An improved lower bound for (1,<=2)-identifying codes in the king grid

We call a subset $C$ of vertices of a graph $G$ a $(1,\leq \ell)$-identifying code if for all subsets $X$ of vertices with size at most $\ell$, the sets $\{c\in C |\exists u \in X, d(u,c)\leq 1\}$ are distinct. The concept of identifying codes was introduced in 1998 by Karpovsky, Chakrabarty and Levitin. Identifying codes have been studied in various grids. In particular, it has been shown that there exists a $(1,\leq 2)$-identifying code in the king grid with density 3/7 and that there are no such identifying codes with density smaller than 5/12. Using a suitable frame and a discharging procedure, we improve the lower bound by showing that any $(1,\leq 2)$-identifying code of the king grid has density at least 47/111

Generalized iterated wreath products of symmetric groups and generalized rooted trees correspondence

Consider the generalized iterated wreath product $S_{r_1}\wr \ldots \wr S_{r_k}$ of symmetric groups. We give a complete description of the traversal for the generalized iterated wreath product. We also prove an existence of a bijection between the equivalence classes of ordinary irreducible representations of the generalized iterated wreath product and orbits of labels on certain rooted trees. We find a recursion for the number of these labels and the degrees of irreducible representations of the generalized iterated wreath product. Finally, we give rough upper bound estimates for fast Fourier transforms.Comment: 18 pages, to appear in Advances in the Mathematical Sciences. arXiv admin note: text overlap with arXiv:1409.060

An approach for designing on-line testable state machines

Synthesis of state machines have attracted the attention of researchers for more than two decades. Several state assignment techniques that result in efficient implementation of the next state logic have been developed [1-3]. However, none of these addresses the testability of an implemented machine. A popular approach for enhancing the testabilit

Robust Residue Codes for Fault-Tolerant Public-Key Arithmetic

We present a scheme for robust multi-precision arithmetic over the positive integers, protected by a novel family of non-linear arithmetic residue codes. These codes have a very high probability of detecting arbitrary errors of any weight. Our scheme lends itself well for straightforward implementation of standard modular multiplication techniques, i.e. Montgomery or Barrett Multiplication, secure against active fault injection attacks. Due to the non-linearity of the code the probability of successfully injecting an error does not depend on the error pattern itself, but also on the data, which is not known to the adversary a priori. We give a proof of the robustness of these codes by providing an upper bound on the number of undetectable errors.