Theory Of Error-correcting Codes Research Articles

Digital System Design for Quantum Error Correction Codes.

Quantum computing is a computer development technology that uses quantum mechanics to perform the operations of data and information. It is an advanced technology, yet the quantum channel is used to transmit the quantum information which is sensitive to the environment interaction. Quantum error correction is a hybrid between quantum mechanics and the classical theory of error-correcting codes that are concerned with the fundamental problem of communication, and/or information storage, in the presence of noise. The interruption made by the interaction makes transmission error during the quantum channel qubit. Hence, a quantum error correction code is needed to protect the qubit from errors that can be caused by decoherence and other quantum noise. In this paper, the digital system design of the quantum error correction code is discussed. Three designs used qubit codes, and nine-qubit codes were explained. The systems were designed and configured for encoding and decoding nine-qubit error correction codes. For comparison, a modified circuit is also designed by adding Hadamard gates.

Generic BCH codes. Polynomial-norm error decoding

The classic Bose – Chaudhuri – Hocquenghem (BCH) codes is famous and well-studied part in the theory of error-correcting codes. Generalization of BCH codes allows us to expand the range of activities in the practical correction of errors. Some generic BCH codes are able to correct more errors than classic BCH code in one message block. So it is important to provide appropriate method of error correction. After our investigation it was found that polynomial-norm method is most convenient and effective for that task. The result of the study was a model of a polynomial-norm decoder for a generic BCH code at length 65.

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes.

Locally Testable and Locally Correctable Codes approaching the Gilbert-Varshamov Bound

One of the most important open problems in the theory of error-correcting codes is to determine the tradeoff between the rate $R$ and minimum distance $\delta $ of a binary code. The best known tradeoff is the Gilbert–Varshamov bound, and says that for every $\delta \in (0, 1/2)$ , there are codes with minimum distance $\delta $ and rate $R = {R_{\mathsf {GV}}}(\delta) > 0$ (for a certain simple function ${R_{\mathsf {GV}}}(\cdot)$ ). In this paper, we show that the Gilbert–Varshamov bound can be achieved by codes, which support local error-detection and error-correction algorithms. Specifically, we show the following results. 1) Local testing: for all $\delta \in (0,1/2)$ and all $R , there exist codes with length $n$ , rate $R$ , and minimum distance $\delta $ that are locally testable with $\mathsf {quasipolylog}(n)$ query complexity. 2) Local correction: for all $\epsilon > 0$ , for all $\delta sufficiently large, and all $R , there exist codes with length $n$ , rate $R$ , and minimum distance $\delta $ that are locally correctable from $({\delta }/{2}) - o(1)$ fraction errors with $O(n^{\epsilon })$ query complexity. Furthermore, these codes have an efficient randomized construction, and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity. Our results on locally correctable codes also immediately give locally decodable codes with the same parameters. Our local testing result is obtained by combining Thommesen’s random concatenation technique and the best known locally testable codes by Kopparty et al. Our local correction result, which is significantly more involved, also uses random concatenation, along with a number of further ideas: the Guruswami–Sudan–Indyk list decoding strategy for concatenated codes, Alon–Edmonds–Luby distance amplification, and the local list-decodability, local list-recoverability, and local testability of Reed–Muller codes. Curiously, our final local correction algorithms go via local list-decoding and local testing algorithms; this seems to be the first time local testability is used in the construction of a locally correctable code.

From Theory to Practice: G. David Forney, Jr. and the Innovation of Information Theory [Scanning our Past

In 2016, the IEEE conferred its Medal of Honor on G. David Forney, Jr., “for pioneering contributions to the theory of error-correcting codes and the development of reliable high-speed data communications.” The cited combination of theory and practice is rare: How does someone reduce a mathematical theory-in this case, Claude Shannon's information theory 1 -to practice?

