution of the inputs over X {\displaystyle X} . The goal of a learning algorithm A {\displaystyle {\mathcal {A}}} is to choose the best function h ∈ H {\displaystyle h\in {\mathcal {H}}} such that it minimizes e r r o r ( h ) = P x ∼ D ( h ( x ) ≠ f ( x ) ) {\displaystyle error(h)=P_{x\sim {\mathcal {D}}}(h(x)\neq f(x))} . Let us suppose we have a function s i z e ( f ) {\displaystyle size(f)} that can measure the complexity of f {\displaystyle f} . Let Oracle ( x ) {\displaystyle {\text{Oracle}}(x)} be an oracle that, whenever called, returns an example x {\displaystyle x} and its correct label f ( x ) {\displaystyle f(x)} . When no noise corrupts the data, we can define learning in the Valiant setting: Definition: We say that f {\displaystyle f} is efficiently learnable using H {\displaystyle {\mathcal {H}}} in the Valiant setting if there exists a learning algorithm A {\displaystyle {\mathcal {A}}} that has access to Oracle ( x ) {\displaystyle {\text{Oracle}}(x)} and a polynomial p ( ⋅ , ⋅ , ⋅ , ⋅ ) {\displaystyle p(\cdot ,\cdot ,\cdot ,\cdot )} such that for any 0 < ε ≤ 1 {\displaystyle 0<\varepsilon \leq 1} and 0 < δ ≤ 1 {\displaystyle 0<\delta \leq 1} it outputs, in a number of calls to the oracle bounded by p ( 1 ε , 1 δ , n , size ( f ) ) {\displaystyle p\left({\frac {1}{\varepsilon }},{\frac {1}{\delta }},n,{\text{size}}(f)\right)} , a function h ∈ H {\displaystyle h\in {\mathcal {H}}} that satisfies with probability at least 1 − δ {\displaystyle 1-\delta } the condition error ( h ) ≤ ε {\displaystyle {\text{error}}(h)\leq \varepsilon } . In the following we will define learnability of f {\displaystyle f} when data have suffered some modification. == Classification noise == In the classification noise model a noise rate 0 ≤ η < 1 2 {\displaystyle 0\leq \eta <{\frac {1}{2}}} is introduced. Then, instead of Oracle ( x ) {\displaystyle {\text{Oracle}}(x)} that returns always the correct label of example x {\displaystyle x} , algorithm A {\displaystyle {\mathcal {A}}} can only call a faulty oracle Oracle ( x , η ) {\displaystyle {\text{Oracle}}(x,\eta )} that will flip the label of x {\displaystyle x} with probability η {\displaystyle \eta } . As in the Valiant case, the goal of a learning algorithm A {\displaystyle {\mathcal {A}}} is to choose the best function h ∈ H {\displaystyle h\in {\mathcal {H}}} such that it minimizes e r r o r ( h ) = P x ∼ D ( h ( x ) ≠ f ( x ) ) {\displaystyle error(h)=P_{x\sim {\mathcal {D}}}(h(x)\neq f(x))} . In applications it is difficult to have access to the real value of η {\displaystyle \eta } , but we assume we have access to its upperbound η B {\displaystyle \eta _{B}} . Note that if we allow the noise rate to be 1 / 2 {\displaystyle 1/2} , then learning becomes impossible in any amount of computation time, because every label conveys no information about the target function. Definition: We say that f {\displaystyle f} is efficiently learnable using H {\displaystyle {\mathcal {H}}} in the classification noise model if there exists a learning algorithm A {\displaystyle {\mathcal {A}}} that has access to Oracle ( x , η ) {\displaystyle {\text{Oracle}}(x,\eta )} and a polynomial p ( ⋅ , ⋅ , ⋅ , ⋅ ) {\displaystyle p(\cdot ,\cdot ,\cdot ,\cdot )} such that for any 0 ≤ η ≤ 1 2 {\displaystyle 0\leq \eta \leq {\frac {1}{2}}} , 0 ≤ ε ≤ 1 {\displaystyle 0\leq \varepsilon \leq 1} and 0 ≤ δ ≤ 1 {\displaystyle 0\leq \delta \leq 1} it outputs, in a number of calls to the oracle bounded by p ( 1 1 − 2 η B , 1 ε , 1 δ , n , s i z e ( f ) ) {\displaystyle p\left({\frac {1}{1-2\eta _{B}}},{\frac {1}{\varepsilon }},{\frac {1}{\delta }},n,size(f)\right)} , a function h ∈ H {\displaystyle h\in {\mathcal {H}}} that satisfies with probability at least 1 − δ {\displaystyle 1-\delta } the condition e r r o r ( h ) ≤ ε {\displaystyle error(h)\leq \varepsilon } . == Statistical query learning == Statistical Query Learning is a kind of active learning problem in which the learning algorithm A {\displaystyle {\mathcal {A}}} can decide if to request information about the likelihood P f ( x ) {\displaystyle P_{f(x)}} that a function f {\displaystyle f} correctly labels example x {\displaystyle x} , and receives an answer accurate within a tolerance α {\displaystyle \alpha } . Formally, whenever the learning algorithm A {\displaystyle {\mathcal {A}}} calls the oracle Oracle ( x , α ) {\displaystyle {\text{Oracle}}(x,\alpha )} , it receives as feedback probability Q f ( x ) {\displaystyle Q_{f(x)}} , such