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Journal articles on the topic 'Mersenne'

Author: Grafiati

Published: 4 June 2021

Last updated: 22 February 2023

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1

Granger, Robert. "Could, or should, the ancient Greeks have discovered the Lucas-Lehmer test?" Mathematical Gazette 97, no. 539 (July 2013): 242–55. http://dx.doi.org/10.1017/s0025557200005830.

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The Lucas-Lehmer (LL) test is the most efficient known for testing the primality of Mersenne numbers, i.e. the integers Ml = 2l − 1, for l ≥ 1. The Mersenne numbers are so-called in honour of the French scholar Marin Mersenne (1588-1648), who in 1644 published a list of exponents l ≤ 257 which he conjectured produced all and only those Ml which are prime, for l in this range, namely l = 2,3,5,7, 13, 17, 19,31,67, 127 and 257 [1]. Mersenne's list turned out to be incorrect, omitting the prime-producing l = 61, 89 and 107 and including the composite-producing l = 67 and 257, although this was not finally confirmed until 1947, using both the LL test and contemporary mechanical calculators [2]. The LL test is based on the following theorem.

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2

Palmerino, Carla Rita. "Infinite Degrees of Speed Marin Mersenne and the Debate Over Galileo's Law of Free Fall." Early Science and Medicine 4, no. 4 (1999): 269–328. http://dx.doi.org/10.1163/157338299x00076.

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AbstractThis article analyzes the evolution of Mersenne's views concerning the validity of Galileo's theory of acceleration. After publishing, in 1634, a treatise designed to present empirical evidence in favor of Galileo's odd-number law, Mersenne developed over the years the feeling that only the elaboration of a physical proof could provide sufficient confirmation of its validity. In the present article, I try to show that at the center of Mersenne's worries stood Galileo's assumption that a falling body had to pass in its acceleration through infinite degrees of speed. His extensive discussions with, or his reading of, Descartes, Gassendi, Baliani, Fabri, Cazre, Deschamps, Le Tenneur, Huygens, and Torricelli led Mersenne to believe that the hypothesis of a passage through infinite degrees of speed was incompatible with any mechanistic explanation of free fall.

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3

Malcolm, Noel. "The Title of Hobbes's Refutation of Thomas White's De Mundo." Hobbes Studies 24, no. 2 (2011): 179–88. http://dx.doi.org/10.1163/187502511x597694.

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AbstractHobbes's manuscript refutation of Thomas White bears no title. Some modern scholars have proposed, on the basis of references to it by Mersenne, that the work was entitled 'De motu, loco et tempore', and the abbreviated version of this, 'De motu', has become current in modern scholarship. This research note analyses Mersenne's references, and concludes that this apparent title was a descriptive phrase introduced by Mersenne himself. The full description included the term 'philosophia' (thus: Hobbes's 'philosophy concerning motion, place and time'); this suggests a double focus, not only on the manuscript text, but also on Hobbes's 'body' of natural philosophy more generally.

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4

RAPHAEL, RENEE. "GALILEO'S DISCORSI AND MERSENNE'S NOUVELLES PENSEES: MERSENNE AS A READER OF GALILEAN 'EXPERIENCE'." Nuncius 23, no. 1 (2008): 7–36. http://dx.doi.org/10.1163/182539108x00012.

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Abstracttitle ABSTRACT /title This study examines Marin Mersenne's 1639 Nouvelles Pensees de Galilee, a translation and adaptation of Galileo Galilei's 1638 Discorsi. I use the translation as a window into how Mersenne, a reader trained in natural philosophy, read and understood Galileo's text and, in particular, Galileo's use of experience to support his claims. This analysis reveals that Mersenne drew on a variety of techniques and conceptions of experience in rendering Galileo's individual accounts of experience and experiment. The differences in the way the two authors relate discourse and experience is shown to be linked to their choices of genre and the varying motivations each brought to their texts.

