# Algebraic-Geometric Codes (Mathematics and its Applications) by M. Tsfasman, S.G. Vladut

By M. Tsfasman, S.G. Vladut

'Et moi, ..., si j'avait su remark en revenir, One provider arithmetic has rendered the je n'y serais aspect aIle.' human race. It has placed good judgment again Jules Verne the place it belongs, at the topmost shelf subsequent to the dusty canister labelled 'discarded non sense'. The sequence is divergent; hence we are able to do anything with it. Eric T. Bell O. Heaviside arithmetic is a device for idea. A hugely useful software in an international the place either suggestions and non linearities abound. equally, every kind of components of arithmetic function instruments for different components and for different sciences. using an easy rewriting rule to the quote at the correct above one reveals such statements as: 'One carrier topology has rendered mathematical physics .. .'; 'One provider good judgment has rendered com puter technology .. .'; 'One carrier classification conception has rendered arithmetic .. .'. All arguably real. And all statements available this manner shape a part of the raison d' etre of this sequence.

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Example text

47 . 44 from Lemma • The next idea is to use spectra. Let C be a linear code. 15 we know that W Cl. (x: y) q -k x - 'Wc(x + (q - l)y y) or, in terms of coefficients: q -k A'. (x) Note that n P. (q Pi(x) - l)i . The generating function of is L i=O P. 1 Since are the A'. ~ coefficients of non-negative integers, i,e. I. (x:y) C i A). Pi (j) ?! e. • p (j) ?! 0 J for (j) . (q ) 1 + n E x. ~ ?! 0 (note that A.?! 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).

Dl. = n with C dl. k + dl. 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.

Qk-i - 1 )}. • 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.

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