scholarly journals Properties of triple error orbits G and their invariants in Bose – Chaudhuri – Hocquenghem codes C7

2019 ◽  
Vol 64 (1) ◽  
pp. 110-117
Author(s):  
V. A. Lipnitski ◽  
A. U. Serada
Keyword(s):  
Bch Codes ◽  
Polynomial Invariants ◽  
Group Of Automorphisms ◽  
Decoding Algorithms ◽  
Basic Properties ◽  
One To One ◽  
Further Development ◽  
Vector Polynomial

This work is the further development of the theory of norms of syndromes: the theory of polynomial invariants of G-orbits of errors expands with the group G of automorphisms of binary cyclic BCH codes obtained by joining the degrees of cyclotomic permutation to the group Γ and practically exhausting the group of automorphisms of BCH codes. It is determined that polynomial invariants, like the norms of syndromes, have a scalar character and are one-to-one characteristics of their orbits for BCH codes with a constructive distance of five. The paper introduces the corresponding vector polynomial invariants for primitive cyclic BCH codes with a constructive distance of seven, next to the norms of the syndromes that are already vector quantities; the basic properties of the vector polynomial invariants are investigated. It is established that the property of mutual unambiguity is violated: there are G-orbit-isomers, which are different, but have the same vector polynomial invariants. It is substantiated and demonstrated by examples that this circumstance greatly complicates error decoding algorithms based on polynomial invariants

scholarly journals The properties and parameters of generic Bose – Chaudhuri – Hocquenghem codes

2020 ◽  
Vol 56 (2) ◽  
pp. 157-165
Author(s):  
A. V. Kushnerov ◽  
V. A. Lipinski ◽  
M. N. Koroliova
Keyword(s):  
Error Correction ◽  
Cyclic Codes ◽  
Error Correcting Code ◽  
Error Correcting Codes ◽  
Bch Codes ◽  
Close Connection ◽  
Design Parameters ◽  
Polynomial Invariants ◽  
Accurate Evaluation

The Bose – Chaudhuri – Hocquenghem type of linear cyclic codes (BCH codes) is one of the most popular and widespread error-correcting codes. Their close connection with the theory of Galois fields gave an opportunity to create a theory of the norms of syndromes for BCH codes, namely, syndrome invariants of the G-orbits of errors, and to develop a theory of polynomial invariants of the G-orbits of errors. This theory as a whole served as the basis for the development of effective permutation polynomial-norm methods and error correction algorithms that significantly reduce the influence of the selector problem. To date, these methods represent the only approach to error correction with non-primitive BCH codes, the multiplicity of which goes beyond design boundaries. This work is dedicated to a special error-correcting code class – generic Bose – Chaudhuri – Hocquenghem codes or simply GBCH-codes. Sufficiently accurate evaluation of the quantity of such codes in each length was produced during our work. We have investigated some properties and connections between different GBCH-codes. Special attention was devoted to codes with constructive distances of 3 and 5, as those codes are usual for practical use. Their almost complete description is given in the range of lengths from 7 to 107. The paper contains a fairly clear theoretical classification of GBCH-codes. Special attention is paid to the corrective capabilities of the codes of this class, namely, to the calculation of the minimal distances of these codes with various parameters. The codes are found whose corrective capabilities significantly exceed those of the well-known GBCH-codes with the same design parameters.

scholarly journals PROPERTIES OF GROUPS G OF DOUBLE ERRORS AND ITS INVARIANTS IN BCH CODES

2018 ◽  
pp. 40-46
Author(s):  
V. A. Lipnitskij ◽  
A. V. Serada
Keyword(s):  
Irreducible Polynomial ◽  
Galois Field ◽  
Bch Codes ◽  
Bch Code ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
The Core ◽  
Complex Process ◽  
Scope Of Application ◽  
Complete Set

