scholarly journals A Topological View of Reed–Solomon Codes

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 578
Author(s):  
Alberto Besana ◽  
Cristina Martínez
Keyword(s):  
Linear Group ◽  
Error Correcting Codes ◽  
Rational Curve ◽  
Rs Codes ◽  
Reed Solomon Codes ◽  
Normal Rational Curve ◽  
Algebraic Codes ◽  
Algebraic Code ◽  
Rational Normal Curve ◽  
Geometric Codes

We studied a particular class of well known error-correcting codes known as Reed–Solomon codes. We constructed RS codes as algebraic-geometric codes from the normal rational curve. This approach allowed us to study some algebraic representations of RS codes through the study of the general linear group GL(n,q). We characterized the coefficients that appear in the decompostion of an irreducible representation of the special linear group in terms of Gromov–Witten invariants of the Hilbert scheme of points in the plane. In addition, we classified all the algebraic codes defined over the normal rational curve, thereby providing an algorithm to compute a set of generators of the ideal associated with any algebraic code constructed on the rational normal curve (NRC) over an extension Fqn of Fq.

scholarly journals Error correction by Reed–Solomon codes using its automorphisms

2021 ◽  
Vol 66 (1) ◽  
pp. 110-116
Author(s):  
V. A. Lipnitsky ◽  
S. I. Semyonov
Keyword(s):  
Error Correction ◽  
Galois Field ◽  
Correction Method ◽  
New Method ◽  
Group Of Automorphisms ◽  
Rs Codes ◽  
Reed Solomon Codes ◽  
Rs Code ◽  
Error Correction Method ◽  
Joint Group

The article explores the syndrome invariants of АГ-group of automorphisms of Reed–Solomon codes (RS-codes) that are a joint group of affine and cyclic permutations. The found real invariants are a set of norms of N Г-orbits that make up one or another АГ-orbit. The norms of Г-orbits are vectors with 2 1 Cδ− coordinates from the Galois field, that are determined by all kinds of pairs of components of the error syndromes. In this form, the invariants of the АГ-orbits were cumbersome and difficult to use. Therefore, their replacement by conditional partial invariants is proposed. These quasi-invariants are called norm-projections. Norm-projection uniquely identifies its АГ-orbit and therefore serves as an adequate way for formulating the error correction method by RS-codes based on АГ-orbits. The power of the АГ-orbits is estimated by the value of N2, equal to the square of the length of the RS-code. The search for error vectors in transmitted messages by a new method is reduced to parsing the АГ‑orbits, but actually their norm-projections, with the subsequent search for these errors within a particular АГ-orbit. Therefore, the proposed method works almost N2 times faster than traditional syndrome methods, operating on the basic of the “syndrome – error” principle, that boils down to parsing the entire set of error vectors until a specific vector is found.

THE BASICS OF SELECTING THE OPTIMAL PARAMETERS OF ERROR-CORRECTING CODES FOR TELECOMMUNICATION SYSTEMS

2020 ◽  
Author(s):  
М.А. Быховский
Keyword(s):  
Communication Systems ◽  
Ldpc Code ◽  
Satellite Communications ◽  
Error Correcting Codes ◽  
Rs Codes ◽  
Transmission Reliability ◽  
Multidimensional Signal ◽  
Large Length ◽  
Parity Check Codes ◽  
Communications System