Injective choosability of subcubic planar graphs with girth 6

An injective coloring of a graph G is an assignment of colors to the vertices of G so that any two vertices with a common neighbor have distinct colors. A graph G is injectively k-choosable if for any list assignment L, where |L(v)|≥k for all v∈V(G), G has an injective L-coloring. Injective colorings have applications in the theory of error-correcting codes and are closely related to other notions of colorability. In this paper, we show that subcubic planar graphs with girth at least 6 are injectively 5-choosable. This strengthens the result of Lužar, Škrekovski, and Tancer that subcubic planar graphs with girth at least 7 are injectively 5-colorable. Our result also improves several other results in particular cases.

On the roots and minimum rank distance of skew cyclic codes

Skew cyclic codes play the same role as cyclic codes in the theory of error-correcting codes for the rank metric. In this paper, we give descriptions of these codes by root spaces, cyclotomic spaces and idempotent generators. We prove that the lattice of skew cyclic codes is anti-isomorphic to the lattice of root spaces, study these two lattices and extend the rank-BCH bound on their minimum rank distance to rank-metric versions of the van Lint-Wilson's shift and Hartmann-Tzeng bounds. Finally, we study skew cyclic codes which are linear over the base field, proving that these codes include all Hamming-metric cyclic codes, giving then a new relation between these codes and rank-metric skew cyclic codes.

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness and randomness extraction. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes.

Algebraic algorithms for LWE problems

No abstract available.

BIBDs algorithm

In this paper, we explore the planar near-rings to build BIBDs (Balanced Incomplete Block Designs) which is an important tool in the field of the theory of error-correcting code. Our purpose is to supply a program which determines BIBDs and their incidence matrices from a finite field, what gives us the possibility of choosing the best BIBD among the BIBDs obtained according to the values of its parameters.

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, and pseudorandomness. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes.

CYCLIC CODES THROUGH B[X], $B[X;\frac{1}{kp}Z_{0}]$ AND $B[X;\frac{1}{p^{k}}Z_{0}]$: A COMPARISON

It is very well known that algebraic structures have valuable applications in the theory of error-correcting codes. Blake [Codes over certain rings, Inform. and Control 20 (1972) 396–404] has constructed cyclic codes over ℤm and in [Codes over integer residue rings, Inform. and Control 29 (1975), 295–300] derived parity check-matrices for these codes. In [Linear codes over finite rings, Tend. Math. Appl. Comput. 6(2) (2005) 207–217]. Andrade and Palazzo present a construction technique of cyclic, BCH, alternant, Goppa and Srivastava codes over a local finite ring B. However, in [Encoding through generalized polynomial codes, Comput. Appl. Math. 30(2) (2011) 1–18] and [Constructions of codes through semigroup ring [Formula: see text] and encoding, Comput. Math. Appl. 62 (2011) 1645–1654], Shah et al. extend this technique of constructing linear codes over a finite local ring B via monoid rings [Formula: see text], where p = 2 and k = 1, 2, respectively, instead of the polynomial ring B[X]. In this paper, we construct these codes through the monoid ring [Formula: see text], where p = 2 and k = 1, 2, 3. Moreover, we also strengthen and generalize the results of [Encoding through generalized polynomial codes, Comput. Appl. Math.30(2) (2011) 1–18] and [Constructions of codes through semigroup ring [Formula: see text]] and [Encoding, Comput. Math. Appl.62 (2011) 1645–1654] to the case of k = 3.

Chains of mean-field models

We consider a collection of Curie–Weiss (CW) spin systems, possibly with a random field,each of which is placed along the positions of a one-dimensional chain. The CW systemsare coupled together by a Kac-type interaction in the longitudinal direction of the chainand by an infinite-range interaction in the direction transverse to the chain. Ourmotivations for studying this model come from recent findings in the theory oferror-correcting codes based on spatially coupled graphs. We find that, althoughmuch simpler than the codes, the model studied here already displays similarbehavior. We are interested in the van der Waals curve in a regime where the sizeof each Curie–Weiss model tends to infinity, and the length of the chain andrange of the Kac interaction are large but finite. Below the critical temperature,and with appropriate boundary conditions, there appears a series of equilibriumstates representing kink-like interfaces between the two equilibrium states of theindividual system. The van der Waals curve oscillates periodically around theMaxwell plateau. These oscillations have a period inversely proportional to the chainlength and an amplitude exponentially small in the range of the interaction; inother words, the spinodal points of the chain model lie exponentially close to thephase transition threshold. The amplitude of the oscillations is closely related to aPeierls–Nabarro free energy barrier for the motion of the kink along the chain. Analogiesto similar phenomena and their possible algorithmic significance for graphicalmodels of interest in coding theory and theoretical computer science are pointedout.

