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

By T. Ericson, V. Zinoviev

Codes on Euclidean spheres are frequently often called 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 thought, geometry, combinatorial idea, and - in fact - to algebraic coding concept. The connections to physics happen inside parts like crystallography and nuclear physics. In engineering round codes are of critical significance in reference to error-control in communique platforms. In that context using round codes is frequently often called "coded modulation.The ebook deals a primary entire remedy of the mathematical thought 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 through many examples. The ebook additionally comprises an in depth desk of top identified round codes in dimensions 3-24, together with specified buildings.

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Additional info for Codes on Euclidean Spheres

Example text

Then the following inequality holds where p = 2 — 2 cos 9. 1 the codewords in X can be surrounded by disjoint caps with angle 0/2. This fact implies Cn(ir) > M • Cn(9/2). The final result follows by observing that M is an integer. A spherical code with the property that it cannot be extended without decreasing the minimum distance is said to be maximal. Any maximal code X with minimal squared Euclidean distance p — 2 — 2 cos 9 must satisfy the relation where the overbar indicates set closure. By the sphere packing bound it is clear that maximal codes with parameters (n, p) exist for any dimension n > 2 and any value of the squared minimum distance p.

Let the orthonormal basis {vij(x} : 1 < j < r^} be given and let U £ On be an arbitrary orthogonal matrix. 2. Spherical polynomials. 33 the family {vij(xU} : 1 < j < r^} is another orthonormal basis in Harm(n,i). It follows that we have where V — [Vmj] is an Ti x TI orthogonal matrix. Using this observation it is easy to show that the sum is independent of the orthogonal matrix U and so depends only on the inner product ( x , y ) . Indeed, 17n is a so called 2-point homogeneous space. This means that to any two pairs x,y € On and x',y' £ fin such that (x, y) = (or', y'} there exists an orthogonal matrix U such that x' = xU and y' = yU.

Now suppose n and p are given. All Simplex codes Si with i < n and such that 2 + 2,/i > p generate lower bounds on An(p). For 2 < p < 4 we get This means that the combination of Rankin's first and second bounds actually gives the optimum performance. 4 that Rankin's first bound does not depend on the dimension n. e. if we consider the problem of finding the best configuration of M points on a sphere in any dimension—the solution is given by the Simplex code. The solution is unique up to equivalence.

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