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5

Berrizbeitia, Pedro, and Boris Iskra. "Gaussian Mersenne and Eisenstein Mersenne primes." Mathematics of Computation 79, no. 271 (March 3, 2010): 1779–91. http://dx.doi.org/10.1090/s0025-5718-10-02324-0.

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6

Daşdemir, Ahmet, and Göksal Bilgici. "Gaussian Mersenne numbers and generalized Mersenne quaternions." Notes on Number Theory and Discrete Mathematics 25, no. 3 (September 30, 2019): 87–96. http://dx.doi.org/10.7546/nntdm.2019.25.3.87-96.

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7

Özkan, Engin, and Mine Uysal. "Mersenne-Lucas hybrid numbers." Mathematica Montisnigri 52 (2021): 17–29. http://dx.doi.org/10.20948/mathmontis-2021-52-2.

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We introduce Mersenne-Lucas hybrid numbers. We give the Binet formula, the generating function, the sum, the character, the norm and the vector representation of these numbers. We find some relations among Mersenne-Lucas hybrid numbers, Jacopsthal hybrid numbers, Jacopsthal-Lucas hybrid numbers and Mersenne hybrid numbers. Then we present some important identities such as Cassini identities for Mersenne-Lucas hybrid numbers

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8

Descotes, Dominique. "Mersenne polémiste." Littératures classiques N° 59, no. 1 (2006): 93. http://dx.doi.org/10.3917/licla.059.0093.

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9

Matsumoto, Makoto, and Takuji Nishimura. "Mersenne twister." ACM Transactions on Modeling and Computer Simulation 8, no. 1 (January 1998): 3–30. http://dx.doi.org/10.1145/272991.272995.

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10

Haidar, Riad. "Marin Mersenne." Photoniques, no. 72 (July 2014): 17–19. http://dx.doi.org/10.1051/photon/20147217.

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11

Chelgham, Mourad, and Ali Boussayoud. "On the k-Mersenne–Lucas numbers." Notes on Number Theory and Discrete Mathematics 27, no. 1 (March 2021): 7–13. http://dx.doi.org/10.7546/nntdm.2021.27.1.7-13.

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In this paper, we will introduce a new definition of k-Mersenne–Lucas numbers and investigate some properties. Then, we obtain some identities and established connection formulas between k-Mersenne–Lucas numbers and k-Mersenne numbers through the use of Binet’s formula.

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12

TASCI, DURSUN. "ON GAUSSIAN MERSENNE NUMBERS." Journal of Science and Arts 21, no. 4 (December 30, 2021): 1021–28. http://dx.doi.org/10.46939/j.sci.arts-21.4-a13.

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In this paper, we define Gaussian Mersenne numbers and we give some properties of them. Moreover we present some relations among Gaussian Mersenne numbers, Gaussian Jacobsthal numbers and Gaussian Jacobsthal-Lucas numbers. We also present some results with matrices involving Gaussian Mersenne numbers.

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13

Sergeev, A. "Interrelation of Symmetry and Antisymmetry of Quasi-Orthogonal Cyclic Matrices with Prime Numbers." Proceedings of Telecommunication Universities 8, no. 4 (January 5, 2023): 14–19. http://dx.doi.org/10.31854/1813-324x-2022-8-4-14-19.

