On Distributed Storage Codes

Distributed storage systems are studied. The interest in such system has become relatively wide due to the increasing amount of information needed to be stored in data centers or different kinds of cloud systems. There are many kinds of solutions for storing the information into distributed devices regarding the needs of the system designer. This thesis studies the questions of designing such storage systems and also fundamental limits of such systems. Namely, the subjects of interest of this thesis include heterogeneous distributed storage systems, distributed storage systems with the exact repair property, and locally repairable codes. For distributed storage systems with either functional or exact repair, capacity results are proved. In the case of locally repairable codes, the minimum distance is studied. Constructions for exact-repairing codes between minimum bandwidth regeneration (MBR) and minimum storage regeneration (MSR) points are given. These codes exceed the time-sharing line of the extremal points in many cases. Other properties of exact-regenerating codes are also studied. For the heterogeneous setup, the main result is that the capacity of such systems is always smaller than or equal to the capacity of a homogeneous system with symmetric repair with average node size and average repair bandwidth. A randomized construction for a locally repairable code with good minimum distance is given. It is shown that a random linear code of certain natural type has a good minimum distance with high probability. Other properties of locally repairable codes are also studied.Siirretty Doriast

Thuật toán xác định tính chất mã của ngôn ngữ chính quy

We present an extension of the test proposed by Sardinas and Patterson (1953) for deciding whether a set of words is a code. As a consequence, we obtain an effective algorithm that decides in ${\cal O}(k)$ time whether a given regular language is a code, where $k$ is the finite index of the syntactic congruence of this language.Trong bài báo này, chúng tôi trình bày một thuật toán mới mở rộng thuật toán Sardinas-Patterson xác định tính chất mã của một ngôn ngữ. Từ đó nhận được một thuật toán với độ phức tạp cỡ ${\cal O}(k)$ để nhận biết một ngôn ngữ chính quy cho trước là mã hay không, với $k$ là chỉ số hữu hạn của tương đẳng cú pháp thỏa ngôn ngữ đó

B-Codes and their relationship with alternative codes

Codes of bounded words (b-codes,  -codes), an extension of the notion of ordinary codes, have been first introduced and considered by P. T. Huy et al. in 2009. In this note, we consider some new subclasses of b-codes. In particular, characteristic properties of such codes are established. Also, relationship between b-codes, alternative codes and their subclasses are considered

Geometry and Arithmetic around Hypergeometric Functions

[no abstract available

Kolme ekvivalenttia koodia autosegmentaalisille esitysmuodoille

Otsikossa esiintyvä "autosegmentaalinen esitysmuoto" viittaa fonologiseen esitysmuotoon, joka erottelee fonologiset piirteet kahdelle tai useammalle lineaariselle aikalinjalle. Vaikka sellainen erottelu on ideoitu fonologiassa, sillä on sovelluksia myös käännösten lineaarisessa kohdistamisessa ja äärellistilaisessa syntaksissa, jota projektini tutkii. Tutkin vastaavia esitysmuotoja dependenssijäsentämisen yhteydessä, jossa tavoittelen enemmän Linear Unit Grammar (Mauranen 2012) -lähestymistapaa kuin syvää syntaksipuuta. Proceeding volume: 12Kaksijakoisten verkkojen merkkijonokoodaus mahdollistaa sen, että toonifonologian autosegmentaalisessa kuvausessa käytettävä verkkojen toisinkirjoittaminen voidaan toteuttaa käyttäen olemassa olevia ja runsaasti optimoituja äärellistilaisia transduktorityökaluja (Yli-Jyrä 2013). Käsillä oleva työ kuvailee ankarasti tämän kooditeoreettisen lähestymistavan ja yleistää siihen liittyvän metodiikan kaikkiin sellaisiin kaksijakoisiin verkkoihin, joissa ei ole ristikkäisiä kaaria eikä järjestämättömiä solmuja. Työssä esitetään kolme bijektiivisesti toisiinsa suhteutettua koodia, joista kullakin on erityiset luonteenpiirteet ja vapaus rikkoa tai ilmaista nk. Obligatory Contour Principle -rajoite. Koodit ovat äärettömiä, äärellistilaisesti esitettyjä ja optimaalisia (tehokkaasti laskettavia, käännettäviä, paikallisesti ikonisia ja kompositionaalisia) Kornain (1995) määrittelmän mukaisesti. Nämä kolme koodia laajentavat koodaukseen perustuvaa lähestymistapaa visualisoinnilla, yleisyydellä ja joustavuudella ja ne tekevät koodatuista graafeista vahvan ehdokkaan silloin, kun autosegmentaalisen fonologian formaali semantiikka tai risteämättömät kohdistusrelaatiot toteutetaan säännöllisen kieliopin puitteissa.A string encoding for a subclass of bipartite graphs enables graph rewriting used in autosegmental descriptions of tone phonology via existing and highly optimized finite- state transducer toolkits (Yli-Jyrä 2013). The current work offers a rigorous treatment of this code-theoretic approach, generalizing the methodology to all bipartite graphs having no crossing edges and un- ordered nodes. We present three bijectively related codes each of which exhibit unique characteristics while preserving the freedom to violate or express the OCP constraint. The codes are infinite, finite-state representable and optimal (efficiently computable, invertible, locally iconic, compositional) in the sense of Kornai (1995). They extend the encoding approach with visualisation, generality and flexibility and they make encoded graphs a strong candidate when the formal semantics of autosegmental phonology or non-crossing alignment relations are implemented within the con- fines of regular grammar.Peer reviewe