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 yes codes.- A.1.3. Parameters of yes 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. definite families.- A.4.3. Asymptotic results.- writer index.- record of symbols.

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Kl. were more than kl. + (do - 1) [n, k, d] q - + d ~ n . If (since the additional therefore + d = genus at most 1 (since i. e. ~ do' dl. 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.

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.

25 (see Exercise x 2 + (q - l)·y 2 are actually enumerators of self-dual codes (or of any codes) is very subtle. Let us present some other results (without proofs) . 31. y. (x (the same is valid for a formally self-dual all its weights self-dual 3 Y . (x 3 code are code 3 - y ) 3 such a is (the that even). is its valid weights 22 - y ) code such that of enumerator polynomial same all The 2 a 3-ary and in for a are formally divisible self-dual by The 3). enumerator of a 4-ary formally self-dual code such that all its X 2 .

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

by Ronald

4.4