40 research outputs found
Multi-Sidon spaces over finite fields
Sidon spaces have been introduced by Bachoc, Serra and Z\'emor in 2017 in
connection with the linear analogue of Vosper's Theorem. In this paper, we
propose a generalization of this notion to sets of subspaces, which we call
multi-Sidon space. We analyze their structures, provide examples and introduce
a notion of equivalnce among them. Making use of these results, we study a
class of linear sets in PG determined by points and we
investigate multi-orbit cyclic subspace codes
Weight Distributions, Automorphisms, and Isometries of Cyclic Orbit Codes
Cyclic orbit codes are subspace codes generated by the action of the Singer subgroup Fqn* on an Fq-subspace U of Fqn. The weight distribution of a code is the vector whose ith entry is the number of codewords with distance i to a fixed reference space in the code. My dissertation investigates the structure of the weight distribution for cyclic orbit codes. We show that for full-length orbit codes with maximal possible distance the weight distribution depends only on q,n and the dimension of U. For full-length orbit codes with lower minimum distance, we provide partial results towards a characterization of the weight distribution, especially in the case that any two codewords intersect in a space of dimension at most 2. We also briefly address the weight distribution of a union of full-length orbit codes with maximum distance.
A related problem is to find the automorphism group of a cyclic orbit code, which plays a role in determining the isometry classes of the set of all cyclic orbit codes. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of Fqn. We then generalize to orbits under the normalizer of the Singer subgroup, although in this setup there is a remaining exceptional case. Finally, we can characterize linear isometries between such codes
Tables of subspace codes
One of the main problems of subspace coding asks for the maximum possible
cardinality of a subspace code with minimum distance at least over
, where the dimensions of the codewords, which are vector
spaces, are contained in . In the special case of
one speaks of constant dimension codes. Since this (still) emerging
field is very prosperous on the one hand side and there are a lot of
connections to classical objects from Galois geometry it is a bit difficult to
keep or to obtain an overview about the current state of knowledge. To this end
we have implemented an on-line database of the (at least to us) known results
at \url{subspacecodes.uni-bayreuth.de}. The aim of this recurrently updated
technical report is to provide a user guide how this technical tool can be used
in research projects and to describe the so far implemented theoretic and
algorithmic knowledge.Comment: 44 pages, 6 tables, 7 screenshot
On 4-general sets in finite projective spaces
A -general set in is a set of points of
spanning the whole and such that no four of them are on a
plane. Such a pointset is said to be complete if it is not contained in a
larger -general set of . In this paper upper and lower
bounds for the size of the largest and the smallest complete -general set in
, respectively, are investigated. Complete -general sets in
, , whose size is close to the theoretical upper
bound are provided. Further results are also presented, including a description
of the complete -general sets in projective spaces of small dimension over
small fields and the construction of a transitive -general set of size in ,
Multivariate Public Key Cryptosystem from Sidon Spaces
A Sidon space is a subspace of an extension field over a base field in which
the product of any two elements can be factored uniquely, up to constants. This
paper proposes a new public-key cryptosystem of the multivariate type which is
based on Sidon spaces, and has the potential to remain secure even if quantum
supremacy is attained. This system, whose security relies on the hardness of
the well-known MinRank problem, is shown to be resilient to several
straightforward algebraic attacks. In particular, it is proved that the two
popular attacks on the MinRank problem, the kernel attack, and the minor
attack, succeed only with exponentially small probability. The system is
implemented in software, and its hardness is demonstrated experimentally.Comment: Appeared in Public-Key Cryptography - PKC 2021, 24th IACR
International Conference on Practice and Theory of Public Key Cryptograph
Conditions for matchability in groups and vector spaces II
We present sufficient conditions for the existence of matchings in abelian
groups and their linear counterparts. These conditions lead to extensions of
existing results in matching theory. Additionally, we classify subsets within
abelian groups that cannot be matched. We introduce the concept of Chowla
subspaces and formulate and conjecture a linear analogue of a result originally
attributed to Y. O. Hamidoune [20] concerning Chowla sets. If proven true, this
result would extend matchings in primitive subspaces. Throughout the paper, we
emphasize the analogy between matchings in abelian groups and field extensions.
We also pose numerous open questions for future research. Our approach relies
on classical theorems in group theory, additive number theory and linear
algebra. As the title of the paper suggests, this work is the second sequel to
a previous paper [5] with a similar theme. This paper is self-contained and can
be read independently.Comment: Comments are welcom