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. rate reductions 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 yes codes.- A.1.3. Parameters of convinced 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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Whenever q n-k there exists a linear < [n,k,d]q-code c Proof: We are going to construct a dual [n,k,d]qsystem Q c W, we take any dim W = n - k For non-zero vector. Suppose that we have already constructed a system consisting of i vectors Ql' ••• , Q i such that any (d - 1) of them are linearly independent. consider the set Si of vectors which are linear combinations of at most (d - 2) elements from {Ql, ... ). (q - 1) j If Hi < IWI we are able to choose Qi+l II! Si . Then any (d - 1) vectors from the set {Ql' ••• ,Qi+l} are linearly independent.

3. [n,k,d]q-code C. H d Show that any (d - 1) columns of H n-k IFq and there linearly dependent columns. According to our definition (n - k) is also called a ~ n - k H' such has H linearly independent rows. language, any matrix r H·x = 0 ). be a parity-check matrix of an are linearly independent (as vectors in exist (since the C is equivalent to the equality e C is H) that Sometimes, H'·x parity-check matrix. rows, only (n - k) n = 0 columns and by abuse of iff Any such x e C , H' of which are independent.

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 .. · pkH. ~ ~ Therefore g' (P j) = 0 for any j e I , i. e. solution (up to a multiplicative constant). Decomposing locators I. L e . p~ ~ ~ = s J. 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 ).

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