Read e-book online Algebraic-Geometric Codes PDF

By M. Tsfasman, S.G. Vladut

ISBN-10: 1402003358

ISBN-13: 9781402003356

ISBN-10: 9401138109

ISBN-13: 9789401138109

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.

Show description

Read or Download Algebraic-Geometric Codes PDF

Best algebraic geometry books

Iterated Integrals and Cycles on Algebra by Bruno Harris PDF

This topic has been of significant curiosity either to topologists and to quantity theorists. the 1st a part of this publication describes a number of the paintings of Kuo-Tsai Chen on iterated integrals and the basic staff of a manifold. the writer makes an attempt to make his exposition obtainable to starting graduate scholars.

Download e-book for kindle: Ramanujan's Lost Notebook: Part IV by George E. Andrews, Bruce C. Berndt

​​​​In the spring of 1976, George Andrews of Pennsylvania kingdom collage visited the library at Trinity collage, Cambridge, to envision the papers of the past due G. N. Watson. between those papers, Andrews came upon a sheaf of 138 pages within the handwriting of Srinivasa Ramanujan. This manuscript was once quickly detailed, "Ramanujan's misplaced laptop.

Additional resources for Algebraic-Geometric Codes

Example text

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.

Download PDF sample

Algebraic-Geometric Codes by M. Tsfasman, S.G. Vladut

by Joseph

Rated 5.00 of 5 – based on 25 votes