By M. Tsfasman, S.G. Vladut

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. savings and schemes.- three. AG-Codes.- 3.1. structures 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 convinced codes.- A.1.3. Parameters of definite 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. sure families.- A.4.3. Asymptotic results.- writer index.- record of symbols.

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For any linear subspace for v ~ el. for v e el. L (x: y) • "f(u) Let us calculate . -l - y) ~ . y) ~ i=l (x + (q - 1) 'y)n-lIuli. (x _ y) lIuli and the theorem is proved . )) ~ ~ q ) , vi *- 0 v 2 elFq n n ,un) IIvll i=l - '··· )·x 1 q-1 q-1 1-V. 16 (for an arbitrary linear code) . 26. dL • Then xn + dL _ 1 where for i ~ n-d LB . (x 0 ~ B. = (~) . 27. ~ an in 1) dL ~ B. B. - 1) i ~ i=O ~ max {0, (n)i . (qk-i - 1 )}. • Check the following interpretation of [n,k,d]q-systems. Let 1> = {P l , · · · ,Pn } be [n,k,d] -system, P.

2 we construct algebraic-geometric codes that give some lower bounds for mq(g,k) when g is small. The question about the precise value of mq(g,k) is most likely very difficult; bounds. 3. Spectra and duality An important invariant of a code is its weight enumerator or spectrum. We are going to study spectra of linear codes. Let e be a linear [n,k,d]q-code, define as the number of code vectors of weight r for course, Ar 2! y r=O r it is easy to see that '\' L. y IIvll Sometimes non-homogeneous coordinates convenient, then we consider polynomials n n r '\' L.

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) . Note that =L SJ. ) L l l=O ieI 1. +l l=O where looking g(P i ) = 0 s j s t = 0, - 1 , are indeterminants. +l y{e .. p~ L L y{S j+l l=O 1. e'P {Y e } g' (x) 1. 1. e . ) 1. we are 1. 0 1. n. Fj(X) L k=O ~eI,~*J for any j e I b k k ·X we have e ..