Unruly codes with unruly distances raise (combinatorial) problems

Abstract For the 70-th birthday of Gyula Katona we offer him a list of open combinatorial problems raised by a general theory of error-correcting codes derived from Shannonʼs zero-error information theory.

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes.

Error-Correcting Codes and Phase Transitions

The theory of error-correcting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous curve in the space of parameters. The main goal of this paper is to relate the asymptotic bound to phase diagrams of quantum statistical mechanical systems. We first identify the code parameters with Hausdorff and von Neumann dimensions, by considering fractals consisting of infinite sequences of code words. We then construct operator algebras associated to individual codes. These are Toeplitz algebras with a time evolution for which the KMS state at critical temperature gives the Hausdorff measure on the corresponding fractal. We extend this construction to algebras associated to limit points of codes, with non-uniform multi-fractal measures, and to tensor products over varying parameters.

Provable Secured Hash Password Authentication

techniques such as secured socket layer (SSL) with client- side certificates are well known in the security research community, most commercial web sites rely on a relatively weak form of password authentication, the browser simply sends a plaintext password to a remote web server, often using SSL. Even when used over an encrypted connection, this form of password authentication is vulnerable to attack. In common password attacks, hackers exploit the fact that web users often use the same password at many different sites. This allows hackers to break into a low security site that simply stores username/passwords in the clear and use the retrieved passwords at a high security site. While password authentication could be abandoned in favor of hardware tokens or client certificates, both options are difficult to adopt because of the cost and inconvenience of hardware tokens and the overhead of managing client certificates. Recently, some collisions have been exposed for a variety of cryptographic hash functions including some of the most widely used today. Many other hash functions using similar constructions can however still be considered secure. Nevertheless, this has drawn attention on the need for new hash function designs. This work developed an improved secure hash function, whose security is directly related to the syndrome decoding problem from the theory of error-correcting codes. The proposal design and develop a user interface, and implementation of a browser extension, password hash, that strengthens web password authentication. Providing customized passwords, can reduce the threat of password attacks with no server changes and little or no change to the user experience. The proposed techniques are designed to transparently provide novice users with the benefits of password practices that are otherwise only feasible for security experts. Experimentation are done with Internet Explorer and Fire fox implementations and report the result of initial user. The hash is implemented using a Pseudo Random Function keyed by the password. Since the hash output is tailored to meet server password requirements, the resulting hashed password is handled normally at the server with no server modifications are required. This technique deters password phishing since the password received at a phishing site is not useful at any other domain. The cryptographic hash makes it difficult to compute hash(pwd,dom2) from hash(pwd,dom1) for any domain dom2 distinct from dom1. For the same reason, passwords gathered by breaking into a low security site are not useful at any other site. The hash attack is always exponential in terms of the length of the hash value. We also study the work-factor of this attack, along with other attacks from coding theory, for non asymptotic range, i.e. for practical values. Accordingly, we propose a few sets of parameters giving a good security and either a faster hashing or a shorter description for the function.

Overlapping pools for high-throughput targeted resequencing

Resequencing genomic DNA from pools of individuals is an effective strategy to detect new variants in targeted regions and compare them between cases and controls. There are numerous ways to assign individuals to the pools on which they are to be sequenced. The naïve, disjoint pooling scheme (many individuals to one pool) in predominant use today offers insight into allele frequencies, but does not offer the identity of an allele carrier. We present a framework for overlapping pool design, where each individual sample is resequenced in several pools (many individuals to many pools). Upon discovering a variant, the set of pools where this variant is observed reveals the identity of its carrier. We formalize the mathematical framework for such pool designs and list the requirements from such designs. We specifically address three practical concerns for pooled resequencing designs: (1) false-positives due to errors introduced during amplification and sequencing; (2) false-negatives due to undersampling particular alleles aggravated by nonuniform coverage; and consequently, (3) ambiguous identification of individual carriers in the presence of errors. We build on theory of error-correcting codes to design pools that overcome these pitfalls. We show that in practical parameters of resequencing studies, our designs guarantee high probability of unambiguous singleton carrier identification while maintaining the features of naïve pools in terms of sensitivity, specificity, and the ability to estimate allele frequencies. We demonstrate the ability of our designs in extracting rare variations using short read data from the 1000 Genomes Pilot 3 project.

