By John Bell, Moshe Machover

A complete one-year graduate (or complicated undergraduate) direction in mathematical good judgment and foundations of arithmetic. No past wisdom of common sense is needed; the e-book is appropriate for self-study. Many routines (with tricks) are included.

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**Example text**

9Which codes are normal? What are possible extensions of the notion of normality and how can they be applied to construct good coverings? 9What is the maximal covering radius of a linear code with given length and size? 9What connections are there between covering radius and minimum distance? Is it possible to construct codes that at the same time have small covering radius and large minimum distance? 9W h a t is the covering radius of the well-known families of error-correcting codes? 9What is the asymptotic behaviour of the parameters of the best coverings?

Let x - (x 1 , . . , x~ + . . + x , ~2 _ 1. We pick K codewords that are in a sense uniformly distributed on the surface, and represent any x by the index of the closest codeword (in euclidean distance). One thus uses log 2 K bits to store (with some distortion) the coordinates of an arbitrary 14 Chapter 1. Introduction point on the sphere. ,c,~) by ( ( - 1 ) e ' / v / n , . . , (-1)e~/v/'n). 9 C e l l u l a r t e l e c o m m u n i c a t i o n s : The following problem originates in cellular telecommunication technology.

IF~, and d(r r d(c~, r all i and j. Then [B~(e~) n B~(e2) n Bt(e3)[ >_ [B~(e~) n B~(e~) n Bt(eg)l. ~. Hamming spheres 35 P r o o f . First consider the case d(e~, c~) d(el, e2), d(c~, c~) - d(e2, c3) and d(e~,c~) - d(cl, e 3 ) + 2 . We may assume that cl = e~ and e2 - e~. 0. Because there also exists a vector e~ of weight w ( c 3 ) + 2 such that d(c~, e2) d(c3, c2), we know that c~ has exactly one more 1 in common with c2 than c3 and therefore there is a coordinate (the first coordinate in the figure) in which c2 and c3 both have 0 and a coordinate (the second coordinate in the figure) in which c2 has 1 and c3 has 0.