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Quasi-orthogonal Hadamard matrices and Mersenne matrices with two and three values of the elements, used in digital data processing, are considered, as well as the basis of error-correcting codes and algorithms for transforming orthogonal images. Attention is paid to the structures of cyclic matrices with symmetries and antisymmetries. The connection between symmetry and antisymmetry of structures of cyclic Hadamard and Mersenne matrices on a orders equal to prime numbers, products of close primes, composite numbers, powers of a prime number is shown. Separately, orders equal to the degrees of the prime number 2 are distinguished, both the orders of Hadamard matrices and the basis of the composite orders of Mersenne matrices of block structures with two element values. It is shown that symmetric Hadamard matrices of cyclic and bicyclic structures, according to the extended Riser boundary, do not exist on orders above 32. Mersenne matrices of composite orders belonging to the sequence of Mersenne numbers 2k ‒ 1 nested in the sequence of orders of the main family of Mersenne matrices 4t ‒ 1 exist in a symmetric and antisymmetric form. For orders equal to the powers of a prime number, Mersenne matrices exist in the form of block-diagonal constructions with three element values. The value of prime power determines the number of blocks along the diagonal of the matrix on which the elements with the third value are located. The cyclic blocks are symmetrical and antisymmetric.

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14

Wollock, Jeffrey. "John Bulwer (1606–1656) and Some British and French Contemporaries." Historiographia Linguistica 40, no. 3 (September 3, 2013): 331–76. http://dx.doi.org/10.1075/hl.40.3.02wol.

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Summary John Bulwer’s (1606–1656) work was unknown in 17th–18th century France. In 1827, when Joseph-Marie Degérando (1772–1842) became curious about the relation between the methods respectively of Bulwer and John Wallis (1616–1703), the pioneer oral instructor of the deaf in Britain, he had to query Charles Orpen, M. D. (1791–1856) in Dublin because no copy of Bulwer’s Philocophus (1648) could be found in Paris. In fact, Theodore Haak (1605–1690) had sent a copy of this book from London to Père Marin Mersenne (1588–1648) in Paris in July 1648, but none of Mersenne’s circle could read English, and Mersenne died several weeks later. In that context, this paper presents a comparison of Bulwer’s views with those of the Cartesians and Port-Royalists. Wallis claimed he knew of no work on speech for the deaf prior to his own, but he must have known about the Philocophus from the time of its publication, five years before his De Loquela (1653) and nearly 14 years before he began teaching the deaf.

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15

Kim, Shin-Wook. "Mersenne prime's inducement." International Journal of Algebra 12, no. 1 (2018): 15–23. http://dx.doi.org/10.12988/ija.2018.71257.

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16

Kalashnіkova, N. V. "Some properties of prime numbers of special form and Carmichael numbers." Researches in Mathematics 24 (September 1, 2016): 41. http://dx.doi.org/10.15421/241607.

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We study some properties of structure of the multiplicative group $Z^*_m$, in case when m is Mersenne prime, Fermat or Carmichael number. Using the results of these studies, we obtain properties of Mersenne primes, Fermat and Carmichael numbers.

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17

Nenashev, V. A., A. M. Sergeev, and E. A. Kapranova. "Research and Analysis of Autocorrelation Functions of Code Sequences Formed on the Basis of Monocyclic Quasi-Orthogonal Matrices." Information and Control Systems, no. 4 (September 23, 2018): 9–14. http://dx.doi.org/10.31799/1684-8853-2018-4-9-14.

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Introduction: Barker codes representing binary sequences (codes) of finite lengths 2, 3, 4, 5, 7, 11 and 13 are widely used in solving the problem of increasing the noise immunity of radar channels. However, the code sequences for n > 13 are unknown. Sequences derived from quasi-orthogonal Mersenne matrices also have not been used for these purposes.Purpose: Studying the ways to compress a complex modulated signal by Mersenne sequences obtained from the first rows of a monocyclic quasi-orthogonal Mersenne matrix, as an alternative to Barker codes.Results:It has been found out that the characteristics of autocorrelation functions for Mersenne codes 3, 7 and 11 exceed those for Barker codes. This is a basis for ensuring greater noise immunity of probing signals in radar channels, as well as for increasing the probability of their correct detection, proving the expediency of their application for amplitude and phase modulation of radio signals.Practical relevance:The obtained results allow you to increase the compression characteristics in radar systems when solving the problem of detecting targets under noise and interference. The wide application of Barker codes of length 3, 7 and 11 in digital data transmission systems provides a special interest in similar Mersenne codes when implementing noise-resistant data transmission in radio channels in a complex electromagnetic environment. Discussion: An unresolved problem is the non-symmetry of elements in a coding Mersenne sequence. This problem can be solved either by special synthesis of a phase-modulated signal or by finding new approaches to their compression.

