Indexes of spherical codes (Q2717194)

Let \({\mathbf S}^{n-1}= \{x= (x_1,x_2,\dots, x_n)\mid x^2_1+ x^2_2+\cdots +x^2_n= 1\}\) be the \(n\)-dimensional Euclidean sphere with the usual inner product \([x,y]= \sum^n_{i=1} x_iy_i\) where \(x= (x_1,x_2,\dots, x_n)\) and \(y= (y_1, y_2,\dots, y_n)\). A finite non-empty set \(C\subset{\mathbf S}^{n-1}\) is said to be an \((n,M,s)\) spherical code, \(M= |C|\), if the maximal inner product of any two different vectors from \(C\) does not exceed \(s\). A spherical code \(C\) is called antipodal if \(C=-C\). A spherical code \(C\subset{\mathbf S}^{n-1}\) is called a spherical \(\gamma\)-design if and only if the formula for the numerical integration NEWLINE\[NEWLINE\int_{{\mathbf S}^{n-1}} f(x) d\mu(x)={1\over|C|} \sum_{x\in C} f(x)NEWLINE\]NEWLINE (\(\mu\) is the Lebesgue measure and \(\mu({\mathbf S}^{n-1})= 1\)) is true for all real polynomials \(f(x_1,x_2,\dots, x_n)\) of the total degrees \(1,2,\dots,\gamma\). A spherical code \(C\subset{\mathbf S}^{n-1}\) is said to have an index \(k\) if and only if the formula above holds for all real homogeneous harmonic polynomials \(f(x)= f(x_1,x_2,\dots, x_n)\) of total degree \(k\). Actually both sides of the formula then are equal to zero. It is clear that any spherical \(\gamma\)-design has indexes \(1,2,\dots,\gamma\), and that any antipodal spherical code has the odd numbers as indexes. Such indexes are called trivial.NEWLINENEWLINENEWLINEIn this paper the authors study nontrivial indexes of certain good (in most of the cases maximal) spherical codes. They described three methods for finding indexes, while giving applications to some known optimal spherical codes. The authors investigate some interesting spherical codes. Among them the most interesting results show that the icosahedron (which is a 5-design and an antipodal code) has non-trivial indexes 8 and 14 and both the 600-cell and the 120-cell have non-trivial indexes 14, 16, 18, 22, 26, 28, 34, 38, 46, and 56. Thus they have characterized some good spherical codes by finding their indexes.NEWLINENEWLINENEWLINEIn the end an elementary approach for upper-bounding nontrivial indexes of concrete codes has been explained. Lastly all the results have been summarized in a tabular form.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00079].