that Q f ( x ) − α ≤ P f ( x ) ≤ Q f ( x ) + α {\displaystyle Q_{f(x)}-\alpha \leq P_{f(x)}\leq Q_{f(x)}+\alpha } . Definition: We say that f {\displaystyle f} is efficiently learnable using H {\displaystyle {\mathcal {H}}} in the statistical query learning model if there exists a learning algorithm A {\displaystyle {\mathcal {A}}} that has access to Oracle ( x , α ) {\displaystyle {\text{Oracle}}(x,\alpha )} and polynomials p ( ⋅ , ⋅ , ⋅ ) {\displaystyle p(\cdot ,\cdot ,\cdot )} , q ( ⋅ , ⋅ , ⋅ ) {\displaystyle q(\cdot ,\cdot ,\cdot )} , and r ( ⋅ , ⋅ , ⋅ ) {\displaystyle r(\cdot ,\cdot ,\cdot )} such that for any 0 < ε ≤ 1 {\displaystyle 0<\varepsilon \leq 1} the following hold: Oracle ( x , α ) {\displaystyle {\text{Oracle}}(x,\alpha )} can evaluate P f ( x ) {\displaystyle P_{f(x)}} in time q ( 1 ε , n , s i z e ( f ) ) {\displaystyle q\left({\frac {1}{\varepsilon }},n,size(f)\right)} ; 1 α {\displaystyle {\frac {1}{\alpha }}} is bounded by r ( 1 ε , n , s i z e ( f ) ) {\displaystyle r\left({\frac {1}{\varepsilon }},n,size(f)\right)} A {\displaystyle {\mathcal {A}}} outputs a model h {\displaystyle h} such that e r r ( h ) < ε {\displaystyle err(h)<\varepsilon } , in a number of calls to the oracle bounded by p ( 1 ε , n , s i z e ( f ) ) {\displaystyle p\left({\frac {1}{\varepsilon }},n,size(f)\right)} . Note that the confidence parameter δ {\displaystyle \delta } does not appear in the definition of learning. This is because the main purpose of δ {\displaystyle \delta } is to allow the learning algorithm a small probability of failure due to an unrepresentative sample. Since now Oracle ( x , α ) {\displaystyle {\text{Oracle}}(x,\alpha )} always guarantees to meet the approximation criterion Q f ( x ) − α ≤ P f ( x ) ≤ Q f ( x ) + α {\displaystyle Q_{f(x)}-\alpha \leq P_{f(x)}\leq Q_{f(x)}+\alpha } , the failure probability is no longer needed. The statistical query model is strictly weaker than the PAC model: any efficiently SQ-learnable class is efficiently PAC learnable in the presence of classification noise, but there exist efficient PAC-learnable problems such as parity that are not efficiently SQ-learnable. == Malicious classification == In the malicious classification model an adversary generates errors to foil the learning algorithm. This setting describes situations of error burst, which may occur when for a limited time transmission equipment malfunctions repeatedly. Formally, algorithm A {\displaystyle {\mathcal {A}}} calls an oracle Oracle ( x , β ) {\displaystyle {\text{Oracle}}(x,\beta )} that returns a correctly labeled example x {\displaystyle x} drawn, as usual, from distribution D {\displaystyle {\mathcal {D}}} over the input space with probability 1 − β {\displaystyle 1-\beta } , but it returns with probability β {\displaystyle \beta } an example drawn from a distribution that is not related to D {\displaystyle {\mathcal {D}}} . Moreover, this maliciously chosen example may strategically selected by an adversary who has knowledge of f {\displaystyle f} , β {\displaystyle \beta } , D {\displaystyle {\mathcal {D}}} , or the current progress of the learning algorithm. Definition: Given a bound β B < 1 2 {\displaystyle \beta _{B}<{\frac {1}{2}}} for 0 ≤ β < 1 2 {\displaystyle 0\leq \beta <{\frac {1}{2}}} , we say that f {\displaystyle f} is efficiently learnable using H {\displaystyle {\mathcal {H}}} in the malicious classification model, if there exist a learning algorithm A {\displaystyle {\mathcal {A}}} that has access to Oracle ( x , β ) {\displaystyle {\text{Oracle}}(x,\beta )}
Alexander Y. Tetelbaum
Alexander Y. Tetelbaum (born August 16, 1948) is a Ukrainian American computer scientist, inventor, and academic who has contributed to electronic design automation (EDA) and artificial intelligence (AI) since the late 1960s; and holds 46 U.S. patents in EDA and related fields. Tetelbaum is the founding president of International Solomon University, the first Jewish university in Ukraine, established during a period of renewed efforts to address antisemitism in Ukraine. == Early life and education == He graduated from a Kyiv mathematical high school with a silver medal in 1966. Tetelbaum enrolled at the Kyiv Polytechnic Institute (KPI), now National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" in 1966, graduating in 1972 with an MS in Electronics with honors. He earned his PhD in Electrical and Computer Engineering from KPI in 1975, with a dissertation on electronic design