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18

HERINGA, J. R., H. W. J. BLÖTE, and A. COMPAGNER. "NEW PRIMITIVE TRINOMIALS OF MERSENNE-EXPONENT DEGREES FOR RANDOM-NUMBER GENERATION." International Journal of Modern Physics C 03, no. 03 (June 1992): 561–64. http://dx.doi.org/10.1142/s0129183192000361.

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The list of primitive binary trinomials with a degree equal to a Mersenne exponent is extended. The newly found primitive trinomials have a degree equal to the 29th and 30th Mersenne exponent. These trinomials enable the construction of new, high-performance random-number generators for use in large-scale Monte Carlo simulations.

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19

MacDougall, Jim. "Mersenne composites and cyclotomic primes." Mathematical Gazette 87, no. 508 (March 2003): 71–75. http://dx.doi.org/10.1017/s0025557200172122.

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One of the long-standing problems of number theory, appealing to professional and recreational mathematicians alike, is the existence of Mersenne primes. These puzzling primes, for example 7, 31, 127 and 8191, are of the form 2P - 1, where p is itself a prime. The problem of their existence originated some 2400 years ago with the early Greek mathematicians' quest for the so-called perfect numbers, those like 6 and 28 which are the sum of their proper divisors. The connection was given in Euclid's Elements in 300 BC: if 2P - 1 is prime, then 2P-1(2P - 1) is a perfect number. Much later, Euler proved that all the even perfect numbers correspond to Mersenne primes. So the interest for many years has been in finding Mersenne primes. Only 39 are known, including several monsters discovered in recent years using thousands of PCs coordinated via the internet (see [1] for information on the project). Many of us would like to know if there is any way of predicting which exponents yield Mersenne primes and whether there is an infinite number of them.

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20

Garber, Daniel. "O que Mersenne aprendeu na Itália." Discurso, no. 31 (December 9, 2000): 89–114. http://dx.doi.org/10.11606/issn.2318-8863.discurso.2000.38035.

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Estudos sobre Marin Mersenne enfatizam freqüentemente o serviço prestado por ele à ciência européia, por ajudar na circulação das idéias, tanto pela correspondência como por suas publicações. Mas o próprio Mersenne foi uma figura importante na Revolução Científica com seu próprio programa intelectual. O propósito do artigo é discutir o papel que o contato epistolar com a Itália exerceu no seu próprio desenvolvimento intelectual. Quero discutir também que a transmissão da ciência italiana para a França feita por Mersenne, no final do anos 1620 e início dos anos 1630, precisamente no momento em que Galileu estava em dificuldades em Roma, foi crucial para a derradeira transformação da ciência e filosofia européias. Minha tese é que por causa de seus contatos com a Italia Mersenne continua, de certo modo, a tradição jesuítica das matemáticas mistas que, em virtude da condenação de Galileu em 1633, não poderia por muito tempo ser praticada na Itália, uma tradição que conduzirá a Descartes, Gassendi, e à filosofia mecânica que dominará o restante do século.

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21

Buccolini, Claudio. "Mersenne Translator of Bacon?" Journal of Early Modern Studies 2, no. 1 (January 1, 2013): 33–59. http://dx.doi.org/10.7761/jems.2.1.33.

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22

Besnier, Bernard. "Mersenne à demi-éveillé." Littératures classiques 17, no. 1 (1992): 25–35. http://dx.doi.org/10.3406/licla.1992.1014.

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23

Seares, Margaret. "Mersenne on Vocal Diminutions." Performance Practice Review 6, no. 2 (1993): 141–45. http://dx.doi.org/10.5642/perfpr.199306.02.06.

