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. mark downs 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 definite 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. convinced families.- A.4.3. Asymptotic results.- writer index.- record of symbols.
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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)? Of course, o s k s n , 1 s d s n which . 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) .
Is called an A projective en, + 1]q-S y stem k, n - k n-set. The corresponding code C is called a maximal distance separable code (an HDS-code) . 2 we give some examples of such codes, namely, line; their length Reed-Solomon n ~ codes find the their exists an maximum the proj ective q + 1 . We get a very interesting problem: q on possible length For given mq(k) k and for which [mq(k) , k, mq(k) + 1 - k]q-code. e. 12 (the Prove that for 2 s. k < = . k + 1 main mq(k) s. q + k - 2 conjecture on MDS-codes).
0 ): 0 H are ?! (j) ·Xi for i for j = d, d + 1, ... 1, ... 50 given set of non-negative real numbers for any For a (the linear programming bound). al, ... ,a n such that j = d, d + 1, ... 45. At last we come to an existence theorem. 51 (the Gilbert-Varshamov bound). Whenever q n-k there exists a linear < [n,k,d]q-code c Proof: We are going to construct a dual [n,k,d]qsystem Q c W, we take any dim W = n - k For non-zero vector. Suppose that we have already constructed a system consisting of i vectors Ql' ••• , Q i such that any (d - 1) of them are linearly independent.
Algebraic-Geometric Codes by M. Tsfasman, S.G. Vladut