Рассмотрены основные принципы выбора параметров помехоустойчивых кодов для систем связи. Приведены формулы для вероятности ошибки декодирования разных кодов в зависимости от их длины и от вероятности ошибки демодуляции сигналов. Показано, что коды Рида-Соломона (RS) достаточно большой длины позволяют обеспечить высокую помехоустойчивость приема сообщений. Показано также, что в системе связи, в которой сообщения передаются с помощью многомерных ансамблей сигналов, а помехоустойчивые коды не используются, высокая помехоустойчивость приема сообщений обеспечивается при предельно малом отношении сигнал/шум на входе демодулятора. Выполнено сравнение параметров систем связи с кодами RS и спутниковой связи стандарта DVB-S2 с двоичными кодами с малой плотностью проверочных символов (LDPC). Показано, что высокая надежность передачи в системе обеспечивается при использовании с кода RS, имеющего существенно меньшую длину, чем у кода LDPC. The basic principles of selection of parameters of error- correcting codes for communication systems are considered. Formulas, that allow determining the probability of decoding errors of different codes depending on their length and the probability demodulation errors of signals, are given. It is shown that Reed-Solomon (RS) codes of a sufficiently large length make it possible to ensure high noise immunity of receiving messages. It is also shown that in a communication system in which messages are transmitted using multidimensional signal ensembles, and error-correcting codes are not used, messages can be transmitted at a given speed with an extremely small signal/noise ratio at the input of the demodulator. A comparison of the parameters of communication systems with RS codes and the DVB-S2 satellite communications system with binary low-density parity-check codes (LDPC) is performed. It is shown that high transmission reliability in the system is achieved by using an RS code having a significantly shorter length than the LDPC code.

THE ATTACKERS POWER BOUNDARIES FOR TRACEABILITY OF ALGEBRAIC GEOMETRIC CODES ON SPECIAL CURVES

2021 ◽  
pp. 55-74
Author(s):  
V. M. Deundyak ◽  
◽  
D. V. Zagumennov ◽  
◽  
Keyword(s):  
Distributed Storage ◽  
Single Point ◽  
Data Distribution ◽  
Error Correcting Codes ◽  
Equivalence Classes ◽  
Broadcast Encryption ◽  
Network Data ◽  
Distributed Data ◽  
Algebraic Geometric Codes ◽  
Geometric Codes

Broadcast encryption is a data distribution protocol which can prevent malefactor parties from unauthorized accessing or copying the distributed data. It is widely used in distributed storage and network data protection schemes. To block the socalled coalition attacks on the protocol, classes of error-correcting codes with special properties are used, namely c-FP and c-TA properties. We study the problem of evaluating the lower and the upper boundaries on coalition power, within which the algebraic geometry codes possess these properties. Earlier, these boundaries were calculated for single-point algebraic-geometric codes on curves of the general form. Now, we clarified these boundaries for single-point codes on curves of a special form; in particular, for codes on curves on which there are many equivalence classes after factorization by equality of the corresponding points coordinates relation.

scholarly journals METHODOLOGICAL OPPORTUNITIES OF INCLUDING TOPIC «ERROR CORRECTING CODES OF REEDSOLOMON» IN TEACHER TRAINING HIGHER EDUCATION

2020 ◽  
Vol 34 (4) ◽  
pp. 116-124
Author(s):  
Natalya V. Glukhova ◽  
◽  
Elina E. Safina ◽  
Keyword(s):  
Higher Education ◽  
Mathematical Method ◽  
Teacher Training ◽  
Error Correction ◽  
Scientific Literature ◽  
Error Correcting Codes ◽  
Information Protection ◽  
Reed Solomon Codes ◽  
Check Matrix

In this article we are going to demonstrate how the topic “error correcting codes” can be used in training teachers of mathematics and IT. This topic is useful as teachers will know more about actual problems of information protection. The attempt to improve some shortcomings in scientific literature on Reed-Solomon codes (concerning students’ difficulties in understanding this topic) is presented here. The article describes the example of mathematical method of solving the problem of Reed-Solomon coding and decoding with error correction. Systematic coding is based on code seed, decoding is based on check matrix: this approach helps to avoid some difficulties in calculations.

scholarly journals Geometry of a simplex inscribed in a normal rational curve

1967 ◽  
Vol 7 (1) ◽  
pp. 17-22
Author(s):  
Sahib Ram Mandan
Keyword(s):  
Rational Curve ◽  
Twisted Cubic ◽  
Natural Home ◽  
Normal Rational Curve ◽  
Pass Through

In 1959, Professor N. A. Court [2] generated synthetically a twisted cubic C circumscribing a tetrahedron T as the poles for T of the planes of a coaxal family whose axis is called the Lemoine axis of C for T. Here is an analytic attempt to relate a normal rational curve rn of order n, whose natural home is an n-space [n], with its Lemoine [n—2] L such that the first polars of points in L for a simplex S inscribed to rn pass through rn anf the last polars of points on rn for S pass through L. Incidently we come across a pair of mutually inscribed or Moebius simplexes but as a privilege of odd spaces only. In contrast, what happens in even spaces also presents a case, not less interesting, as considered here.