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24

Mangueira, Milena Carolina dos Santos, Francisco Régis Vieira Alves, and Paula Maria Machado Cruz Catarino. "Números híbridos de Mersenne." Iniciação Científica 18, no. IC (July 2020): 01–11. http://dx.doi.org/10.21167/cqdvol18ic202023169664mcsmfrvapmmcc0111.

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25

Malet, Antoni, and Daniele Cozzoli. "Mersenne and Mixed Mathematics." Perspectives on Science 18, no. 1 (May 2010): 1–8. http://dx.doi.org/10.1162/posc.2010.18.1.1.

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26

Colquitt, W. N., and L. Welsh. "A new Mersenne prime." Mathematics of Computation 56, no. 194 (May 1, 1991): 867. http://dx.doi.org/10.1090/s0025-5718-1991-1068823-9.

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27

Balonin, N. A., and M. B. Sergeev. "Mersenne and Hadamard Matrices." Informatsionno-upravliaiushchie sistemy (Information and Control Systems) 1, no. 80 (January 2016): 2–15. http://dx.doi.org/10.15217/issn1684-8853.2016.1.2.

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28

Shirali, Shailesh A. "Marin Mersenne, 1588–1648." Resonance 18, no. 3 (March 2013): 226–40. http://dx.doi.org/10.1007/s12045-013-0034-2.

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29

Hoque, Azizul, and Helen K. Saikia. "On generalized Mersenne prime." SeMA Journal 66, no. 1 (September 16, 2014): 1–7. http://dx.doi.org/10.1007/s40324-014-0019-4.

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30

Granger, Robert, and Andrew Moss. "Generalised Mersenne numbers revisited." Mathematics of Computation 82, no. 284 (May 8, 2013): 2389–420. http://dx.doi.org/10.1090/s0025-5718-2013-02704-4.

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31

Bravo, Jhon J., and Carlos A. Gómez. "Mersenne k-Fibonacci numbers." Glasnik Matematicki 51, no. 2 (December 2, 2016): 307–19. http://dx.doi.org/10.3336/gm.51.2.02.

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32

Einsiedler, Manfred, Graham Everest, and Thomas Ward. "Primes in Sequences Associated to Polynomials (After Lehmer)." LMS Journal of Computation and Mathematics 3 (2000): 125–39. http://dx.doi.org/10.1112/s1461157000000255.

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AbstractIn a paper of 1933, D. H. Lehmer continued Pierce's study of integral sequences associated to polynomials generalizing the Mersenne sequence. He developed divisibility criteria, and suggested that prime apparition in these sequences — or in closely related sequences — would be denser if the polynomials were close to cyclotomic, using a natural measure of closeness.We review briefly some of the main developments since Lehmer's paper, and report on further computational work on these sequences. In particular, we use Mossinghoff's collection of polynomials with smallest known measure to assemble evidence for the distribution of primes in these sequences predicted by standard heuristic arguments.The calculations lend weight to standard conjectures about Mersenne primes, and the use of polynomials with small measure permits much larger numbers of primes to be generated than in the Mersenne case.

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33

Casinillo, Leomarich F. "Some New Notes on Mersenne Primes and Perfect Numbers." Indonesian Journal of Mathematics Education 3, no. 1 (April 30, 2020): 15. http://dx.doi.org/10.31002/ijome.v3i1.2282.

