Weighted coverings and packings

On Integral Zeros of Krawtchouk Polynomials

“…In Problem 2 only the even roots are relevant since the switching of the subset of vertices coincides with the switching of the complement subset. For perfect multiple coverings the presented condition is a particular case of a more general theorem in [8].…”

“…If P n s (x) has less than N integer roots then there is no perfect +-fold covering of radius s in F n+1 (see [8]), where N is the minimum integer such that Regarding the reconstruction Problems 2 4 note that the, so called, balance equations (see [22]) for all the three problems are the same. The graphs are reconstructible if (but not only if ) those equations have a unique solution.…”

“…Namely, they are problems of existence of perfect L-codes [11,21], and perfect weighted coverings [8,12]. In these cases we are interested in integral roots of linear combinations of Krawtchouk polynomials of different degrees.…”

Integer Zeros ofq-Krawtchouk Polynomials in Classical Combinatorics

“…Note that we do not need q to be a prime power, a condition that is often required. This proof can easily adapted to perfect weighted coverings [3]. Let m = m 0 m n a n + 1 -uple of nonnegative rational numbers, with m n = 0.…”

Constant 2-labellings and an application to (r,a,b)-covering codes

“…The (r, a, b)-codes have already been studied in some graphs under the name of weighted covering codes by Cohen et al [4]. Their work corresponds to a study of these codes in the Hamming metric.…”