By Dino Lorenzini
During this quantity the writer offers a unified presentation of a few of the elemental instruments and ideas in quantity idea, commutative algebra, and algebraic geometry, and for the 1st time in a booklet at this point, brings out the deep analogies among them. The geometric perspective is under pressure in the course of the publication. large examples are given to demonstrate every one new notion, and lots of attention-grabbing routines are given on the finish of every bankruptcy. lots of the vital leads to the one-dimensional case are proved, together with Bombieri's evidence of the Riemann speculation for curves over a finite box. whereas the publication isn't meant to be an creation to schemes, the writer exhibits what percentage of the geometric notions brought within the publication relate to schemes with a purpose to reduction the reader who is going to the following point of this wealthy topic
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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.
An invitation to arithmetic geometry by Dino Lorenzini