Partitions Of Vector Space Research Articles

Affine vector space partitions and spreads of quadrics

An affine spread is a set of subspaces of AG(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{AG}(n, q)$$\\end{document} of the same dimension that partitions the points of AG(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{AG}(n, q)$$\\end{document}. Equivalently, an affine spread is a set of projective subspaces of PG(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PG}(n, q)$$\\end{document} of the same dimension which partitions the points of PG(n,q)\\H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PG}(n, q) \\setminus H_{\\infty }$$\\end{document}; here H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{\\infty }$$\\end{document} denotes the hyperplane at infinity of the projective closure of AG(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{AG}(n, q)$$\\end{document}. Let Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {Q}$$\\end{document} be a non-degenerate quadric of H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_\\infty $$\\end{document} and let Π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi $$\\end{document} be a generator of Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {Q}$$\\end{document}, where Π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi $$\\end{document} is a t-dimensional projective subspace. An affine spread P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {P}$$\\end{document} consisting of (t+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(t+1)$$\\end{document}-dimensional projective subspaces of PG(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PG}(n, q)$$\\end{document} is called hyperbolic, parabolic or elliptic (according as Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {Q}$$\\end{document} is hyperbolic, parabolic or elliptic) if the following hold:Each member of P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {P}$$\\end{document} meets H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_\\infty $$\\end{document} in a distinct generator of Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {Q}$$\\end{document} disjoint from Π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Pi $$\\end{document};Elements of P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {P}$$\\end{document} have at most one point in common;If S,T∈P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S, T \\in \\mathcal {P}$$\\end{document}, |S∩T|=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|S \\cap T| = 1$$\\end{document}, then ⟨S,T⟩∩Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle S, T \\rangle \\cap \\mathcal {Q}$$\\end{document} is a hyperbolic quadric of Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {Q}$$\\end{document}. In this note it is shown that a hyperbolic, parabolic or elliptic affine spread of PG(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PG}(n, q)$$\\end{document} is equivalent to a spread of Q+(n+1,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {Q}^+(n+1, q)$$\\end{document}, Q(n+1,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {Q}(n+1, q)$$\\end{document} or Q-(n+1,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {Q}^-(n+1, q)$$\\end{document}, respectively.

Affine vector space partitions

AbstractAn affine vector space partition of$${{\,\textrm{AG}\,}}(n,q)$$AG(n,q)is a set of proper affine subspaces that partitions the set of points. Here we determine minimum sizes and enumerate equivalence classes of affine vector space partitions for small parameters. We also give parametric constructions for arbitrary field sizes.

Vector space partitions of GF(2)^8

A vector space partition P of the projective space PG(v-1,q) is a set of subspaces in PG(v-1,q) which partitions the set of points. We say that a vector space partition P has type (v-1)^{m_{v-1}} ... 2^{m_2}1^{m_1} if precisely m_i of its elements have dimension i, where 1 <= i <= v-1. Here we determine all possible types of vector space partitions in PG(7,2).

On the infiniteness of a family of APN functions

APN functions play a fundamental role in cryptography against attacks on block ciphers. Several families of quadratic APN functions have been proposed in the recent years, whose construction relies on the existence of specific families of polynomials. A key question connected with such constructions is to determine whether such APN functions exist for infinitely many dimensions or do not.In this paper we consider a family of functions recently introduced by Li et al. in 2021 showing that for any dimension m≥3 there exists an APN function belonging to such a family.Our main result is proved by a combination of different techniques arising from both algebraic varieties over finite fields connected with linearized permutation rational functions and partial vector space partitions, together with investigations on the kernels of linearized polynomials.

Reconfiguration of subspace partitions

AbstractLet be a fixed prime power and let denote a vector space of dimension over the Galois field with elements. A subspace partition (also called “vector space partition”) of is a collection of subspaces of with the property that every nonzero element of appears in exactly one of these subspaces. Given positive integers such that , we say a subspace partition of has type if it is composed of subspaces of dimension and subspaces of dimension . Let . In this paper, we prove that if divides , then one can (algebraically) construct every possible subspace partition of of type whenever , where and . Our construction allows us to sequentially reconfigure batches of subspaces of dimension into batches of subspaces of dimension . In particular, this accounts for all numerically allowed subspace partition types of under some additional conditions, for example, when .