automation, and his Doctor of Engineering Science in 1986. == Academic career == Tetelbaum began his academic career at KPI in 1973 as a junior scientist, becoming a professor in the Computer and Electrical Engineering Department in 1980. Later, he founded and served as president of International Solomon University in Kyiv from 1991 to 1996, the first Jewish university in Ukraine. The university became a major academic center for computer science and Jewish studies in the post-Soviet era. He was a visiting and adjunct professor at Michigan State University from 1993 to 1996. == Professional career == Tetelbaum worked as an engineer at the Kiev Institute of Cybernetics from 1972 to 1973, and later, he led the Design Automation Lab at Kyiv Polytechnic Institute from 1975 to 1987. In the United States, he served as EDA manager at Silicon Graphics Corporation from 1996 to 1998 and principal engineer at LSI Corporation from 1998 to 2012. He founded and served as CEO of Abelite Design Automation, Inc., from 2012 to 2022. == Contributions in computer science == Tetelbaum has contributed to electronic design automation (EDA) and artificial intelligence (AI) since the 1960s. His early work included methods for EDA, particularly physical design automation and mathematical optimization; and he developed force-directed placement and topological routing methods. Tetelbaum generalized Rent's rule for hierarchical systems and large blocks, proposing a graph-based framework that extends applicability to arbitrary partition sizes with improved accuracy. Additional IEEE and related conference contributions from the mid-1990s include: "Path Search for Complicated Function", 1995 IEEE International Symposium on Circuits and Systems "A Performance-driven Placement Approach of Standard Cells" (International Conference on Intelligent Systems, 1995) "Framework of a New Methodology for Behavioral to Physical Design Linkage" (38th Midwest Symposium on Circuits and Systems, 1996) Statistical timing design and variations Test Methodologies These and other works and patents contributed to timing-driven placement, crosstalk reduction, clock tree synthesis, and interconnect optimization in VLSI design. == Patents == Tetelbaum holds 46 U.S. patents in EDA and related fields. Notable examples include: For the full list of patents, see Justia Patents or Google Patents. == Publications == === Early publications in the Soviet Union === Before the appearance of American books on electronic design automation (EDA), Tetelbaum published several scientific books and monographs on the subject in Russian/Ukrainian. Electronic Design Automation, Kiev: Znanie Publisher, 1975. Planar Design of Electronic Circuits, Kiev: Znanie Publisher, 1977. Formal Design of Computer Systems, Moscow: Sovetskoe Radio, 1979. CAD of Electronic Equipment: Topological Approach, Kiev: Vyssha Shkola, 1980; 2nd ed. 1981. Automated Design of Electronic Circuits (1981) CAD of VLSI Circuits, Kiev: Vyssha Shkola, 1983. Topological Algorithms of Multilayer Printed Circuit Boards Routing, Moscow: Radio i Svyaz, 1983. CAD of VLSI Circuits on Master Slice Chips, Moscow: Radio i Svyaz, 1988. Increasing the Effectiveness of CAD Systems, Kiev: UMKVO, 1991. === Scientific Monographs (English) === Minimum Number of Timing Signoff Corners (2022) Interviewing AI (2026) The AI Debate (2026) New Nostradamus Predictions: 2026: The Next Decade & Beyond (2035–2050+) (2026) For a consolidated record of Tetelbaum's publications, see Alexander Y. Tetelbaum, Wikidata Q4720205. === Other publications === Tetelbaum also published educational books on problem-solving methods: Yes-No Puzzles-Games Puzzle Games for Kids Solving Non-Standard Problems Solving Non-Standard Very Hard Problems Additionally, Tetelbaum published three thrillers: Omerta Operations Executive Director Eruption Yacht Finally, he published his memoir and an entertaining book: Unfinished Equations Artificially Intelligent Humor
European Grid Infrastructure