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<p>Mersenne primes are specific type of prime numbers that can be derived using the formula <img title="\large M_p=2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&space;M_p=2^{p}-1" alt="" />, where <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&space;p" alt="" /> is a prime number. A perfect number is a positive integer of the form <img title="\large P(p)=2^{p-1}(2^{p}-1)" src="https://latex.codecogs.com/gif.latex?\large&space;P(p)=2^{p-1}(2^{p}-1)" alt="" /> where <img title="\large 2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&space;2^{p}-1" alt="" /> is prime and <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&space;p" alt="" /> is a Mersenne prime, and that can be written as the sum of its proper divisor, that is, a number that is half the sum of all of its positive divisor. In this note, some concepts relating to Mersenne primes and perfect numbers were revisited. Further, Mersenne primes and perfect numbers were evaluated using triangular numbers. This note also discussed how to partition perfect numbers into odd cubes for odd prime <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&space;p" alt="" />. Also, the formula that partition perfect numbers in terms of its proper divisors were constructed and determine the number of primes in the partition and discuss some concepts. The results of this study is useful to better understand the mathematical structure of Mersenne primes and perfect numbers.</p>

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34

De la Croix, David, and Julie Duchêne. "Scholars and Literati at the “Mersenne” Academy (1635–1648)." Repertorium eruditorum totius Europae 2 (January 11, 2021): 7–12. http://dx.doi.org/10.14428/rete.v2i0/mersenne.

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35

De la Croix, David, and Julie Duchêne. "Scholars and Literati at the “Mersenne” Academy (1635–1648)." Repertorium eruditorum totius Europae 2 (January 11, 2021): 7–12. http://dx.doi.org/10.14428/rete.v2i0/mersenne.

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36

ASBOEI, ALIREZA KHALILI. "CHARACTERIZING PSL2(r) BY ITS ORDER AND THE NUMBER OF ITS SYLOW r-SUBGROUPS." Journal of Algebra and Its Applications 13, no. 01 (August 20, 2013): 1350065. http://dx.doi.org/10.1142/s0219498813500655.

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We show that if r is a prime number that is not a Mersenne prime, then PSL2 (r) is determined up to isomorphism by its order and by the number of its Sylow r-subgroups. We then show that if r is a Mersenne prime other than 7, then PSL2 (r) is determined up to isomorphism by its order, the number of its Sylow r-subgroups, and the fact that r is an isolated vertex of the prime graph of the group.

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37

Shiu, Peter. "Pseudoperfect numbers with no small prime divisors." Mathematical Gazette 93, no. 528 (November 2009): 404–9. http://dx.doi.org/10.1017/s0025557200185146.

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A perfect number is a number which is the sum of all its divisors except itself, the smallest such number being 6. By results due to Euclid and Euler, all the even perfect numbers are of the form 2P-1(2p - 1) where p and 2p - 1 are primes; the latter one is called a Mersenne prime. Whether there are infinitely many Mersenne primes is a notoriously difficult problem, as is the problem of whether there is an odd perfect number.

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38

Szynal-Liana, Anetta, and Iwona Włoch. "On generalized Mersenne hybrid numbers." Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica 74, no. 1 (October 20, 2020): 77. http://dx.doi.org/10.17951/a.2020.74.1.77-84.

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The hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper we consider a special kind of hybrid numbers, namely the Mersenne hybrid numbers and we give some of their properties.

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39

Jullien, Vincent. "Gassendi, Roberval à l'académie Mersenne." Dix-septième siècle 233, no. 4 (2006): 601. http://dx.doi.org/10.3917/dss.064.0601.

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40

Alves, Francisco Regis Vieira. "Bivariate Mersenne polynomials and matrices." Notes on Number Theory and Discrete Mathematics 26, no. 3 (September 2020): 83–95. http://dx.doi.org/10.7546/nntdm.2020.26.3.83-95.

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41

Willstrop, R. V. "The flat-field Mersenne-Schmidt." Monthly Notices of the Royal Astronomical Society 216, no. 2 (September 1, 1985): 411–27. http://dx.doi.org/10.1093/mnras/216.2.411.

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42

Silva, Paulo Tadeu da. "A harmonia mecanicista em Mersenne." Discurso, no. 37 (December 8, 2007): 75–102. http://dx.doi.org/10.11606/issn.2318-8863.discurso.2007.62919.

