Peter Boyvalenkov

Upper bounds on the minimum distance of spherical codes

We use linear programming techniques to obtain new upper bounds on the maximal squared minimum distance of spherical codes with fixed cardinality. Functions Q/sub j/(n,s) are introduced with the property that Q/sub j/(n,s) m if and only if the Levenshtein bound L/sub m/(n,s) on A(n,s)=max{|W|:W is an (n,|W|,s) code} can be improved by a polynomial of degree at least m+1. General conditions on the existence of new bounds are presented. We prove that for fixed dimension n/spl ges/5 there exists a constant k=k(n) such that all Levenshtein bounds L/sub m/(n, s) for m/spl ges/2k-1 can be improved. An algorithm for obtaining new bounds is proposed and discussed.

Extremal polynomials for obtaining bounds for spherical codes and designs

We investigate two extremal problems for polynomials giving upper bounds for spherical codes and for polynomials giving lower bounds for spherical designs, respectively. We consider two basic properties of the solutions of these problems. Namely, we estimate from below the number of double zeros and find zero Gegenbauer coefficients of extremal polynomials. Our results allow us to search effectively for such solutions using a computer. The best polynomials we have obtained give substantial improvements in some cases on the previously known bounds for spherical codes and designs. Some examples are given in Section 6.

Necessary Conditions for Existence of Some Designs in Polynomial Metric Spaces

In this paper we consider designs in polynomial metric spaces with relatively small cardinalities (near to the classical bounds). We obtain restrictions on the distributions of the inner products of points of such designs. These conditions turn out to be strong enough to ensure obtaining nonexistence results already for the first open cases.

Nonexistence of Certain Spherical Designs of Odd Strengths and Cardinalities

Abstract. A spherical τ -design on Sn-1 is a finite set such that, for all polynomials f of degree at most τ , the average of f over the set is equal to the average of f over the sphere Sn-1 . In this paper we obtain some necessary conditions for the existence of designs of odd strengths and cardinalities. This gives nonexistence results in many cases. Asymptotically, we derive a bound which is better than the corresponding estimation ensured by the Delsarte—Goethals—Seidel bound. We consider in detail the strengths τ =3 and τ =5 and obtain further nonexistence results in these cases. When the nonexistence argument does not work, we obtain bounds on the minimum distance of such designs.

Uniqueness of the 120-point spherical 11-design in four dimensions

Abstract. We prove that on the Euclidean sphere S3 there exist a unique up to isometry 120-point spherical 11-design and a maximal (4, 120,

Nonexistence of certain symmetric spherical codes

cos(\pi/5)

On Maximal Codes in Polynominal Metric Spaces

)-code. Both these are nothing but copies of a famous regular polytope in four dimensions – the 600-cell.

On Maximal Spherical Codes I

A spherical 1-codeW is any finite subset of the unit sphere inn dimensionsSn−1, for whichd(u, v)≥1 for everyu, v fromW, u≠v. A spherical 1-code is symmetric ifu∈W implies −u∈W. The best upper bounds in the size of symmetric spherical codes onSn−1 were obtained in [1]. Here we obtain the same bounds by a similar method and improve these bounds forn=5, 10, 14 and 22.

On the extremality of the polynomials used for obtaining the best known upper bounds for the kissing numbers

We study the possibilities for attaining the best known universal linear programming bounds on the cardinality of codes in polynomial metric spaces (finite or infinite). We show that in many cases these bounds cannot be attained. Applications in different antipodal polynomial metric spaces are considered with special emphasis on the Euclidean sphere and the binary Hamming space.

New lower bounds for some spherical designs

We investigate the possibilities for attaining two Levenshtein upper bounds for spherical codes. We find the distance distributions of all codes meeting these bounds. Then we show that the fourth Levenshtein bound can be attained in some very special cases only. We prove that no codes with an irrational maximal scalar product meet the third Levenshtein bound. So in dimensions 3 ≤ n ≤ 100 exactly seven codes are known to attain this bound and ten cases remain undecided. Moreover, the first two codes (in dimensions 5 and 6) are unique up to isometry. Nonexistence of maximal codes in all dimensions n with cardinalities between 2n+1 and 2n+[7√n] is shown as well. We prove nonexistence of several infinite families of maximal codes whose maximal scalar product is rational. The distance distributions of the only known nontrivial infinite family of maximal codes (due to Levenshtein) are given.