EGI (originally an initialism for European Grid Infrastructure) is a federation of computing and storage resource providers that deliver advanced computing and data analytics services for research and innovation. The Federation is governed by its participants represented in the EGI Council and coordinated by the EGI Foundation. As of 2024, the EGI Federation supports 160 scientific communities worldwide and over 95,000 users in their intensive data analysis. The most significant scientific communities supported by EGI in 2022 were Medical and Health Sciences, High Energy Physics, and Engineering and Technology. The EGI Federation provideds services through over 150 data centres, of which 25 are cloud sites, in 43 countries and 64 Research Infrastructures (4 of which are members of the Federation). == Name == Originally, EGI stood for European Grid Infrastructure. This reflected its focus on providing access to high-throughput computing resources across Europe using Grid computing techniques. However, as EGI's service offerings expanded beyond traditional grid computing, particularly with the incorporation of federated cloud services, the original meaning of the acronym became less accurate. To emphasise the broader scope of EGI's services and avoid any confusion associated with the outdated term "grid," it is recommended to refer to EGI simply as EGI. == Structure == === EGI Federation === The EGI Federation delivers a scalable digital research infrastructure (e-infrastructure), empowering tens of thousands of researchers across diverse scientific disciplines. Through the EGI Federation, researchers gain access to advanced computing and data analytics capabilities, including large-scale data analysis, while benefiting from the collaborative efforts of hundreds of service providers from both public and private sectors, consolidating resources from Europe and beyond. Overall, the EGI Federation offers a range of services, encompassing distributed high-throughput computing and cloud computing, storage and data management capabilities, co-development of new solutions, expert support, and comprehensive training opportunities. This ecosystem propels collaboration, scientific progress and innovation. === EGI Foundation === The EGI Foundation is the coordinating body of the EGI Federation. It was established in 2010 with headquarters in Amsterdam, Netherlands. The Foundation coordinates the research and innovation efforts of its members, spanning technical areas critical to data-intensive science, including large-scale data processing and analysis, distributed Artificial Intelligence/Machine Learning, federated Identity and access management and the application of digital twins for research. The day-to-day running of the EGI Foundation is supervised by the Executive Board. The board’s members work closely with the EGI Director on operational, technical and financial issues. The Executive Board’s members are appointed by the EGI Council for a two-year term. === EGI Council === The EGI Council is responsible for defining the strategic direction of the EGI Federation. The Council acts as the senior decision-making and supervisory authority of the EGI Foundation, with a mandate to define the strategic direction of the entire EGI ecosystem. === EGI Services === EGI offers a suite of services to support data-intensive research. These services include compute resources, orchestration tools, storage and data management solutions, training programmes, security and identity services, and applications. Compute resources encompass cloud compute, cloud container compute, high-throughput compute, and software distribution. Orchestration tools include the Workload Manager and infrastructure manager. Storage and data management solutions include online storage, data transfer, and DataHub. Training programmes cover FitSM, ISO 27001, and general training infrastructure. EGI Check-in and Secrets Store are key security and identity services, while applications such as Notebooks and Replay enhance research productivity. In addition to services for Research, EGI also provides services for Federation and Business. Services for Federation are designed to help resource providers and user communities collaborate and share resources. EGI also offers a range of services to support businesses in their digital transformation. Through the EGI Digital Innovation Hub (EGI DIH), companies can access advanced computing resources, networking, funding and training opportunities, collaborate with research institutions, and test solutions before investing. == History == In 2002, the first large-scale experimental facility was successfully demonstrated by the DataGrid project under the lead of CERN with tens of technical architects from the major High Energy Physics institutes in the world. For the first time, distributed computing was applied to data-intensive processing. It aimed at developing a large-scale computational grid to facilitate distributed data-intensive scientific computing across High Energy Physics, Earth Observation, and Biology science applications. On 28 February 2003, the first software release of LCG-MW was published. gLite, the Lightweight Middleware for Grid Computing and LCG, Large Hadron Collider Computing Grid, are the cornerstone of the Worldwide LHC Computing Grid, which expanded over time towards the EGI Federation. 