Multilevel secret sharing schemes arising from the normal rational curve

2020 ◽  
Vol 284 ◽  
pp. 158-165
Author(s):  
Stefania Caputo ◽  
Gábor Korchmáros ◽  
Angelo Sonnino
Keyword(s):  
Secret Sharing ◽  
Rational Curve ◽  
Secret Sharing Schemes ◽  
Normal Rational Curve

scholarly journals Forward Error Correction For Gigabit Automotive Ethernet using RS(450,406) Encoder

2019 ◽  
Vol 9 (2S) ◽  
pp. 117-123
Keyword(s):  
Error Correction ◽  
Data Transmission ◽  
Communication System ◽  
Forward Error Correction ◽  
Rs Codes ◽  
Reed Solomon Codes ◽  
Rs Code ◽  
Correction Technique ◽  
Forward Error ◽  
Rs Coding

Error correction and detection during data transmission is a major issue. For resolving this, many error correction techniques are available. The Reed-Solomon coding is the most powerful forward error correction technique used in Gigabit Automotive Ethernet to compact channel noise during data transmission. The car becomes more smarter day by day and more new advanced electronics is being used in-vehicle. Gigabit Automotive Ethernet(1000BASE-T1) provide fast bandwidth for many kinds of applications and connect different functional parts in the car. The Reed Solomon(RS) coding is the powerful forward error correction(FEC) technique used in 1000BASE-T1 Automotive Ethernet. RS(450,406) coding is also known as shortened Reed Solomon codes. The Reed Solomon(RS) codes are generally used in communication system due to its ability of correcting both random and burst errors. Reed Solomon codes are no-binary systematic linear block codes. RS coding is widely used in high speed communication system. This RS code is implemented using Galois field(GF). The Automotive Ethernet is encoded using RS(450,406) codes through GF(512) for FEC. This RS codes can corrects the error up to t=22 symbol, while other encoding techniques corrects the error in t bits. In this paper we implemented the RS(Reed Solomon) code in Cadence ncsim Verilog software and used Cadence Simvision for showing timing diagrams. This RS code uses 9-bit based shortened (450,406) code.

scholarly journals On the Properties of Algebraic Geometric Codes as Copy Protection Codes

2020 ◽  
Vol 27 (1) ◽  
pp. 22-38
Author(s):  
Vladimir M. Deundyak ◽  
Denis V. Zagumennov
Keyword(s):  
Lower Bound ◽  
Error Correcting Code ◽  
Error Correcting Codes ◽  
Broadcast Encryption ◽  
Distributed Data ◽  
Algebraic Geometric Codes ◽  
Authorized User ◽  
Infinite Point ◽  
Encryption Schemes ◽  
Geometric Codes

Traceability schemes which are applied to the broadcast encryption can prevent unauthorized parties from accessing the distributed data. In a traceability scheme a distributor broadcasts the encrypted data and gives each authorized user unique key and identifying word from selected error-correcting code for decrypting. The following attack is possible in these schemes: groups of c malicious users are joining into coalitions and gaining illegal access to the data by combining their keys and identifying codewords to obtain pirate key and codeword. To prevent this attacks, classes of error-correcting codes with special c-FP and c-TA properties are used. In particular, c -FP codes are codes that make direct compromise of scrupulous users impossible and c -TA codes are codes that make it possible to identify one of the a‹ackers. We are considering the problem of evaluating the lower and the upper boundaries on c, within which the L-construction algebraic geometric codes have the corresponding properties. In the case of codes on an arbitrary curve the lower bound for the c-TA property was obtained earlier; in this paper, the lower bound for the c-FP property was constructed. In the case of curves with one infinite point, the upper bounds for the value of c are obtained for both c-FP and c-TA properties. During our work, we have proved an auxiliary lemma and the proof contains an explicit way to build a coalition and a pirate identifying vector. Methods and principles presented in the lemma can be important for analyzing broadcast encryption schemes robustness. Also, the c-FP and c-TA boundaries monotonicity by subcodes are proved.

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