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. discounts 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 definite codes.- A.1.3. Parameters of sure constructions.- A.1.4. Binary codes from AG-codes.- A.2. Asymptotic bounds.- A.2.1. record 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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**Example text**

G is the only into factors we find the set of error system solve the us now Let o :S j :S a (in indeterminants e i ). The g ieI values of ei we are looking for satisfy this system. If {e'. } is another solution, then L (e. p~ = 0 i. e. ~ ieI ~ ~ ~ the vector but its weight is at most e - e' e C 2t :S d - 1. The contradiction we get shows that e is found uniquely. 2. Some families of codes Let us discuss some other interesting examples.

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 .. F . (P . ) . ) ~ = L ieI t-1 'r' L k=O J ~ ~ t-1 L e ..

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 ..