On November 14th a workshop was held at DTU-compute, Technical University of Denmark. The workshop was financed by the research project "How secret is a secret?" under the Danish Council for Independent Research - Natural Sciences.
The program of the conference was based on a series of talks that gave rise to fruitful research discussions:
- 10:30-11:00 Ignacio Cascudo: "Some results on squares of cyclic codes and their application in multiplicative secret sharing"
- 11:15-12:00 Johan S. H. Rosenkilde: "Efficient Padé Approximations with Applications in Coding Theory"
- 12:00-13:00 Lunch
- 13:00-13:30 Johan P. Hansen: "Multiplicative Structures on Toric Codes (work in progress)"
- 13:45-14:15 Diego Ruano: "Subfield subcodes in network coding"
- 14:30-15:00 Olav Geil: "Improved constructions of nested code pairs"
- 15:15-15:45 Peter Beelen: "Progress on the generalized Hamming weights of projective Reed-Muller codes"
Talks and abstracts:
- Peter Beelen: "Progress on the generalized Hamming weights of projective Reed-Muller codes".
Abstract: The q-ary projective Reed-Muller code of degree d and m variables is denoted by PRM_q(d,m). In this talk, recent progress on the determination of the generalized Hamming weights such codes will be discussed. Concretely, in this talk it will be discussed how the first (m+2)(m+1)/2 generalized Hamming weights of PRM_q(d,m) can be determined in case d is strictly less than q. These results are joint work with Mrinmoy Datta and Sudhir Ghorpade.
- Ignacio Cascudo: "Some results on squares of cyclic codes and their application in multiplicative secret sharing".
Abstract: The square of a linear code C is the linear code spanned by all coordinatewise products of two words in C. Understanding the behaviour of the parameters of the square of a code is important in order to get a grasp on the multiplicative properties of the associated secret sharing schemes. I will discuss some partial results I have about squares of *cyclic* codes, the motivation for studying them, and the consequences and obstacles of the results regarding their application to secret sharing and some other cryptographic tools.
- Olav Geil: "Improved constructions of nested code pairs". Joint work with Carlos Galindo, Fernando Hernando and Diego Ruano.
Abstract: Two new constructions of linear code pairs $C2 \subset C1$ are given for which the codimension and the relative minimum distances $M_1(C_1,C_2)$, $M_1(C_2^\perp,C_1\perp ) are good. By this we mean that for any two out of the three parameters the third parameter of the constructed code pair is large. Such pairs of nested codes are indispensable for the determination of good linear ramp secret sharing schemes . They can also be used to ensure reliable communication over asymmetric quantum channels . The new constructions result from carefully applying the Feng-Rao bounds [18,27] to a family of codes defined from multivariate polynomials and Cartesian product point sets.
 G. L. Feng and T. R. N. Rao. A simple approach for construction of algebraic-geometric codes from affine plane curves. IEEE Trans. Inform. Theory, 40(4):1003?1012, 1994.
 O. Geil, R. Matsumoto, and D. Ruano. Feng-Rao decoding of primary codes. Finite Fields Appl., 23:35-52, 2013.
 J. Kurihara, T. Uyematsu, and R. Matsumoto. Secret sharing schemes based on linear codes can be precisely characterized by the relative generalized Hamming weight. IEICE Trans. Fundamentals, E95-A(11):2067-2075, Nov. 2012.
 A. M. Steane. Simple quantum error-correcting codes. Phys. Rev. A, 54(6):4741, 1996.
- Johan P. Hansen: "Multiplicative Structures on Toric Codes (work in progress)".
Abstract: We consider linear codes constructed from toric varieties over a finite field $\Fq$ with $q$ elements. The length of the code is $(q-1)^r-1$ for $r\geq 1$. The theme is the multiplicative structure on toric codes. We used this structure to construct linear secret sharing schemes with strong multiplication via Massey's construction generalizing the Shamir LSSS-scheme constructed from Reed-Solomon codes. We will discuss decoding of toric codes, resembling the decoding of Reed-Solomon codes, using their multiplicative structure.
- Johan S. H. Rosenkilde: "Efficient Padé Approximations with Applications in Coding Theory".
Abstract: PadPadé approximations over F[x] for F a finite field, as well as their Hermitian and Simultaneous generalisations, have numerous applications
in coding theory, and have recently been identified as the computational bottleneck of many decoding algorithms.
We describe this purely symbolic-algebraic problem, exemplify the applications in coding theory and describe a recent algorithm which
solves this problem faster than previously possible.
- Diego Ruano: "Subfield subcodes in network coding". Joint work with Olav Geil and Daniel Lucani.
Abstract: We will consider an application of subfield subcodes in network coding.
List of participants:
- Peter Beelen, Technical University of Denmark
- Ignacio Cascudo, Aalborg University
- Mrinmoy Datta, Technical University of Denmark
- Olav Geil, Aalborg University
- Jaron Skovsted Gundersen, Aalborg University
- Johan P. Hansen, Aarhus University
- Tom Høholdt, Aalborg University
- Johan S. H. Rosenkilde, Technical University of Denmark
- Diego Ruano, Aalborg University