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. discount rates 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 definite codes.- A.1.3. Parameters of yes 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. definite families.- A.4.3. Asymptotic results.- writer index.- record of symbols.

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Suppose formally self-dual integer t > 1 . 33. th respec t w~ exists q-ary + y2)n/2, «q - 1) ·x2 • x. ; such that such that x·e Yi = is 2 Xi ' then there exists some self-dual. Check that any Part 1 CODES 26 element of a finite field of characteristic 2 is a square, and hence if q = 2 m then for any quasi-self-dual code C c. 4. Bounds We have already explained that a good code should have large and k d for a given n. Let q be fixed. For n, k, d does there exist a linear [n,k,d]q-COde (or just some [n,k,dJq-code)?

In many cases it is difficult to calculate precise values of parameters but it is possible to bound them. The spoiling lemma makes it possible to pass from an [:s n, d]q-code C to an [n,k,d]q-code. In such a situation we say that up to a spoiling the code C is an [n,k,d]q-code. 2: k, 2: So we can always spoil parameters, but of course we cannot always make them better. Here are some restrictions. 36 (the Singleton bound). linear [n,k,d]q-code For any Part 1 CODES 28 Proof: Let us argue in terms of [n,k,d]q-systems.

24. 1) =L f(v) L Xv(u) + veel. If v e el. L f(v) L Xv(u) v~cl. uee for all u e e uee o u-v then L x 1 (O) lei • uee v Let , thus be such that there exists Then x (u) - L v since (3 e 1 uee (u) X u0 + e = e v ; = L uee X v (u 0 + u) L Xv (u) = hence uee It is left to prove that i f then =X 0 (0:- * IF q get a v e el.. (u-v» = 1 fact, In for such that Xl «(3) contradiction, i. e. 1) proves the lemma . 25. For any linear subspace for v ~ el. for v e el. L (x: y) • "f(u) Let us calculate . -l - y) ~ .