3:00pm - 5:00pmCoding theory and cryptography
Chair(s): Alessio Caminata (University of Neuchâtel, Switzerland), Alberto Ravagnani (University College Dublin, Ireland)
The focus of this proposal is on coding theory and cryptography, with emphasis on the algebraic aspects of these two research fields. Error-correcting codes are mathematical objects that allow reliable communications over noisy/lossy/adversarial channels. Constructing good codes and designing efficient decoding algorithms for them often reduces to solving algebra problems, such as counting rational points on curves, solving equations, and classifying finite rings and modules. Cryptosystems can be roughly defined as functions that are easy to evaluate, but whose inverse is difficult to compute in practice. These functions are in general constructed using algebraic objects and tools, such as polynomials, algebraic varieties, and groups. The security of the resulting cryptosystem heavily relies on the mathematical properties of these. The sessions we propose feature experts of algebraic methods in coding theory and cryptography. All levels of experience are represented, from junior to very experienced researchers.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
New results on graph-based codes
Christine Kelley
University of Nebraska-Lincoln
Codes over graphs have become widespread in industry applications due to their excellent performance with low-complexity decoders. Since long block lengths are desirable in practice, constructing codes using lifts of well-designed base graphs has become a standard technique. In this talk, we will present some recent results on how the permutations chosen affect the parameters of the resulting codes.
Large constant dimension subspace codes consisting of k-dimensional subspaces, pairwise intersecting in at least (k-2)-dimensional subspaces
Leo Storme
Ghent University
Within the theory of subspace codes, a constant dimension code C is a set of k-dimensional subspaces in the vector space V(n,q) of dimension n over the finite field of order q.
One of the goals in the theory of subspace codes is to characterize large subspace codes, satisfying specific conditions, such as intersection conditions or lower bounds on the minimum distance.
There are two types of constant dimension codes consisting of k-dimensional subspaces, pairwise intersecting in (k-1)-dimensional subspaces. They are either: (1) a sunflower: a set of k-dimensional subspaces passing through a common (k-1)-dimensional subspace, or (2) a set of k-dimensional subspaces lying in a common (k+1)-dimensional subspace.
The next step would be to investigate the largest sets of k-dimensional subspaces in the vector space V(n,q), pairwise intersecting in exactly
(k-2)-dimensional subspaces, or pairwise intersecting in at least (k-2)-dimensional subspaces.
In this talk, we present classification results on the largest examples of sets of k-dimensional subspaces, pairwise intersecting in exactly (k-2)-dimensional subspaces, or pairwise intersecting in at least (k-2)-dimensional subspaces.
These classification results are obtained via geometrical techniques in the corresponding (n-1)-dimensional projective space PG(n-1,q) corresponding to the n-dimensional vector space V(n,q).
Algebraic properties of codes with symmetries
Martino Borello
Université Paris 8 - LAGA
We will illustrate some new results and properties of codes with symmetries. Whenever a linear code over K has a non-trivial group of (permutation) automorphisms G, it can be viewed as a KG-module. Many well-studied families of codes are characterized by this property: cyclic, quasi-cyclic, abelian, quasi-abelian, group codes, etc. We will show how the algebraic structure of these codes allow to deduce properties on their parameters and to construct optimal codes. Moreover, we will show new asymptotic results for group codes in odd characteristic.
Quantum codes coming from J-affine variety codes
Carlos Galindo
Universidad Jaume I
We will introduce J-affine variety codes and we will give conditions for their self-orthogonality with respect to Euclidean and Hermitian inner products. Parameters of stabilizer codes coming from subfield-subcodes of J-affine variety codes will be showed. Many of these codes turn to exceed the known Gilbert-Varshamov bounds and improve some quantum codes given in the literature. Finally, we will show how to use hyperbolic codes to provide stabilizer codes with designed distance.