Distance spectra of lattice-based codes (Q1310753)

The author defines the distance spectra of a lattice-based code \(X\) of length \(n\) as the set of numbers \(\{A_ m\}\) where \[ A_ m= \frac{\left\{\{x,x'\}\mid x,x' \in X,| x-x|^ 2 =m\right\}}{| X|}. \] He introduces the generating function of spectra as the formal sum \(A(s)= \sum_{m=0}^ \infty A_ m s^ m\), and writes the coefficients \(A_ m\) in another more usable geometric form. In the case of spherical codes a combinatorial formula is given for this quantity and some interesting concrete example for the calculation can be found. The last three paragraphs contain an analytic and asymptotic treatment of the distance spectra.