Families of nested completely regular codes and distance-regular graphs

In this paper infinite families of linear binary nested completely regular codes are constructed. They have covering radius $\rho$ equal to $3$ or $4$, and are $1/2^i$-th parts, for $i\in\{1,\ldots,u\}$ of binary (respectively, extended binary) Hamming codes of length $n=2^m-1$ (respectively, $2^m$), where $m=2u$. In the usual way, i.e., as coset graphs, infinite families of embedded distance-regular coset graphs of diameter $D$ equal to $3$ or $4$ are constructed. In some cases, the constructed codes are also completely transitive codes and the corresponding coset graphs are distance-transitive

On the number of nonequivalent propelinear extended perfect codes

The paper proves that there exist an exponential number of nonequivalent propelinear extended perfect binary codes of length growing to infinity. Specifically, it is proved that all transitive extended perfect binary codes found by Potapov are propelinear. All such codes have small rank, which is one more than the rank of the extended Hamming code of the same length. We investigate the properties of these codes and show that any of them has a normalized propelinear representation

A Fair and Secure Cluster Formation Process for Ad Hoc Networks

An efficient approach for organizing large ad hoc networks is to divide the nodes into multiple clusters and designate, for each cluster, a clusterhead which is responsible for holding intercluster control information. The role of a clusterhead entails rights and duties. On the one hand, it has a dominant position in front of the others because it manages the connectivity and has access to other node¿s sensitive information. But on the other hand, the clusterhead role also has some associated costs. Hence, in order to prevent malicious nodes from taking control of the group in a fraudulent way and avoid selfish attacks from suitable nodes, the clusterhead needs to be elected in a secure way. In this paper we present a novel solution that guarantees the clusterhead is elected in a cheat-proof manner