Genuinely multipartite entangled states and orthogonal arrays

A pure quantum state of N subsystems with d levels each is called k-multipartite maximally entangled state, written k-uniform, if all its reductions to k qudits are maximally mixed. These states form a natural generalization of N-qudits GHZ states which belong to the class 1-uniform states. We establish a link between the combinatorial notion of orthogonal arrays and k-uniform states and prove the existence of several new classes of such states for N-qudit systems. In particular, known Hadamard matrices allow us to explicitly construct 2-uniform states for an arbitrary number of N>5 qubits. We show that finding a different class of 2-uniform states would imply the Hadamard conjecture, so the full classification of 2-uniform states seems to be currently out of reach. Additionally, single vectors of another class of 2-uniform states are one-to-one related to maximal sets of mutually unbiased bases. Furthermore, we establish links between existence of k-uniform states, classical and quantum error correction codes and provide a novel graph representation for such states.Comment: 24 pages, 7 figures. Comments are very welcome

Binary doubly-even self-dual codes of length 72 with large automorphism groups

We study binary linear codes constructed from fifty-four Hadamard 2-(71,35,17) designs. The constructed codes are self-dual, doubly-even and self-complementary. Since most of these codes have large automorphism groups, they are suitable for permutation decoding. Therefore we study PD-sets of the obtained codes. We also discuss error-correcting capability of the obtained codes by majority logic decoding. Further, we describe a construction of a strongly regular graph with parameters (126,25,8,4) from a binary [35,8,4] code related to a derived 2-(35,17,16) design

Higher-order CIS codes

We introduce {\bf complementary information set codes} of higher-order. A binary linear code of length $tk$ and dimension $k$ is called a complementary information set code of order $t$ ($t$-CIS code for short) if it has $t$ pairwise disjoint information sets. The duals of such codes permit to reduce the cost of masking cryptographic algorithms against side-channel attacks. As in the case of codes for error correction, given the length and the dimension of a $t$-CIS code, we look for the highest possible minimum distance. In this paper, this new class of codes is investigated. The existence of good long CIS codes of order $3$ is derived by a counting argument. General constructions based on cyclic and quasi-cyclic codes and on the building up construction are given. A formula similar to a mass formula is given. A classification of 3-CIS codes of length $\le 12$ is given. Nonlinear codes better than linear codes are derived by taking binary images of $\Z_4$-codes. A general algorithm based on Edmonds' basis packing algorithm from matroid theory is developed with the following property: given a binary linear code of rate $1/t$ it either provides $t$ disjoint information sets or proves that the code is not $t$-CIS. Using this algorithm, all optimal or best known $[tk, k]$ codes where $t=3, 4, \dots, 256$ and $1 \le k \le \lfloor 256/t \rfloor$ are shown to be $t$-CIS for all such $k$ and $t$, except for $t=3$ with $k=44$ and $t=4$ with $k=37$.Comment: 13 pages; 1 figur

Design and Implementation of an Economy Plane for the Internet

The Internet has been very successful in supporting many network applications. As the diversity of uses for the Internet has increased, many protocols and services have been developed by the industry and the research community. However, many of them failed to get deployed in the Internet. One challenge of deploying these novel ideas in operational network is that the network providers need to be involved in the process. Many novel network protocols and services, like multicast and end-to-end QoS, need the support from network providers. However, since network providers are typically driven by business reasons, if they can not get economic profit from supporting new protocols and services, they will not deploy them. Therefore, we conclude that the lack of explicit economic relationship in the current Internet hinders the innovation of itself, and it is critical that a network architecture intrinsically considers economic relationships. ChoiceNet is an NSF funded Future Internet Architecture (FIA) project that aims to address these challenges. ChoiceNet proposes an ``economy plane\u27\u27 of the Internet to explicitly represent economic relationship within the architecture. This economy plane enables entities in the network to dynamically set up fine-grained, short-term economic contracts for network services. A marketplace can be established for advertising and selling services. The services can be simple path services ( pathlets ) between end-points, or more complex processing and storage services (e.g., transcoding and caching). ChoiceNet is a comprehensive project, and its architecture is designed by researchers from several institutes. This work will not cover every aspect of it. Instead, this work will focus on five aspects of ChoiceNet: 1) service definition and protocol design, 2) marketplace design, 3) use plane design, 4) path finding algorithm design, and 5) access control for services. Service definition aims at a unified and extensible description of services, and the method to compose them. Marketplace design discusses the protocols used to advertise and request services. The use plane design describes how network providers and users will access the Marketplace while preserving the existing infrastructure and applications, it also discusses how to progressively deploy ChoiceNet in the current Internet. The path finding algorithm design proposes ParetoBFS, an algorithm finding all the Pareto-optimal paths in a multi-criteria network. The access control discusses how to prevent unauthorized usage of the services, we present OrthCredential, an algorithm for high-performance access control in ChoiceNet. To prove the feasibility of such an economy plane, this work presents a Software Defined Networking (SDN) based implementation of ChoiceNet. The implementation has been deployed and tested on GENI, a global test bed for network architectures. By designing and implementing ChoiceNet, this work tries to offer a network architecture that users can select from several different network services rather than being limited to a single choice. By enabling greater choice, ChoiceNet can promote competition among providers for price and quality. This competition will lead to lower prices and higher quality services, which are beneficial for consumers and eventually help bring sustained innovation into the Internet