Classification of large partial plane spreads in $${{\,\mathrm{PG}\,}}(6,2)$$ PG ( 6 , 2 ) and related combinatorial objects

In this article, the partial plane spreads in $PG(6,2)$ of maximum possible size $17$ and of size $16$ are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: Vector space partitions of $PG(6,2)$ of type $(3^{16} 4^1)$, binary $3\times 4$ MRD codes of minimum rank distance $3$, and subspace codes with parameters $(7,17,6)_2$ and $(7,34,5)_2$.

Heden’s bound on the tail of a vector space partition

A vector space partition of Fqv is a collection of subspaces such that every non-zero vector is contained in a unique element. We improve a lower bound of Heden, in a subcase, on the number of elements of the smallest occurring dimension in a vector space partition. To this end, we introduce the notion of qr-divisible sets of k-subspaces in Fqv. By geometric arguments we obtain non-existence results for these objects, which then imply the improved result of Heden.

Projective partitions of vector spaces

Given infinite-dimensional real vector spaces $V,W$ with $|W| \leq |V|$, it is shown that there exists a collection of subspaces of $V$ that are isomorphic to $W$, mutually intersect only at 0, and altogether cover $V$.

Partitions of finite vector spaces over [formula omitted] into subspaces of dimensions 2 and s

A vector space partition of a finite vector space V over the field of q elements is a collection of subspaces whose union is all of V and whose pairwise intersections are trivial. While a number of necessary conditions have been proved for certain types of vector space partitions to exist, the problem of the existence of partitions meeting these conditions is still open. In this note, we consider vector space partitions of a finite vector space over the field GF(2) into subspaces of dimensions 2 and s.While certain cases have been done previously (s=1, s=3, and s even), in our main theorem we build upon these general results to give a constructive proof for the existence of vector space partitions over GF(2) into subspaces of dimensions s and 2 of almost all types. In doing so, we introduce techniques that identify subsets of our vector space which can be viewed as the union of subspaces having trivial pairwise intersection in more than one way. These subsets are used to transform a given partition into another partition of a different type. This technique will also be useful in constructing further partitions of finite vector spaces.

Partitions of V(n, q) into 2‐ and s‐Dimensional Subspaces

AbstractLet denote a vector space of dimension n over the field with q elements. A set of subspaces of V is a (vector space) partition of V if every nonzero element of V is contained in exactly one subspace in . Suppose that is a partition of V with subspaces of dimension for . Then we call the type of the partition . Which possible types correspond to actual partitions is in general an open question. We prove that for any odd integer and for any integer , the existence of partitions of across a suitable range of types guarantees the existence of partitions of of essentially all the types for any integer . We then apply this result to construct new classes of partitions of V. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 467‐482, 2012

A SURVEY OF THE DIFFERENT TYPES OF VECTOR SPACE PARTITIONS

A vector space partition is here a collection [Formula: see text] of subspaces of a finite vector space V(n, q), of dimension n over a finite field with q elements, with the property that every non-zero vector is contained in a unique member of [Formula: see text]. Vector space partitions relate to finite projective planes, design theory and error correcting codes. In the first part of the paper I will discuss some relations between vector space partitions and other branches of mathematics. The other part of the paper contains a survey of known results on the type of a vector space partition, more precisely: the theorem of Beutelspacher and Heden on T-partitions, rather recent results of El-Zanati et al. on the different types that appear in the spaces V(n, 2), for n ≤ 8, a result of Heden and Lehmann on vector space partitions and maximal partial spreads including their new necessary condition for the existence of a vector space partition, and furthermore, I will give a theorem of Heden on the length of the tail of a vector space partition. Finally, I will also give a few historical remarks.

Some necessary conditions for vector space partitions

Some new necessary conditions for the existence of vector space partitions are derived. They are applied to the problem of finding the maximum number of spaces of dimension t in a vector space partition of V ( 2 t , q ) that contains m d spaces of dimension d , where t / 2 < d < t , and also spaces of other dimensions. It is also discussed how this problem is related to maximal partial t -spreads in V ( 2 t , q ) . We also give a lower bound for the number of spaces in a vector space partition and verify that this bound is tight.

