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. 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 convinced 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. definite families.- A.4.3. Asymptotic results.- writer index.- checklist of symbols.

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S v < n - d is a code vector of would Ho (that is so already The parameters of the corresponding Ho). code of which is not a linear combination of v We get a matrix Ho. 8, + 1 dl. Hence k+d=n + n - n which row k < then we we have is not of kl. < n ). Consider a [6,4,2] 2 -code C generated by the matrix [ ~ i ~ ~ ~ ~l 000 110 000 011 Calculate the spectra of (Ansver: B 4 = 6 , B B' 0 = 0 3 15, B' 1 C B 1 = B'2 = and cl. 42 6 in terms of B B' 3 2 2 ) . 15, recall that an additive character of homomorphism X ~q from the additive group of 3 B.

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. y are more r r=O r A code e has only one vector of weight 0 and has no other vectors of weight less that d. 1 15 Since in many cases we know not the precise value of d but only some lower estimate for it, the following form is rather convenient.

State and prove the spoiling lemma for non-linear codes. (Hint: In this case (k - 1) is changed by logq(lqk-l Exercise r) ). 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.