Related papers: Recent advances on minimal codes
We will show that a theorem of Rudin \cite{wr1}, \cite{wr}, permits us to determine minimal projections not only with respect to the operator norm but with respect to quasi-norms in operators ideals and numerical radius in many concrete…
Recently a systematic investigation of monoids of sequences of plus-minus weighted zero-sum sequences had been started, which is among others motivated by applications to monoids of norms of algebraic integers. In the current paper these…
In this paper we initiate the study of cyclic algebraic geometry codes. We give conditions to construct cyclic algebraic geometry codes in the context of algebraic function fields over a finite field by using their group of automorphisms.…
We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the finite generation of the canonical ring.
Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…
Minimal codes are linear codes where all non-zero codewords are minimal, i.e., whose support is not properly contained in the support of another codeword. The minimum possible length of such a $k$-dimensional linear code over $\mathbb{F}_q$…
Strong blocking sets and their counterparts, minimal codes, attracted lots of attention in the last years. Combining the concatenating construction of codes with a geometric insight into the minimality condition, we explicitly provide…
Linear codes with few weights have been an interesting subject of study for many years, as these codes have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, linear codes…
Finite projective (lattice) geometries defined over rings instead of fields have recently been recognized to be of great importance for quantum information theory. We believe that there is much more potential hidden in these geometries to…
In this article we consider linear codes coming from skew-symmetric determinantal varieties, which are defined by the vanishing of minors of a certain fixed size in the space of skew-symmetric matrices. In odd characteristic, the minimum…
Subspace codes and particularly constant dimension codes have attracted much attention in recent years due to their applications in random network coding. As a particular subclass of subspace codes, cyclic subspace codes have additional…
Finite semisimple group algebras for which all the minimal ideals are easily computable dimension (ECD) are characterized and some lower bounds for the minimum Hamming distance of group codes in these algebras are offered. Examples…
Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. Here we collect the present knowledge on lower and upper bounds for binary subspace codes for…
The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge…
In this paper, we study linear spaces of matrices defined over discretely valued fields and discuss their dimension and minimal rank drops over the associated residue fields. To this end, we take first steps into the theory of rank-metric…
Recently, many good quantum codes over various finite fields $F_q$ have been constructed from codes over extension rings or mixed alphabet rings via some version of a Gray map. We show that most of these codes can be obtained more directly…
The Gilbert type bound for codes in the title is reviewed, both for small and large alphabets. Constructive lower bounds better than these existential bounds are derived from geometric codes, either over Fp or Fp2 ; or over even degree…
The simple interpretation of the minimum distance of a linear code obtained by De Boer and Pellikaan, and later refined by the second author, is further developed through the study of various finitely generated graded modules. We use the…
We discuss a class of binary cyclic codes and their dual codes. The minimum distance is determined using algebraic geometry, and an application of Weil's theorem. We relate the weights appearing in the dual codes to the number of rational…
The article provides a survey on convolutional codes stressing the connections to module theory and systems theory. Constructions of codes with maximal possible distance and distance profile are provided. The article will appear as book…