Enhancing Undergraduate Mathematics Curriculum Via Coding Theory and Cryptography

The theory of error-correcting codes and cryptography are two relatively recent applications of mathematics to information and communication systems. The mathematical tools used in these fields generally come from algebra, elementary number theory, and combinatorics, including concepts from computational complexity. It is possible to introduce the basics of the subjects to undergraduates without requiring a lot of prerequisite knowledge, and at the same time connect several different areas of mathematics and computer science. Moreover, these are branches of mathematics suitable for undergraduate research. In this article, I will describe how I taught a special topics course on coding theory and cryptography, in a non-traditional way, at a small liberal arts college, the pedagogical aspects and the results and implications of the course.

Coding Theory

The workshop on Coding Theory has brought together leading researchers in several key areas of mathematical coding theory. On the side of many mathematicians there were computer scientist and electrical engineers present. Participants came from many countries and the group included both senior and junior researchers. Ever since its conception in the late 1940's, the theory of error-correcting codes has established itself as one of the central areas in mathematics. Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete mathematics and combinatorics, complexity theory, and statistical physics, are just but a few areas which have brought about very interesting applications in coding theory in recent years. The multitude of methods and means to construct and analyze codes and their properties suggests that a workshop with the explicit aim of bringing together researchers in different sub-fields of coding theory is necessary for cross-fertilization of ideas and global advancement of the field. The following topics were covered during the workshop. Combinatorial and probabilistic coding theory: This area has experienced a huge revival in recent years because of its success in the design of codes with superior performance. Very roughly, in this area combinatorial structures are used to construct error-correcting codes, and properties of these structures are used to design and analyze efficient encoding and decoding algorithms for the codes. One of the most prominent examples in this area is furnished by the class of LDPC codes. These codes are constructed from sparse bipartite graphs. More generally Michael Tanner showed in the 80's how to construct ‘general codes on graphs’. The sparsity of the graph provides methods for construction of low complexity encoders and decoders. The graphs need to be designed in such a way as to facilitate an optimal operation of the algorithms. To achieve this goal researchers have developed and applied methods from probability theory and statistics, algebra, discrete mathematics, number theory, and statistical physics. Algebraic coding theory: Algebraic coding theory primarily investigates codes obtained from algebraic constructions. Prime examples of this area of coding theory are codes from algebraic geometry, and codes obtained from algebraically constructed expander graphs. This discipline is almost as old as the coding theory itself, and has attracted (and continues to attract) some of the brightest minds in the field. Among the most exciting advances in this field in recent years has been the invention of list-decoding algorithms for various classes of algebraic codes. Such decoding algorithms yield for a received word a short list of codewords that have at most a given distance \tau to the received word. The size of the list depends on the distance \tau . The methods in this field are mostly algebraic and make use of various properties of multivariate polynomials, or more generally, the properties of “well-behaved” functions in the function field of an irreducible variety. Methods from algebraic geometry are very important in this area. On the computational side the field naturally embedds in the theory of Gröbner bases. There are emerging relationships between this area and codes on graphs, the leading question being whether or not it is possible to match the superior performance of graph-based codes with list-decoding algorithms, or at least with algorithms that are derived from list-decoding algorithms. Theoretical computer science: Theoretical computer science has cotributed a large number of ideas to coding theory. The above mentioned analysis and design of LDPC codes, and the conception of list-decoding algorithms are two prime examples of such contributions. The reader will find it interesting to study in more details the summary of the talks collected in this report.