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. savings 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 yes 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. sure families.- A.4.3. Asymptotic results.- writer index.- checklist of symbols.

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**Example text**

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 ).

Projective systems have an advantage of dispensing with a choice of some particular code in its equivalence class; they also look more natural than codes since there is no choice of basis involved. Besides the problem of possible parameters of a projective system looks quite natural, being just a question of how general a position of n points in k 1 IP can be. 8. Sometimes it is indeed necessary to consider systems with multiplicities (for example when constructing codes whose parameters lie on the Griesmer bound, cf.

41 (the sphere-packing Hamming bound). ) . (q ~ or the - --,d-211 n Proof: Consider spheres in Wq of radius t I centered at the code vectors (by definition a sphere of radius centered at t a is Bi (a) = {X e These spheres do not W~ I IIx - all s intersect, t} ). hence the product of lei = qk and the volume (the number of elements) of such a sphere is at most Iwnl = qn . Now the theorem is implied by q the following statement. 42. 43. The following t L i=O (~). (q - 1) i • Prove the lemma. theorem is obtained by averaging procedure with the sphere-packing argument.

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

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