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A relação entre música e ciência é um capítulo importante na história da ciência e da filosofia. Este artigo procura discutir alguns aspectos das investigações de Marin Mersenne sobre a música e a acústica, tendo em vista o desenvolvimento da teoria da coincidência da consonância e seu compromisso com uma visão mecânica da natureza, pela qual ele estabeleceu as propriedades físicas do som e proporções matemáticas dos intervalos musicais.

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43

Balonin, N. A., and M. B. Sergeev. "Mersenne and Hadamard Matrices, Products." Informatsionno-upravliaiushchie sistemy (Information and Control Systems) 5, no. 84 (November 2016): 2–14. http://dx.doi.org/10.15217/issn1684-8853.2016.5.2.

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44

Banks, William D., Alessandro Conflitti, John B. Friedlander, and Igor E. Shparlinski. "Exponential sums over Mersenne numbers." Compositio Mathematica 140, no. 01 (January 2004): 15–30. http://dx.doi.org/10.1112/s0010437x03000022.

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45

Wolf, Mrek. "Computer Experiments with Mersenne Primes." Computational Methods in Science and Technology 19, no. 3 (July 31, 2013): 157–65. http://dx.doi.org/10.12921/cmst.2013.19.03.157-165.

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46

Boussakta, S., M. T. Hamood, and N. Rutter. "Generalized New Mersenne Number Transforms." IEEE Transactions on Signal Processing 60, no. 5 (May 2012): 2640–47. http://dx.doi.org/10.1109/tsp.2012.2186131.

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47

Gallardo, Luis H., and Olivier Rahavandrainy. "On Mersenne polynomials over F2." Finite Fields and Their Applications 59 (September 2019): 284–96. http://dx.doi.org/10.1016/j.ffa.2019.06.006.

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Andrade, Doherty. "Números Primos e Números de Mersenne." Jornal Eletrônico de Ensino e Pesquisa de Matemática 2, no. 1 (July 1, 2018): 81–89. http://dx.doi.org/10.4025/jeepema.v2.n1.art3.

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Sergeev, M. B., V. A. Nenashev, and A. M. Sergeev. "Nested code sequences of Barker — Mersenne — Raghavarao." Information and Control Systems, no. 3 (June 21, 2019): 71–81. http://dx.doi.org/10.31799/1684-8853-2019-3-71-81.

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Abstract:

Introduction: The problem of noise-free encoding for an open radio channel is of great importance for data transfer. The results presented in this paper are aimed at stimulating scientific interest in new codes and bases derived from quasi-orthogonal matrices, as a basis for the revision of signal processing algorithms.Purpose: Search for new code sequences as combinations of codes formed from the rows of Mersenne and Raghavarao quasi-orthogonal matrices, as well as complex and more efficient Barker — Mersenne — Raghavarao codes.Results: We studied nested code sequences derived from the rows of quasi-orthogonal cyclic matrices of Mersenne, Raghavarao and Hadamard, providing estimates for the characteristics of the autocorrelation function of nested Barker, Mersenne and Raghavarao codes, and their combinations: in particular, the ratio between the main peak and the maximum positive and negative “side lobes”. We have synthesized new codes, including nested ones, formed on the basis of quasi-orthogonal matrices with better characteristics than the known Barker codes and their nested constructions. The results are significant, as this research influences the establishment and development of methods for isolation, detection and processing of useful information. The results of the work have a long aftermath because new original code synthesis methods need to be studied, modified, generalized and expanded for new application fields.Practical relevance: The practical application of the obtained results guarantees an increase in accuracy of location systems, and detection of a useful signal in noisy background. In particular, these results can be used in radar systems with high distance resolution, when detecting physical objects, including hidden ones.

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Francis, Richard L. "New Worlds to Conquer." Mathematics Teacher 98, no. 3 (October 2004): 166–70. http://dx.doi.org/10.5951/mt.98.3.0166.

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