10:00am - 12:00pmNetwork coding and subspace designs
Chair(s): Daniele Bartoli (University of Perugia), Anna-Lena Horlemann-Trautmann (University of St. Gallen, Switzerland)
This symposium collects presentations about results on codes for linear network coding, either in the rank metric or in the subspace metric. Codes in the rank metric are usually subsets of the matrix space F_q^{m x n}, where F_q is a finite field; codes in the subspace metric are usually subsets of a finite Grassmann variety. Many interesting questions arise in this topic, e.g., about good packings in these two spaces, as well as fast encoding and decoding algorithms.
(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)
Sum-Rank Codes and Linearized Reed-Solomon Codes
Umberto Martinez-Penas
University of Toronto
The sum-rank metric naturally extends both the Hamming and rank metrics in Coding Theory. In this talk, we will present some of their applications and general properties. We will also present linearized Reed-Solomon codes, which constitute the first general family of maximum sum-rank distance (MSRD) linear codes whose field sizes are subexponential in the code length. These codes are tightly connected to skew Reed-Solomon codes, and are natural hybrids between generalized Reed-Solomon codes and Gabidulin codes.
On some automorphisms of polynomial rings and their applications in rank metric codes
Tovohery Randrianarisoa
IIT Bombay
Recently, there is a growing interest in the study of rank metric codes. These codes have applications in network coding and cryptography. In this talk, I investigate some automorpshisms on polynomial rings over finite fields. We will show how the linear operators from these automorphisms can be used to construct some maximum rank distance (MRD) codes. First we will work on rank metric codes over arbitrary extension and then we will reduce these to finite fields extension. Some particular constructions give MRD codes which are not equivalent to twisted Gabidulin codes. Another application is to use these linear operators to construct some optimal rank metric codes from some Ferrers diagram. In fact we will give some examples of Ferrers diagrams for which there was no known construction of optimal rank metric codes.
Invariants of rank-metric codes via Galois group action
Alessandro Neri
University of Zurich
Codes in the rank metric have been introduced in 1978, but only in the last ten years they significantly gained interest due to their many applications in communications and security. These codes are linear subspaces of the space of matrices over a finite field, but they can be also seen as vectors over an extension field. There are not many explicit constructions of families of rank-metric codes with good parameters, and finding new ones hase become an important ongoing research question. However, when one considers new rank-metric codes constructions, it is important to check whether the new codes are equivalent to any other known construction. For this purpose, one wants to develop some criteria to check code equivalence.
In this talk we introduce a new series of invariants of rank-metric codes obtained via the action of the Galois group of the underlying field extension. In particular, we consider the subspaces generated by the code and the application of several automorphisms to itself and show that their dimensions are invariant under code equivalence. This tool provides an easy checkable criterion for determining code inequivalence. We derive lower bounds on the number of equivalence classes of Gabidulin and twisted Gabidulin codes using this new invariant. In some special cases, the exact number of such equivalence classes is provided.