Exploring invariant linear codes through generators and centralizers

Partha Pratim Dey

Displaying similar documents to “Exploring invariant linear codes through generators and centralizers”

The nonexistence of robust codes for subsets of ω₁

David Asperó (2005)

Fundamenta Mathematicae

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Several results are presented concerning the existence or nonexistence, for a subset S of ω₁, of a real r which works as a robust code for S with respect to a given sequence $⟨ S_{α} : α < ω ₁ ⟩$ of pairwise disjoint stationary subsets of ω₁, where “robustness” of r as a code may either mean that $S \in L [r, ⟨ S *_{α} : α < ω ₁ ⟩]$ whenever each $S *_{α}$ is equal to $S_{α}$ modulo nonstationary changes, or may have the weaker meaning that $S \in L [r, ⟨ S_{α} \cap C : α < ω ₁ ⟩]$ for every club C ⊆ ω₁. Variants of the above theme are also considered which result when the requirement that...

On sets of vectors of a finite vector space in which every subset of basis size is a basis

Simeon Ball (2012)

Journal of the European Mathematical Society

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It is shown that the maximum size of a set $S$ of vectors of a $k$ -dimensional vector space over $𝔽_{q}$ , with the property that every subset of size $k$ is a basis, is at most $q + 1$ , if $k \leq p$ , and at most $q + k - p$ , if $q \geq k \geq p + 1 \geq 4$ , where $q = p^{k}$ and $p$ is prime. Moreover, for $k \leq p$ , the sets $S$ of maximum size are classified, generalising Beniamino Segre’s “arc is a conic” theorem. These results have various implications. One such implication is that a $k \times (p + 2)$ matrix, with $k \leq p$ and entries from $𝔽_{p}$ , has $k$ columns which are linearly dependent....

The natural operators of general affine connections into general affine connections

Jan Kurek, Włodzimierz M. Mikulski (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We reduce the problem of describing all $ℳ f_{m}$ -natural operators transforming general affine connections on $m$ -manifolds into general affine ones to the known description of all $G L (𝐑^{m})$ -invariant maps $𝐑^{m *} \otimes 𝐑^{m} \to \otimes^{k} 𝐑^{m *} \otimes \otimes^{k} 𝐑^{m}$ for $k = 1, 3$ .

On the Hausdorff dimension of certain self-affine sets

Abercrombie Alex G.., Nair R. (2002)

Studia Mathematica

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A subset E of ℝⁿ is called self-affine with respect to a collection ϕ₁,...,ϕₜ of affinities if E is the union of the sets ϕ₁(E),...,ϕₜ(E). For S ⊂ ℝⁿ let $Φ (S) = ⋃_{1 \leq j \leq t} ϕ_{j} (S)$ . If Φ(S) ⊂ S let $E_{Φ} (S)$ denote $⋂_{k \geq 0} Φ^{k} (S)$ . For given Φ consisting of contracting “pseudo-dilations” (affinities which preserve the directions of the coordinate axes) and subject to further mild technical restrictions we show that there exist self-affine sets $E_{Φ} (S)$ of each Hausdorff dimension between zero and a positive number depending on Φ. We also...

Self-affine measures and vector-valued representations

Qi-Rong Deng, Xing-Gang He, Ka-Sing Lau (2008)

Studia Mathematica

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Let A be a d × d integral expanding matrix and let $S_{j} (x) = A^{- 1} (x + d_{j})$ for some $d_{j} \in ℤ^{d}$ , j = 1,...,m. The iterated function system (IFS) ${S_{j}}_{j = 1}^{m}$ generates self-affine measures and scale functions. In general this IFS has overlaps, and it is well known that in many special cases the analysis of such measures or functions is facilitated by expressing them in vector-valued forms with respect to another IFS that satisfies the open set condition. In this paper we prove a general theorem on such representation. The proof...

