Ilia Krasikov

On Integral Zeros of Krawtchouk Polynomials

We derive new conditions for the nonexistence of integral zeros of binary Krawtchouk polynomials. Upper bounds for the number of integral roots of Krawtchouk polynomials are presented.

On spectra of BCH codes

Estimates for accuracy of binomial approximation to the spectra of BCH codes are derived. These estimates are better than known results.

Uniform bounds for Bessel functions

Abstract For ν > –1/2 and x real we shall establish explicit bounds for the Bessel function Jν (x) which are uniform in x and ν. This work and the recent result of L. J. Landau [J. Londom Math. Soc. 61: 197–215, 2000] provide relatively sharp inequalities for all real x.

Combinatorial reconstruction problems

Abstract A general technique for tackling various reconstruction problems is presented and applied to some old and some new instances of such problems.

Nonnegative Quadratic Forms and Bounds on Orthogonal Polynomials

We show that some nonnegative quadratic forms containing orthogonal polynomials, such as e.g. the Christoffel-Darboux kernel for x=y in the classical case, provide a lot of information about behavior of the polynomials on the real axis. We illustrate the method for the case of Hermite polynomials and use it to derive new explicit bounds for binary Krawtchouk polynomials.

On a Reconstruction Problem for Sequences

It is shown that any word of lengthnis uniquely determined by all itsformula]subwords of lengthk, providedk??167n?+5. This improves the boundk??n/2? given in B. Manvelet al.(Discrete Math.94(1991), 209?219).

Finding next-to-shortest paths in a graph

We study the problem of finding the next-to-shortest paths in a graph. A next-to-shortest (u,v)-path is a shortest (u,v)-path amongst (u,v)-paths with length strictly greater than the length of the shortest (u,v)-path. In contrast to the situation in directed graphs, where the problem has been shown to be NP-hard, providing edges of length zero are allowed, we prove the somewhat surprising result that there is a polynomial time algorithm for the undirected version of the problem.

Estimates for the range of binomiality in codes' spectra

We derive new estimates for the range of binomiality in a codes spectra, where the distance distribution of a code is upperbounded by the corresponding normalized binomial distribution. The estimates depend on the codes dual distance.

Approximations for the Bessel and Airy functions with an explicit error term

We show how one can obtain an asymptotic expression for some special functions with a very explicit error term starting from appropriate upper bounds. We will work out the details for the Bessel function Jν(x) and the Airy function Ai(x). In particular, we answer a question raised by Olenko and find a sharp bound on the difference between Jν(x) and its standard asymptotics. We also give a very simple and surprisingly precise approximation for the zeros Ai(x).

Bounds on Spectra of Codes with Known Dual Distance

We estimate the interval where the distance distribution of a code of length n and of given dual distance is upperbounded by the binomial distribution. The binomial upper bound is shown to be sharp in this range in the sense that for every subinterval of size about √n ln n there exists a spectrum component asymptotically achieving the binomial bound. For self-dual codes we give a better estimate for the interval of binomiality.