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. 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. 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.- checklist of symbols.

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N + k + d codes 1) form an Unfortunately, • extended to codes and n. Moreover, embedded family, their length cannot exceed basis = the the projective 2 a {l,x,x , ... ,x } generator = i O,l, ... ,a; The dual code. line with is also a matrix in c of = IFq "P Reed-Solomon code with a Let 'P [n, n - a - 1, a + 2]q . i is Let us start with the simplest case. or for L(a) 1,2, ... ,n . 1. parameters L xi = 0 on 1 , n s q + 1 Choose the basis xelF which is In section 3. 1. 1. 36 it cannot be better), a + 1 L(a) c L(a = k + d (Hint: = IF * i = '; - Prove that a - 2 and Use the fact that < q - 1).

We start with a rather strange but quite useful statement that having a good code we can get a lot of worse ones. 34 (the spoiling lemma). exists a [n,k,dJq-code non-degenerate Suppose that there c. 1. 2) . 1 Choose CODES AND THEIR PARAMETERS a hyperplane H o 27 c IPk - 1 such one more point from max IHo n:P1 • Add to H (it does not matter whether it already belongs to not) . Let vectors. Exclude from to Qo (it is possible by 1, and d and (n b) that H o or be a linearly dependent set of d Q any vector which does not belong since d < n ).

1) i ~ i=O ~ max {0, (n)i . (qk-i - 1 )}. • Check the following interpretation of [n,k,d]q-systems. Let 1> = {P l , · · · ,Pn } be [n,k,d] -system, P. e V . By H. e'R Hi) . Then ~ B. 26. Self-dual codes. iff C = CL exists a . ~ '" 0 C i Part 1 CODES 24 1, ... L . Here is called formally self-dual self-dual code if W C = W quasi-self-dual, is quasi-self-dual code is formally self-dual. L . Of and any 2 then any quasi-self-dual code is self-dual. 28. Let q = 2 or q = 3 . Show that the weights of all code vectors of a self-dual divisible by q-ary code are q.