By M. Tsfasman, S.G. Vladut

ISBN-10: 1402003358

ISBN-13: 9781402003356

ISBN-10: 9401138109

ISBN-13: 9789401138109

1. Codes.- 1.1. Codes and their parameters.- 1.2. Examples and constructions.- 1.3. Asymptotic problems.- 2. Curves.- 2.1. Algebraic curves.- 2.2. Riemann-Roch theorem.- 2.3. Rational points.- 2.4. Elliptic curves.- 2.5. Singular curves.- 2.6. rate reductions and schemes.- three. AG-Codes.- 3.1. buildings and properties.- 3.2. Examples.- 3.3. Decoding.- 3.4. Asymptotic results.- four. Modular Codes.- 4.1. Codes on classical modular curves.- 4.2. Codes on Drinfeld curves.- 4.3. Polynomiality.- five. Sphere Packings.- 5.1. Definitions and examples.- 5.2. Asymptotically dense packings.- 5.3. quantity fields.- 5.4. Analogues of AG-codes.- Appendix. precis of effects and tables.- A.1. Codes of finite length.- A.1.1. Bounds.- A.1.2. Parameters of yes codes.- A.1.3. Parameters of sure constructions.- A.1.4. Binary codes from AG-codes.- A.2. Asymptotic bounds.- A.2.1. checklist of bounds.- A.2.2. Diagrams of comparison.- A.2.3. Behaviour on the ends.- A.2.4. Numerical values.- A.3. extra bounds.- A.3.1. consistent weight codes.- A.3.2. Self-dual codes.- A.4. Sphere packings.- A.4.1. Small dimensions.- A.4.2. yes families.- A.4.3. Asymptotic results.- writer index.- record of symbols.

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Consider an [n, n - a - 1, a + 2]q-code C dual to a Reed-Solomon code of degree a. Recall that fd ; decoding up to t = 11 means an algorithm it possible, starting with some v E F~ which is distance t from some code vector u e C, to u. Let is called v - u = e, lIeU:s t , e vector . that makes at most at find this the error CODES 42 v e ~n . ·p~ 1. 1. o Pie'P s j s 2t - 1 . I= {i Ie ... O} e . p~ , where is the 1. ieI 1. 1. (unknown) set of error locators, since for u e C the corresponding sum equals zero by definition (the matrix (Pi) is a parity-check matrix of the code) .

Cut the message into pieces of length k each. Map every word of length k (there are qk possibilities) to an n , C , i . e. f i;x: an embedding element of E:A k ~CS;A and instead of the piece of the message let us a E Ak transmi t the corresponding word E (a) of length n (we encode the message). The transmission is now R times slower, which justifies the term "rate" for R. On the other end of the channel we obtain a distorted word E(a)' E An and we transform it into the nearest word E (a) E C ( i .

54 (the Gilbert bound). prove a slightly weaker result: if q n-k < d-l L i=O (~). (q It is possible to - then there exists a (non-linear) [n,k,d]q-COde; this can be proved by choosing coding vectors in F~ one by one so that each point does not lie in the union of spheres of radius (d - 1) centered in the previolls points. 4) somehow bound the possible values of parameters. still the problem of actually finding out what is possible and what is not is very difficult. 2 EXAMPLES AND CONSTRUCTIONS In this chapter we present several examples of codes.

### Algebraic-Geometric Codes by M. Tsfasman, S.G. Vladut

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