2004 marks the year of the first pilot infrastructure, seeing the participation of CERN and data centres in the United Kingdom, Spain, Germany, the Netherlands, France, Canada, Russia, Bulgaria, the Asia-Pacific region and Switzerland. Over the years, the infrastructure has grown into a federation of 128 data centres and 25 cloud providers serving more than 95,000 users worldwide. In 2004, the first data processing tasks started being formally recorded in a central accounting system. The EGI Accounting Portal provides the accounting data for Compute, Storage and Data services gathered from the data centres of the EGI Federation. A few years later, in 2010, EGI was established as the coordinating body of the EGI Federation to build an integrated pan-European infrastructure to support European research communities primarily. In the same year, EGI launched the flagship project EGI Inspire. That project brought together European organisations to establish a sustainable European Grid Infrastructure for large-scale data analysis. The success of the project was due to the adoption of a distributed computing model to solve big data problems. Moreover, EGI-Inspire harmonised operational policies across its federation of affiliated data centres and cloud service providers worldwide, integrating e-infrastructures from 57 countries. The EGI Federation was the first to apply federation to cloud provisioning, opening a new avenue in large-scale interactive data analysis. In 2015, within EGI Engage, opening a new avenue in large-scale interactive data analysis. The EGI Federated Cloud is an IaaS-type cloud, incorporating academic and private clouds and virtualised resources built using open standards. Its development is driven by the needs of the scientific community, resulting in a novel research e-infrastructure that relies on well-established federated operational services, making EGI a dependable resource for scientific endeavours. In 2015, EGI, EUDAT, GÉANT, LIBER and OpenAIRE published a position paper on a 'European Open Science Cloud for Research'. With the EOSC-hub project in 2016, EGI started contributing in practice to shaping the services for the EOSC. The work continued with a series of projects, like EOSC Enhance, EOSC Life and EOSC Synergy. With EGI-ACE and its contribution to EOSC Future, EGI has continued developing the EOSC Core. In early 2024, EGI started providing services to the EOSC EU Node, and with EOSC Beyond it will provide new EOSC Core capabilities and pilot additional national and thematic nodes. In October 2024, EUDAT, GÉANT, OpenAIRE, PRACE and EGI signed a Memorandum of Understanding establishing the European e-Infrastructures Assembly. This collaboration will bolster the position and promote the services of e-Infrastructures, empowering researchers across Europe to drive innovation and advance scientific discovery.
Consumer relationship system
Consumer relationship systems (CRS) are specialized customer relationship management (CRM) software applications that are used to handle a company's dealings with its customers. Current consumer relationship systems integrate the software with telephone and call recording systems as well as with corporate systems for input and reporting. Customers can provide input from the company's website directly into the CRS. These systems are popular because they can deliver the 'voice of the consumer' that contributes to product quality improvement and that ultimately increases corporate profits. Consumer relationship systems that provide automated support as well as advanced systems may have artificial intelligence (AI) interfaces that can extract and analyse data collected, or handle basic questions and complaints. == History == The first CRS was developed in the 1980s. In 1981 Michael Wilke and Robert Thornton founded Wilke/Thornton, Inc in Columbus, Ohio, to develop new CRS software.
Locally recoverable code
