Applications of finite geometry in coding theory and cryptography

We present in this article the basic properties of projective geometry, coding theory, and cryptography, and show how finite geometry can contribute to coding theory and cryptography. In this way, we show links between three research areas, and in particular, show that finite geometry is not only interesting from a pure mathematical point of view, but also of interest for applications. We concentrate on introducing the basic concepts of these three research areas and give standard references for all these three research areas. We also mention particular results involving ideas from finite geometry, and particular results in cryptography involving ideas from coding theory

Xing-Ling Codes, Duals of their Subcodes, and Good Asymmetric Quantum Codes

A class of powerful $q$-ary linear polynomial codes originally proposed by Xing and Ling is deployed to construct good asymmetric quantum codes via the standard CSS construction. Our quantum codes are $q$-ary block codes that encode $k$ qudits of quantum information into $n$ qudits and correct up to \flr{(d_{x}-1)/2} bit-flip errors and up to \flr{(d_{z}-1)/2} phase-flip errors.. In many cases where the length $(q^{2}-q)/2 \leq n \leq (q^{2}+q)/2$ and the field size $q$ are fixed and for chosen values of $d_{x} \in \{2,3,4,5\}$ and $d_{z} \ge \delta$, where $\delta$ is the designed distance of the Xing-Ling (XL) codes, the derived pure $q$-ary asymmetric quantum CSS codes possess the best possible size given the current state of the art knowledge on the best classical linear block codes.Comment: To appear in Designs, Codes and Cryptography (accepted Sep. 27, 2013

Assessing the visual aspect of rotating virtual rose bushes by a labeled sorting task

Aesthetics is one of the major parameters for consumers when buying a rose bush. Therefore, managing this quality is important for agronomists. Tools are needed to assess visual characteristics and to find links with architectural plant parameters. Sensory analyses were developed using real plants and photographs as stimuli. With technology and modeling improvements, using virtual plants could presents numerous advantages. This study demonstrated the feasibility of using rotating virtual rose bush videos as stimuli for a labeled sorting task. The virtual rose bush reflected a natural within-crop variability of one cultivar based on bud breaks location and axes length. Two panels of subjects closely linked to the horticulture sector sorted and described 40 rotating virtual rose bush videos. Non-metric Multidimensional Scaling (MDS) results for both panels were similar and allowed us to highlight five groups of virtual rose bushes with their specific sensory characteristics and their own most representative products using a combination of the paragons and the most typical products. This approach revealed that subjects detected high visual differences between products, and that by using rotation, they were able to integrate 3D properties about variations around plant facets. Finally, a labeled sorting task is a powerful method for preliminary exploration of the visual aspect of virtual plants

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization