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. 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 sure 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.- record of symbols.
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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). q Then arbitrary let "P S; us IF q find the dual code for the case of an . Set n P . e"P (x - Pi) -1 , ~ Cons ider the vector space n (a) spanned by the functions CODES 40 ge (x) F (x) Part 1 e ~ n - a - 2. Consider a function = f (x) . go (x) , f (x) being a polynomial. Recall the definition of the residue of F at Pi for 0 ~ Res p F = f ( P . ) . n (P. 3. e'P ~ Res'P Prove that for (the residue formula).
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. Note that for formally self-dual codes this statement (1,0) E 1F2 is wrong: a [2,1,l]2-code spanned by the vector 2 is formally self-dual.
3. Spectra and duality An important invariant of a code is its weight enumerator or spectrum. 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.