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. 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 convinced 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. yes families.- A.4.3. Asymptotic results.- writer index.- checklist of symbols.

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**Example text**

1P' =H This ~' = 'P n Ho C 1P' • 0 en' , k' 'd'] -system where n' =n d , q , H' be a hyperplane (of dimension k - 3 I 'P' n H'I = n' - d' There are (q + 1) k-l of IP passing through H' and Therefore Set (q + 1)-(n - d) q+l ~ = L i=1 IHi n ~I IV n + q- (n - d - d') n 'PI , is a projective k' = k - 1 Let in IP' such that H_ hyperplanes IHi n 'PI + q-IH' :5. 1. :: 1 ..!!. q r . Iterating [n (k) ,0, d (k)] -system q The condition k-1 n (k) = n - d - d' - ... · q~ i=O r • The following bounds we prove for any codes.

L L + B '.. (x - i=O ~ q - k . J aLB .. q. i (x - 1) n-~ i=O ~ = x-I we get Hence for B { . n-~ i. e. L + 1 and a - g + 1 k n - k + g - 1 = n - a + 2g - 2 ): B, J { (j)' (qa- j - g +1 _ 1) for j==a-2g+1 (j)' (qa- j - g +1 _ 1) + B' ,'qa-J'-g+l n-J for a-2g+2==j==a . The lower bound and the equality are proved. 27) . 17. e. that in and this case the code is of genus 0 (an MDS-code) . 18. 19. Consider [ 4 , 2 , 1] 2 -code C generated by the matrix (~ The dual code has the o 0 1 0 same ~) parameters.

42 6 in terms of B B' 3 2 2 ) . 15, recall that an additive character of homomorphism X ~q from the additive group of 3 B. = 1. 23. 22. p E aelF Proof: x(a'b) __ { q Let for b for b qO ao choose b '" 0, * ° ° such that X (a 'b) '" 1. o Then x(a 'b)'E o aelF x(a·b) q since the shift by If of 1. b = 0 U'V e IF q E aelF q X( (a + a) ·b) a o maps then always Fix a = x(a·b) E 0 a'elF IFq onto itself = 1 . non-trivial character x(a' 'b) q bijectively. • Xl For be their inner product. p]-mOdule. 1 L Ter nee necl.

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

by Joseph

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