03898nam a22004815i 4500001001800000003000900018005001700027007001500044008004100059020001800100020001900118024003500137082001200172100002800184245010500212250000700317264004200324300004300366336002600409337002600435338003600461347002400497490004400521505020100565520209300766650001702859650001902876650002502895650001902920650001702939650004502956650003503001650004103036650003303077650001903110650002403129700002803153710003403181773002003215776003603235830004403271856010103315978-0-8176-8256-9DE-He21320260521092033.0cr nn 008mamaa110823s2011    xxu|    s    |||| 0|eng d  a9780817682569  a997808176825697 a10.1007/978-0-8176-8256-92doi04a5192231 aJoyner, David.eauthor.10aSelected Unsolved Problems in Coding Theoryh[electronic resource] /cby David Joyner, Jon-Lark Kim.  a1. 1aBoston :bBirkhäuser Boston,c2011.  aXII, 248p. 17 illus.bonline resource.  atextbtxt2rdacontent  acomputerbc2rdamedia  aonline resourcebcr2rdacarrier  atext filebPDF2rda1 aApplied and Numerical Harmonic Analysis0 aPreface -- Background -- Codes and Lattices -- Kittens and Blackjack -- RH and Coding Theory -- Hyperelliptic Curves and QR Codes -- Codes from Modular Curves -- Appendix -- Bibliography -- Index.  aUsing an original mode of presentation and emphasizing the computational nature of the subject, this book explores a number of the unsolved problems that continue to exist in coding theory. A well-established and still highly relevant branch of mathematics, the theory of error-correcting codes is concerned with reliably transmitting data over a 'noisy' channel. Despite its frequent use in a range of contexts-the first close-up pictures of the surface of Mars, taken by the NASA spacecraft Mariner 9, were transmitted back to Earth using a Reed-Muller code-the subject contains interesting problems that have to date resisted solution by some of the most prominent mathematicians of recent decades. Employing SAGE-a free open-source mathematics software system-to illustrate their ideas, the authors begin by providing background on linear block codes and introducing some of the special families of codes explored in later chapters, such as quadratic residue and algebraic-geometric codes. Also surveyed is the theory that intersects self-dual codes, lattices, and invariant theory, which leads to an intriguing analogy between the Duursma zeta function and the zeta function attached to an algebraic curve over a finite field. The authors then examine a connection between the theory of block designs and the Assmus-Mattson theorem and scrutinize the knotty problem of finding a non-trivial estimate for the number of solutions over a finite field to a hyperelliptic polynomial equation of "small" degree, as well as the best asymptotic bounds for a binary linear block code. Finally, some of the more mysterious aspects relating modular forms and algebraic-geometric codes are discussed. Selected Unsolved Problems in Coding Theory is intended for graduate students and researchers in algebraic coding theory, especially those who are interested in finding current unsolved problems. Familiarity with concepts in algebra, number theory, and modular forms is assumed. The work may be used as supplementary reading material in a graduate course on coding theory or for self-study. 0aMATHEMATICS. 0aCODING THEORY. 0aGEOMETRY, ALGEBRAIC. 0aNUMBER THEORY.14aMATHEMATICS.24aINFORMATION AND COMMUNICATION, CIRCUITS.24aCODING AND INFORMATION THEORY.24aSIGNAL, IMAGE AND SPEECH PROCESSING.24aAPPLICATIONS OF MATHEMATICS.24aNUMBER THEORY.24aALGEBRAIC GEOMETRY.1 aKim, Jon-Lark.eauthor.2 aSpringerLink (Online service)0 tSpringer eBooks08iPrinted edition:z9780817682552 0aApplied and Numerical Harmonic Analysis40uhttp://dx.doi.org/10.1007/978-0-8176-8256-9zVer el texto completo en las instalaciones del CICY