Multidimensional self-affine sets: non-empty interior and the set of uniqueness

Kevin G. Hare, Nikita Sidorov (2015)

Studia Mathematica

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Let M be a d × d real contracting matrix. We consider the self-affine iterated function system Mv-u, Mv+u, where u is a cyclic vector. Our main result is as follows: if $| d e t M | \geq 2^{- 1 / d}$ , then the attractor $A_{M}$ has non-empty interior. We also consider the set $_{M}$ of points in $A_{M}$ which have a unique address. We show that unless M belongs to a very special (non-generic) class, the Hausdorff dimension of $_{M}$ is positive. For this special class the full description of $_{M}$ is given as well. This paper continues our...

Sufficient conditions for the spectrality of self-affine measures with prime determinant

Jian-Lin Li (2014)

Studia Mathematica

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Let $μ_{M, D}$ be a self-affine measure associated with an expanding matrix M and a finite digit set D. We study the spectrality of $μ_{M, D}$ when |det(M)| = |D| = p is a prime. We obtain several new sufficient conditions on M and D for $μ_{M, D}$ to be a spectral measure with lattice spectrum. As an application, we present some properties of the digit sets of integral self-affine tiles, which are connected with a conjecture of Lagarias and Wang.

On strongly affine extensions of commutative rings

Nabil Zeidi (2020)

Czechoslovak Mathematical Journal

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A ring extension $R \subseteq S$ is said to be strongly affine if each $R$ -subalgebra of $S$ is a finite-type $R$ -algebra. In this paper, several characterizations of strongly affine extensions are given. For instance, we establish that if $R$ is a quasi-local ring of finite dimension, then $R \subseteq S$ is integrally closed and strongly affine if and only if $R \subseteq S$ is a Prüfer extension (i.e. $(R, S)$ is a normal pair). As a consequence, the equivalence of strongly affine extensions, quasi-Prüfer extensions and INC-pairs is shown....

Association schemes and MacWilliams dualities for generalized Niederreiter-Rosenbloom-Tsfasman posets

Dae San Kim, Hyun Kwang Kim

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Let P be a poset on the set [m]×[n], which is given as the disjoint sum of posets on ’columns’ of [m]×[n], and let P̌ be the dual poset of P. Then P is called a generalized Niederreiter-Rosenbloom-Tsfasman poset (gNRTp) if all further posets on columns are weak order posets of the ’same type’. Let G (resp. Ǧ) be the group of all linear automorphisms of the space $_{q}^{m \times n}$ preserving the P-weight (resp. P̌-weight). We define two partitions of $_{q}^{m \times n}$ , one consisting of ’P-orbits’ and the other of...

An obstruction to $ℓ^{p}$ -dimension

Nicolas Monod, Henrik Densing Petersen (2014)

Annales de l’institut Fourier

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Let $G$ be any group containing an infinite elementary amenable subgroup and let $2 < p < \infty$ . We construct an exhaustion of $ℓ^{p} G$ by closed invariant subspaces which all intersect trivially a fixed non-trivial closed invariant subspace. This is an obstacle to $ℓ^{p}$ -dimension and gives an answer to a question of Gaboriau.

Uniqueness of Cartesian Products of Compact Convex Sets

Zbigniew Lipecki, Viktor Losert, Jiří Spurný (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let $X_{i}$ , i∈ I, and $Y_{j}$ , j∈ J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of $\prod_{i \in I} X_{i}$ onto $\prod_{j \in J} Y_{j}$ . We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of $X_{i}$ onto $Y_{b (i)}$ , i∈ I.

A note on integer translates of a square integrable function on ℝ

Maciej Paluszyński (2010)

Colloquium Mathematicae

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We consider the subspace of L²(ℝ) spanned by the integer shifts of one function ψ, and formulate a condition on the family ${ψ (\cdot - n)}_{n = - \infty}^{\infty}$ , which is equivalent to the weight function $\sum_{n = - \infty}^{\infty} | ψ ̂ (\cdot + n) | ²$ being > 0 a.e.