Locally recoverable codes are a family of error correction codes that were introduced first by D. S. Papailiopoulos and A. G. Dimakis and have been widely studied in information theory due to their applications related to distributive and cloud storage systems. An [ n , k , d , r ] q {\displaystyle [n,k,d,r]_{q}} LRC is an [ n , k , d ] q {\displaystyle [n,k,d]_{q}} linear code such that there is a function f i {\displaystyle f_{i}} that takes as input i {\displaystyle i} and a set of r {\displaystyle r} other coordinates of a codeword c = ( c 1 , … , c n ) ∈ C {\displaystyle c=(c_{1},\ldots ,c_{n})\in C} different from c i {\displaystyle c_{i}} , and outputs c i {\displaystyle c_{i}} . == Overview == Erasure-correcting codes, or simply erasure codes, for distributed and cloud storage systems, are becoming more and more popular as a result of the present spike in demand for cloud computing and storage services. This has inspired researchers in the fields of information and coding theory to investigate new facets of codes that are specifically suited for use with storage systems. It is well-known that LRC is a code that needs only a limited set of other symbols to be accessed in order to restore every symbol in a codeword. This idea is very important for distributed and cloud storage systems since the most common error case is when one storage node fails (erasure). The main objective is to recover as much data as possible from the fewest additional storage nodes in order to restore the node. Hence, Locally Recoverable Codes are crucial for such systems. The following definition of the LRC follows from the description above: an [ n , k , r ] {\displaystyle [n,k,r]} -Locally Recoverable Code (LRC) of length n {\displaystyle n} is a code that produces an n {\displaystyle n} -symbol codeword from k {\displaystyle k} information symbols, and for any symbol of the codeword, there exist at most r {\displaystyle r} other symbols such that the value of the symbol can be recovered from them. The locality parameter satisfies 1 ≤ r ≤ k {\displaystyle 1\leq r\leq k} because the entire codeword can be found by accessing k {\displaystyle k} symbols other than the erased symbol. Furthermore, Locally Recoverable Codes, having the minimum distance d {\displaystyle d} , can recover d − 1 {\displaystyle d-1} erasures. == Definition == Let C {\displaystyle C} be a [ n , k , d ] q {\displaystyle [n,k,d]_{q}} linear code. For i ∈ { 1 , … , n } {\displaystyle i\in \{1,\ldots ,n\}} , let us denote by r i {\displaystyle r_{i}} the minimum number of other coordinates we have to look at to recover an erasure in coordinate i {\displaystyle i} . The number r i {\displaystyle r_{i}} is said to be the locality of the i {\displaystyle i} -th coordinate of the code. The locality of the code is defined as An [ n , k , d , r ] q {\displaystyle [n,k,d,r]_{q}} locally recoverable code (LRC) is an [ n , k , d ] q {\displaystyle [n,k,d]_{q}} linear code C ∈ F q n {\displaystyle C\in \mathbb {F} _{q}^{n}} with locality r {\displaystyle r} . Let C {\displaystyle C} be an [ n , k , d ] q {\displaystyle [n,k,d]_{q}} -locally recoverable code. Then an erased component can be recovered linearly, i.e. for every i ∈ { 1 , … , n } {\displaystyle i\in \{1,\ldots ,n\}} , the space of linear equations of the code contains elements of the form x i = f ( x i 1 , … , x i r ) {\displaystyle x_{i}=f(x_{i_{1}},\ldots ,x_{i_{r}})} , where i j ≠ i {\displaystyle i_{j}\neq i} . == Optimal locally recoverable codes == Theorem Let n = ( r + 1 ) s {\displaystyle n=(r+1)s} and let C {\displaystyle C} be an [ n , k , d ] q {\displaystyle [n,k,d]_{q}} -locally recoverable code having s {\displaystyle s} disjoint locality sets of size r + 1 {\displaystyle r+1} . Then An [ n , k , d , r ] q {\displaystyle [n,k,d,r]_{q}} -LRC C {\displaystyle C} is said to be optimal if the minimum distance of C {\displaystyle C} satisfies == Tamo–Barg codes == Let f ∈ F q [ x ] {\displaystyle f\in \mathbb {F} _{q}[x]} be a polynomial and let ℓ {\displaystyle \ell } be a positive integer. Then f {\displaystyle f} is said to be ( r {\displaystyle r} , ℓ {\displaystyle \ell } )-good if • f {\displaystyle f} has degree r + 1 {\displaystyle r+1} , • there exist distinct subsets A 1 , … , A ℓ {\displaystyle A_{1},\ldots ,A_{\ell }} of F q {\displaystyle \mathbb {F} _{q}} such that – for any i ∈ { 1 , … , ℓ } {\displaystyle i\in \{1,\ldots ,\ell \}} , f ( A i ) = { t i } {\displaystyle f(A_{i})=\{t_{i}\}} for some t i ∈ F q {\displaystyle t_{i}\in \mathbb {F} _{q}} , i.e., f {\displaystyle f} is constant on A i {\displaystyle A_{i}} , – # A i = r + 1 {\displaystyle \#A_{i}=r+1} , – A i ∩ A j = ∅ {\displaystyle A_{i}\cap A_{j}=\varnothing } for any i ≠ j {\displaystyle i\neq j} . We say that { A 1 , … , A ℓ {\displaystyle A_{1},\ldots ,A_{\ell }} } is a splitting covering for f {\displaystyle f} . === Tamo–Barg construction === The Tamo–Barg construction utilizes good polynomials. • Suppose that a ( r , ℓ ) {\displaystyle (r,\ell )} -good polynomial f ( x ) {\displaystyle f(x)} over F q {\displaystyle \mathbb {F} _{q}} is given with splitting covering i ∈ { 1 , … , ℓ } {\displaystyle i\in \{1,\ldots ,\ell \}} . • Let s ≤ ℓ − 1 {\displaystyle s\leq \ell -1} be a positive integer. • Consider the following F q {\displaystyle \mathbb {F} _{q}} -vector space of polynomials V = { ∑ i = 0 s g i ( x ) f ( x ) i : deg ( g i ( x ) ) ≤ deg ( f ( x ) ) − 2 } . {\displaystyle V=\left\{\sum _{i=0}^{s}g_{i}(x)f(x)^{i}:\deg(g_{i}(x))\leq \deg(f(x))-2\right\}.} • Let T = ⋃ i = 1 ℓ A i {\textstyle T=\bigcup _{i=1}^{\ell }A_{i}} . • The code { ev T ( g ) : g ∈ V } {\displaystyle \{\operatorname {ev} _{T}(g):g\in V\}} is an ( ( r + 1 ) ℓ , ( s + 1 ) r , d , r ) {\displaystyle ((r+1)\ell ,(s+1)r,d,r)} -optimal locally coverable code, where ev T {\displaystyle \operatorname {ev} _{T}} denotes evaluation of g {\displaystyle g} at all points in the set T {\displaystyle T} . === Parameters of Tamo–Barg codes === • Length. The length is the number of evaluation points. Because the sets A i {\displaystyle A_{i}} are disjoint for i ∈ { 1 , … , ℓ } {\displaystyle i\in \{1,\ldots ,\ell \}} , the length of the code is | T | = ( r + 1 ) ℓ {\displaystyle |T|=(r+1)\ell } . • Dimension. The dimension of the code is ( s + 1 ) r {\displaystyle (s+1)r} , for s {\displaystyle s} ≤ ℓ − 1 {\displaystyle \ell -1} , as each g i {\displaystyle g_{i}} has degree at most deg ( f ( x ) ) − 2 {\displaystyle \deg(f(x))-2} , covering a vector space of dimension deg ( f ( x ) ) − 1 = r {\displaystyle \deg(f(x))-1=r} , and by the construction of V {\displaystyle V} , there are s + 1 {\displaystyle s+1} distinct g i {\displaystyle g_{i}} . • Distance. The distance is given by the fact that V ⊆ F q [ x ] ≤ k {\displaystyle V\subseteq \mathbb {F} _{q}[x]_{\leq k}} , where k = r + 1 − 2 + s ( r + 1 ) {\displaystyle k=r+1-2+s(r+1)} , and the obtained code is the Reed-Solomon code of degree at most k {\displaystyle k} , so the minimum distance equals ( r + 1 ) ℓ − ( ( r + 1 ) − 2 + s ( r + 1 ) ) {\displaystyle (r+1)\ell -((r+1)-2+s(r+1))} . • Locality. After the erasure of the single component, the evaluation at a i ∈ A i {\displaystyle a_{i}\in A_{i}} , where | A i | = r + 1 {\displaystyle |A_{i}|=r+1} , is unknown, but the evaluations for all other a ∈ A i {\displaystyle a\in A_{i}} are known, so at most r {\displaystyle r} evaluations are needed to uniquely determine the erased component, which gives us the locality of r {\displaystyle r} . To see this, g {\displaystyle g} restricted to A j {\displaystyle A_{j}} can be described by a polynomial h {\displaystyle h} of degree at most deg ( f ( x ) ) − 2 = r + 1 − 2 = r − 1 {\displaystyle \deg(f(x))-2=r+1-2=r-1} thanks to the form of the elements in V {\displaystyle V} (i.e., thanks to the fact that f {\displaystyle f} is constant on A j {\displaystyle A_{j}} , and the g i {\displaystyle g_{i}} 's have degree at most deg ( f ( x ) ) − 2 {\displaystyle \deg(f(x))-2} ). On the other hand | A j ∖ { a j } | = r {\displaystyle |A_{j}\backslash \{a_{j}\}|=r} , and r {\displaystyle r} evaluations uniquely determine a polynomial of degree r − 1 {\displaystyle r-1} . Therefore h {\displaystyle h} can be constructed and evaluated at a j {\displaystyle a_{j}} to recover g ( a j ) {\displaystyle g(a_{j})} . === Example of Tamo–Barg construction === We will use x 5 ∈ F 41 [ x ] {\displaystyle x^{5}\in \mathbb {F} _{41}[x]} to construct [ 15 , 8 , 6 , 4 ] {\displaystyle [15,8,6,4]} -LRC. Notice that the degree of this polynomial is 5, and it is constant on A i {\displaystyle A_{i}} for i ∈ { 1 , … , 8 } {\displaystyle i\in \{1,\ldots ,8\}} , where A 1 = { 1 , 10 , 16 , 18 , 37 } {\displaystyle A_{1}=\{1,10,16,18,37\}} , A 2 = 2 A 1 {\displaystyle A_{2}=2A_{1}} , A 3 = 3 A 1 {\displaystyle A_{3}=3A_{1}} , A 4 = 4 A 1 {\displaystyle A_{4}=4A_{1}} , A 5 = 5 A 1 {\displaystyle A_{5}=5A_{1}} , A 6 = 6 A 1 {\displaystyle A_{6}=6A_{1}}
Arattai
