Parameters of the codes over affine varieties

We derive a lower bound of the generalized Hamming weights of the codes over affine varieties, which are defined by appropriate sequences of rational polynomials over varieties.     

A class of almost MDS codes

MDS codes and almost MDS (AMDS) codes are special classes of linear codes, and have important applications in communications, data storage, combinatorial theory, and secrete sharing. The objective of this paper is to present a class of AMDS codes from some BCH codes and determine their parameters. It turns out the proposed AMDS codes are distance-optimal and dimension-optimal locally repairable codes. The parameters of the duals of this class of AMDS codes are also discussed.     

Construction of almost integral manifolds of affine control systems

We consider the problem of constructing manifolds of a special form, which are said to be almost integral, lie in the state space of an affine control system, and have the following property: the restriction of the control system to such a manifold is an almost trivial control system defining the part of trajectories of the original system lying on that manifold.     

Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps

A quasiplane f(V)$f(V)$ is the image of an n-dimensional Euclidean subspace V   of R^N${\mathbb{R}}^N$ (1≤n≤N−1$1\le n\le N-1$) under a quasiconformal map f:R^N→R^N$f:{\mathbb{R}}^N\to {\mathbb{R}}^N$. We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz n  -manifold and for a quasiplane to have big pieces of bi-Lipschitz images of Rⁿ${\mathbb{R}}^n$. One main novelty of these results is that we analyze quasiplanes in arbitrary codimension N−n$N-n$. To establish the big pieces criterion, we prove new extension theorems for “almost affine” maps, which are of independent interest. This work is related to investigations by Tukia and Väisälä on extensions of quasisymmetric maps with small distortion.     

Defining sets of extended cyclic codes invariant under the affine group

We describe the defining sets of extended cyclic codes of length pⁿ over a field and over the ring of integers modulo p^e admitting the affine group AGL_m(p^t), n=mt, as a permutation group.     

From primary to dual affine variety codes over the Klein quartic

Designs, Codes and Cryptography - In Geil and Özbudak (Cryptogr Commun 11(2):237–257, 2019) a novel method was established to estimate the minimum distance of primary affine variety...     

Flags in affine planes and maximal spreads in a non-singular quadric of PG(4,q)

We give an example of maximal spreadF in a non-singular quadric of PG(4,q), with ¦F¦ = 3q + 1.     

Constructions of almost secure frameproof codes with applications to fingerprinting schemes

This paper presents explicit constructions of fingerprinting codes. The proposed constructions use a class of codes called almost secure frameproof codes. An almost secure frameproof code is a relaxed version of a secure frameproof code, which in turn is the same as a separating code. This relaxed version is the object of our interest because it gives rise to fingerprinting codes of higher rate than fingerprinting codes derived from separating codes. The construction of almost secure frameproof codes discussed here is based on weakly biased arrays, a class of combinatorial objects tightly related to weakly dependent random variables.     

Denominators in cluster algebras of affine type

The Fomin–Zelevinsky Laurent phenomenon states that every cluster variable in a cluster algebra can be expressed as a Laurent polynomial in the variables lying in an arbitrary initial cluster. We give representation-theoretic formulas for the denominators of cluster variables in cluster algebras of affine type. The formulas are in terms of the dimensions of spaces of homomorphisms in the corresponding cluster category, and hold for any choice of initial cluster.     

Canonical bases and affine Hecke algebras of type D

We prove a conjecture of Miemietz and Kashiwara on canonical bases and branching rules of affine Hecke algebras of type D. The proof is similar to the proof of the type B case in Varagnolo and Vasserot (in press) [15].     

On quiver varieties and affine Grassmannians of type A

We construct Nakajima's quiver varieties of type A in terms of affine Grassmannians of type A. This gives a compactification of quiver varieties and a decomposition of affine Grassmannians into a disjoint union of quiver varieties. Consequently, singularities of quiver varieties, nilpotent orbits and affine Grassmannians are the same in type A. The construction also provides a geometric framework for skew (GL(m),GL(n)) duality and identifies the natural basis of weight spaces in Nakajima's construction with the natural basis of multiplicity spaces in tensor products which arises from affine Grassmannians. To cite this article: I. Mirkovi?, M. Vybornov, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

A new concatenated type construction for LCD codes and isometry codes

We give a new concatenated type construction for linear codes with complementary dual (LCD) over small finite fields. In this construction,we need a special class of inner codes that we call isometry codes. Our construction generalizes a recent construction of Carlet et al. (2014–2016) and of Güneri et al. (2016). In particular, it allows us to construct LCD codes with improved parameters directly.