Codes on Euclidean Spheres by T. Ericson, V. Zinoviev

By T. Ericson, V. Zinoviev

Codes on Euclidean spheres are usually known as round codes. they're of curiosity from mathematical, actual and engineering issues of view. Mathematically the subject belongs to the area of algebraic combinatorics, with shut connections to quantity conception, geometry, combinatorial conception, and - in fact - to algebraic coding thought. The connections to physics ensue inside of components like crystallography and nuclear physics. In engineering round codes are of primary significance in reference to error-control in conversation platforms. In that context using round codes is usually often called "coded modulation.The booklet deals a primary entire remedy of the mathematical idea of codes on Euclidean spheres. Many new effects are released right here for the 1st time. Engineering functions are emphasised in the course of the textual content. the idea is illustrated by means of many examples. The e-book additionally includes an intensive desk of top identified round codes in dimensions 3-24, together with distinctive buildings.

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In the same way it follows that any point in Sn(y,t] is outside Sn(x,t). It follows that Sn(x,t) and Sn(y,t] are disjoint. Next suppose that Sn(x,t) and Sn(y,i) are disjoint. Let z be of the form z = Ox + (1 — 0)y, where 0 is chosen so that \\x — z\\ = t. We have and as Sn(x,t) and Sn(y,t) are disjoint we also have \\z — y\\ > t. The inequality \\x — y\\ > 2t follows. An immediate consequence of this lemma is the so called sphere packing bound. It can be stated as follows. 6. The Chabauty-Shannon-Wyner bound.

4. The Rankin bounds. 1: Notice that the bound does not depend on the dimension n. 2: Alternatively the result can also be stated as an inequality for the function p n (M), namely In terms of the inverse function the same result takes the following form Proof: Let X — { x i , X 2 , - . • , #M} De a spherical code with parameters (n,p, M). For i ^ j we have (x^Xj} < 1 — p/2. From this observation we obtain. On the other hand we also have Combining these two inequalities we obtain the result by a simple reordering of the terms.

5. The Levenshtein bound. 2 Let f ( t ) be a polynomial of degree at most 2fc -f-1. , k. ,/- a), and so where As we have already noticed the coefficients pm—which of course depend on the inner product—are always positive. A slight variation of this formula will be very useful in the study of the Levenshtein bound. 5 The Levenshtein bound The key problem in applying the linear programming bound is to find an efficient polynomial f ( t ) = /o + fiQi(t) -\ h fkQk(t] satisfying the following two properties: 48 Chapter 2.

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