The goal of the work is the further extending the scope of application of code automorthism in methods and algorithms of error correction by these codes. The effectiveness of such approach was demonstrated by norm of syndrome theory that was developed by Belarusian school of noiseless coding at the turn of the XX and XXI century. The group Г of the cyclical shift of vector component lies at the core of the theory. Under its action The error vectors are divided into disjoint Г-orbits with definite spectrum of syndromes. This allowed to introduce norms of syndrome of a family of BCH codes that are invariant over action of group Г. Norms of syndrome are unique characteristic of error orbit Г of any decoding set, hence it is the basis of permutation norm methods of error decoding. Looking over the Г-orbits of errors not the errors these methods are faster than classic syndrome methods of error decoding, are avoided from the complex process of solving the algebraic equation in Galois field, are simply implemented.A detailed theory for automorphism group G of BCH codes obtained by adding cyclotomic substitution to the group Г develops in the article. The authors held a detailed study of structure of G-orbit of errors as union of orbits Г of error vectors; one-to-one mapping of this structure on the norm structure of group Г. These norms being interconnected by Frobenius automorphism in the Galois field – field of BCH code constitute the complete set of roots of the only irreducible polynomial. It is a polynomial invariant of its orbit G. The main focus of the work is on the description of properties and specific features of groups G of double errors and its polynomial invariants.

On Blahut's decoding algorithms for two-dimensional BCH codes

1998 ◽  
Vol 44 (1) ◽  
pp. 358-367 ◽  
Author(s):  
H.S. Madhusudhana ◽  
M.U. Siddiqi
Keyword(s):  
Bch Codes ◽  
Two Dimensional ◽  
Decoding Algorithms

scholarly journals Invariants of Finite Reflection Groups

1963 ◽  
Vol 22 ◽  
pp. 57-64 ◽  
Author(s):  
Louis Solomon
Keyword(s):  
Vector Space ◽  
Characteristic Zero ◽  
Polynomial Functions ◽  
Dimensional Vector ◽  
Reflection Groups ◽  
Linear Transformations ◽  
Graded Ring ◽  
Polynomial Invariants ◽  
Dimensional Vector Space ◽  
Group Of Automorphisms

Let K be a field of characteristic zero. Let V be an n-dimensional vector space over K and let S be the graded ring of polynomial functions on V. If G is a group of linear transformations of V, then G acts naturally as a group of automorphisms of S if we defineThe elements of S invariant under all γ ∈ G constitute a homogeneous subring I(S) of S called the ring of polynomial invariants of G.

scholarly journals Index polynomials for virtual tangles

2018 ◽  
Vol 27 (12) ◽  
pp. 1850073 ◽  
Author(s):  
Nicolas Petit
Keyword(s):  
Knot Theory ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Vassiliev Invariants ◽  
Basic Properties ◽  
Virtual Knots

We generalize the index polynomial invariant, originally introduced by Turaev [Cobordism of knots on surfaces, J. Topol. 1(2) (2008) 285–305] and Henrich [A sequence of degree one vassiliev invariants for virtual knots, J. Knot Theory Ramifications 19(4) (2010) 461–487], to the case of virtual tangles. Three polynomial invariants result from this generalization; we give a brief overview of their definition and some basic properties.

A sequence of polynomial invariants for Gauss diagrams

2017 ◽  
Vol 26 (07) ◽  
pp. 1750039 ◽  
Author(s):  
Young Ho Im ◽  
Sera Kim
Keyword(s):  
Polynomial Invariants ◽  
Virtual Knot ◽  
One To One ◽  
Knot Diagrams

We introduce a sequence of polynomial invariants for Gauss diagrams which are one-to-one correspondence with virtual knot diagrams. Also, we give some properties of these polynomials and examples.

scholarly journals Generalizing the Minkowski question mark function to a family of multidimensional continued fractions

2018 ◽  
Vol 14 (09) ◽  
pp. 2473-2516 ◽  
Author(s):  
Thomas Garrity ◽  
Peter Mcdonald
Keyword(s):  
Continued Fractions ◽  
Singular Function ◽  
Question Mark ◽  
Basic Properties ◽  
Naturally Occurring ◽  
Almost Everywhere ◽  
Multidimensional Continued Fractions ◽  
One To One ◽  
Almost All ◽  
Minkowski Question Mark Function

The Minkowski question mark function [Formula: see text] is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of continued fractions. Thus [Formula: see text] is a naturally occurring number theoretic singular function. This paper generalizes the question mark function to the 216 triangle partition (TRIP) maps. These are multidimensional continued fractions which generate a family of almost all known multidimensional continued fractions. We show for each TRIP map that there is a natural candidate for its analog of the Minkowski question mark function. We then show that the analog is singular for 96 of the TRIP maps and show that 60 more are singular under an assumption of ergodicity.

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