In 1968, Golomb and Welch conjectured that there does not exist perfect Lee code in Z with radius r ≥ 2 and dimension n ≥ 3. Besides its own interest in coding theory and discrete geometry, this conjecture is also strongly related to the degree-diameter problems of abelian Cayley graphs. Although there are many papers on this topic, the Golomb-Welch conjecture is far from being solved. In this ...

This article presents a simple algorithm for packing sparse 2-D arrays into minimal I-D arrays in O(r?) time. Retrieving an element from the packed I-D array is O(l). This packing algorithm is then applied to create minimal perfect hashing functions for large word lists. Many existing perfect hashing algorithms process large word lists by segmenting them into several smaller lists. The perfect ...

Our study of perfect spline approximation reveals: (i) it is closely related to Σ∆ modulation used in one-bit quantization of bandlimited signals. In fact, they share the same recursive formulae, although in different contexts; (ii) the best rate of approximation by perfect splines of order r with equidistant knots of mesh size h is hr−1. This rate is optimal in the sense that a function can be...

Let H = (V,E) be an r-uniform hypergraph and let F ⊂ 2 . A matching M of H is (α,F)perfect if for each F ∈ F , at least α|F | vertices of F are covered by M . Our main result is a theorem giving sufficient conditions for an r-uniform hypergraph to have a (1− ,F)-perfect matching. As a special case of our theorem we obtain the following result. Let K(n, r) denote the complete r-uniform hypergrap...

CCC-r control chart is a monitoring technique for high yield processes. It is based on the analysis of the number of inspected items until observing a specific number of defective items.  One of the assumptions in implementing CCC-r chart that has a significant effect on the design of the control chart is that the inspection is perfect. However, in reality, due to the multiple reasons, the...

A binary extended 1-perfect code C folds over its kernel via the Steiner quadruple systems associated with its codewords. The resulting folding, proposed as a graph invariant for C, distinguishes among the 361 nonlinear codes C of kernel dimension κ obtained via Solov’eva-Phelps doubling construction, where 9 ≥ κ ≥ 5. Each of the 361 resulting graphs has most of its nonloop edges expressible in...