000			03260nam a22003377a 4500
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005			20250801152129.0
008			250801b |||||||| |||| 00| 0 eng d
020			_a9780821826287
020			_a9789349750296 (pbk.) _c$ 49.00
040			_bENG _cIISER-BPR _dIISER-BPR
041			_aENG
082			_a516.352 _bWAL _223rd
100			_aWalker, Judy L. _95562
222			_aMathematics
245			_aCodes and curves
250			_a1st Indian ed.
260			_aRhode Island : _bAmerican Mathematical Society, _cc2006.
260			_aHyderabad : _bUniversity Press (India) Pvt. Ltd., _c2025.
300			_ax, 68 p. : _bill. ; _c22cm.
440			_aStudent Mathematical Library _pIAS/PARK City Mathematical Subseries _vVol. 7 _94093
504			_aIncludes illustrations, appendices, and bibliographic references.
520			_aWhen information is transmitted, errors are likely to occur. This problem has become increasingly important as tremendous amounts of information are transferred electronically every day. Coding theory examines efficient ways of packaging data so that these errors can be detected, or even corrected. The traditional tools of coding theory have come from combinatorics and group theory. Since the work of Goppa in the late 1970s, however, coding theorists have added techniques from algebraic geometry to their toolboxes. In particular, by re-interpreting the Reed-Solomon codes as coming from evaluating functions associated to divisors on the projective line, one can see how to define new codes based on other divisors or on other algebraic curves. For instance, using modular curves over finite fields, Tsfasman, Vladut, and Zink showed that one can define a sequence of codes with asymptotically better parameters than any previously known codes. This book is based on a series of lectures the author gave as part of the IAS/Park City Mathematics Institute (Utah) program on arithmetic algebraic geometry. Here, the reader is introduced to the exciting field of algebraic geometric coding theory. Presenting the material in the same conversational tone of the lectures, the author covers linear codes, including cyclic codes, and both bounds and asymptotic bounds on the parameters of codes. Algebraic geometry is introduced, with particular attention given to projective curves, rational functions and divisors. The construction of algebraic geometric codes is given, and the Tsfasman-Vladut-Zink result mentioned above is discussed. No previous experience in coding theory or algebraic geometry is required. Some familiarity with abstract algebra, in particular finite fields, is assumed. However, this material is reviewed in two appendices. There is also an appendix containing projects that explore other codes not covered in the main text. This book is published in cooperation with IAS/Park City Mathematics Institute. Readership: Undergraduates in mathematics; mathematicians interested in coding theory or algebraic geometry and the connections between the two subjects.
650			_aMathematics
650			_aGeometry
650			_aAnalytic geometries
650			_aAlgebraic geometry
650			_aTheory of curves _93217
650			_aCoding theory _94234
942			_2ddc _cBK
999			_c4298 _d4298