Arattai Messenger (or simply Arattai) is an encrypted messaging service for instant messaging, voice calls, and video calls, developed by Zoho Corporation. The name Arattai means "chat" or "conversation" in Tamil. The app was soft-launched in January 2021. The app saw a sharp surge in downloads in September 2025, partially fueled by endorsements from Indian government officials. However, the app dropped from the top rankings in October 2025. == History == Arattai was initially tested internally among Zoho employees before being released publicly in early 2021. The launch coincided with a surge in interest for privacy-focused and messaging services, triggered by concerns over WhatsApp's updated terms of service. In September 2025, Arattai experienced a major surge in adoption, with daily sign-ups reportedly increasing 100-fold, from around 3,000 to more than 350,000 in three days. The surge in downloads was attributed to Zoho products being promoted by Indian government officials as part of their Make in India push for homegrown alternatives to foreign‐owned apps, amid deteriorating India–US relations. The growth temporarily strained Zoho's infrastructure, prompting rapid scaling of servers and capacity expansion. During the same period, the app reached the top position in Apple's App Store charts for the "Social Networking" category in India. The app dropped from the top ranking in late October 2025. == Reception == At launch, Arattai was positioned as a potential domestic rival to WhatsApp in India, but analysts noted that it faced challenges with encryption, ecosystem, and network effect. Critics pointed to occasional sync delays.
Myrinet
Myrinet, ANSI/VITA 26-1998, is a high-speed local area networking system designed by the company Myricom to be used as an interconnect between multiple machines to form computer clusters. == Description == Myrinet was promoted as having lower protocol overhead than standards such as Ethernet, and therefore better throughput, less interference, and lower latency while using the host CPU. Although it can be used as a traditional networking system, Myrinet is often used directly by programs that "know" about it, thereby bypassing a call into the operating system. Earlier versions of Myrinet used a variety of media and connectors: Generation 2 used copper media with DC-37 (Myrinet-LAN, M2L- controllers and switches) or microribbon (Myrinet-SAN, M2M-) connectors. Generation 3 used copper media with HSSDC (Myrinet-Serial, M3S-) or microribbon (Myrinet-SAN, M3M-) connectors, or fiber with LC-connectors (Myrinet-Fiber, M3F-). The later versions of Myrinet physically consist of two fibre optic cables, upstream and downstream, connected to the host computers with a single connector. Machines are connected via low-overhead routers and switches, as opposed to connecting one machine directly to another. Myrinet includes a number of fault-tolerance features, mostly backed by the switches. These include flow control, error control, and "heartbeat" monitoring on every link. The "fourth-generation" Myrinet, called Myri-10G, supported a 10 Gbit/s data rate and can use 10 Gigabit Ethernet on PHY, the physical layer (cables, connectors, distances, signaling). Myri-10G started shipping at the end of 2005. Myrinet was approved in 1998 by the American National Standards Institute for use on the VMEbus as ANSI/VITA 26-1998. One of the earliest publications on Myrinet is a 1995 IEEE article. === Performance === Myrinet is a lightweight protocol with little overhead that allows it to operate with throughput close to the basic signaling speed of the physical layer. For supercomputing, the low latency of Myrinet is even more important than its throughput performance, since, according to Amdahl's law, a high-performance parallel system tends to be bottlenecked by its slowest sequential process, which in all but the most embarrassingly parallel supercomputer workloads is often the latency of message transmission across the network. === Deployment === According to Myricom, 141 (28.2%) of the June 2005 TOP500 supercomputers used Myrinet technology. In the November 2005 TOP500, the number of supercomputers using Myrinet was down to 101 computers, or 20.2%, in November 2006, 79 (15.8%), and by November 2007, 18 (3.6%), a long way behind gigabit Ethernet at 54% and InfiniBand at 24.2%. In the June 2014 TOP500 list, the number of supercomputers using Myrinet interconnect was 1 (0.2%). In November 2013, the assets of Myricom (including the Myrinet technology) were acquired by CSP Inc. In 2016, it was reported that Google had also offered to buy the company.