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

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We discuss them in detail, calculate the spectra, and give a decoding algorithm. 2 we briefly discuss other interesting families of codes. 3 is devoted to a number of rather simple constructions which produce new codes starting with some codes we already know. 1. L (n - k) + d = n + 1 Le. 18). Trivial codes. For any there n are three simple q-ary codes which it is naturally to call trivial. are: [n , n , 1] q -code C1 = IF~ , [n,n - 1,2] q -code C 2 = {(v 1 , ... ,vn ) e IFnq I LVi~ (called the parity-check code), and [n , 1 , n ] q -code These C 3 = {v = ( ex, ••• , ex) e O} IF~} , ex elf q (called the repetition code).

4. There a is ence between the set of classes of set of linear [n,k,d]q-COdes. Sketch of proof: on V , (V * ) * rp : V * ~IF~ . rpi(Q) = Q(P i ) V* Let one-to-one [n,k,d]q-systems and the be the space of linear forms V • Let 'P = (P1 , · · · ,Pn) defined by This map rp(Q) = . 5. particular that the parameters do coincide. Projective systems. object. e. the space ~(V) linear space over V) IFq . [n,k,d]q-system is a finite unordered family of ~ which (note that we write does I 'P I ~ n = I'PI, dim 'P c multiplicities).

Establish this formula for Solomon codes by a direct computation. Check that for Reedi * 0 Decoding. Consider an [n, n - a - 1, a + 2]q-code C dual to a Reed-Solomon code of degree a. Recall that fd ; decoding up to t = 11 means an algorithm it possible, starting with some v E F~ which is distance t from some code vector u e C, to u. Let is called v - u = e, lIeU:s t , e vector . that makes at most at find this the error CODES 42 v e ~n . ·p~ 1. 1. o Pie'P s j s 2t - 1 . I= {i Ie ... O} e . p~ , where is the 1.

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

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