On the distance distribution of duals of BCH codes

We derive upper bounds on the components of the distance distribution of duals of BCH codes. Roughly speaking, these bounds show that the distance distribution can be upper-bounded by the corresponding normal distribution. To derive the bounds we use the linear programming approach along with some estimates on the magnitude of Krawtchouk polynomials of fixed degree in a vicinity of q/

On spectra of BCH codes

Derives an estimate for the error term in the binomial approximation of spectra of BCH codes. This estimate asymptotically improves on the bounds by Sidelnikov (1971), Kasami et al. (1985), and Sole (1990)

Estimates for the range of binomiality in codes' spectra

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

On the accuracy of the binomial approximation to the distance distribution of codes

The binomial distribution is a well-known approximation to the distance spectra of many classes of codes. We derive a lower estimate for the deviation from the binomial approximatio

An improved upper bound on the minimum distance of doubly-even self-dual codes

We derive a new upper bound on the minimum distance d of doubly-even self-dual codes of length n. Asymptotically, for n growing, it gives limn→∞ sup d/n <=(5-5^3/4)/10 <0.165630, thus improving on the Mallows-Odlyzko-Sloane bound of 1/6 and our recent bound of 0.16631

A note on the edge-reconstruction of K1,m-free graphs

AbstractWe show that there exists an absolute constant c such that any K1,m-free graph with the maximum degree Δ > cm(log m)12 is edge reconstructible

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 constrast 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

On a Reconstruction Problem for Sequences

AbstractIt is shown that any word of lengthnis uniquely determined by all its[formula]subwords of lengthk, providedk⩾⌊167n⌋+5. This improves the boundk⩾⌊n/2⌋ given in B. Manvelet al.(Discrete Math.94(1991), 209–219)

Switching reconstruction and diophantine equations

AbstractBased on a result of R. P. Stanley (J. Combin. Theory Ser. B 38, 1985, 132–138) we show that for each s ≥ 4 there exists an integer Ns such that any graph with n > Ns vertices is reconstructible from the multiset of graphs obtained by switching of vertex subsets with s vertices, provided n ≠ 0 (mod 4) if s is odd. We also establish an analog of P. J. Kelly's lemma (Pacific J. Math., 1957, 961–968) for the above s-switching reconstruction problem