On [formula omitted]-fold partitions of finite vector spaces and duality

Vector space partitions of an n-dimensional vector space V over a finite field are considered in Bu (1980) [5], Heden (1984) [10], and more recently in a series of papers Blinco et al. (2008) [3], El-Zanati et al. (2008, 2009) [8,9]. In this paper, we consider the generalization of a vector space partition which we call a λ-fold partition (or simply a λ-partition). In particular, for a given positive integer, λ, we define a λ-fold partition of V to be a multiset of subspaces of V such that every nonzero vector in V is contained in exactly λ subspaces in the given multiset. A λ-fold spread as defined in Hirschfeld (1998) [12] is one example of a λ-fold partition. After establishing some definitions in the introduction, we state some necessary conditions for a λ-fold partition of V to exist, then introduce some general ways to construct such partitions. We also introduce the construction of a dual λ-partition as a way of generating λ′-partitions from a given λ-partition. One application of this construction is that the dual of a vector space partition will, in general, be a λ-partition for some λ>1. In the last section, we discuss a connection between λ-partitions and some designs over finite fields.

Partitions and indivisibility properties of countable dimensional vector spaces

We investigate infinite versions of vector and affine space partition results, and thus obtain examples and a counterexample for a partition problem for relational structures. In particular we provide two (related) examples of an age indivisible relational structure which is not weakly indivisible.

On the length of the tail of a vector space partition

A vector space partition P of a finite dimensional vector space V = V ( n , q ) of dimension n over a finite field with q elements, is a collection of subspaces U 1 , U 2 , … , U t with the property that every non zero vector of V is contained in exactly one of these subspaces. The tail of P consists of the subspaces of least dimension d 1 in P , and the length n 1 of the tail is the number of subspaces in the tail. Let d 2 denote the second least dimension in P . Two cases are considered: the integer q d 2 − d 1 does not divide respective divides n 1 . In the first case it is proved that if 2 d 1 > d 2 then n 1 ≥ q d 1 + 1 and if 2 d 1 ≤ d 2 then either n 1 = ( q d 2 − 1 ) / ( q d 1 − 1 ) or n 1 > 2 q d 2 − d 1 . These lower bounds are shown to be tight and the elements in the subspaces in tails of minimal length will constitute a subspace of V of dimension 2 d 1 respectively d 2 . In case q d 2 − d 1 divides n 1 it is shown that if d 2 < 2 d 1 then n 1 ≥ q d 2 − q d 1 + q d 2 − d 1 and if 2 d 1 ≤ d 2 then n 1 ≥ q d 2 . The last bound is also shown to be tight. The results considerably improve earlier found lower bounds on the length of the tail.

On vector space partitions and uniformly resolvable designs

Let V n (q) denote a vector space of dimension n over the field with q elements. A set $${\mathcal{P}}$$ of subspaces of V n (q) is a partition of V n (q) if every nonzero vector in V n (q) is contained in exactly one subspace in $${\mathcal{P}}$$ . A uniformly resolvable design is a pairwise balanced design whose blocks can be resolved in such a way that all blocks in a given parallel class have the same size. A partition of V n (q) containing a i subspaces of dimension n i for 1 ? i ? k induces a uniformly resolvable design on q n points with a i parallel classes with block size $$q^{n_i}$$ , 1 ? i ? k, and also corresponds to a factorization of the complete graph $$K_{q^n}$$ into $$a_i K_{q^{n_i}}$$ -factors, 1 ? i ? k. We present some sufficient and some necessary conditions for the existence of certain vector space partitions. For the partitions that are shown to exist, we give the corresponding uniformly resolvable designs. We also show that there exist uniformly resolvable designs on q n points where corresponding partitions of V n (q) do not exist.

Partitions of finite vector spaces into subspaces

AbstractLet Vn(q) denote a vector space of dimension n over the field with q elements. A set ${\cal P}$ of subspaces of Vn(q) is a partition of Vn(q) if every nonzero element of Vn(q) is contained in exactly one element of ${\cal P}$. Suppose there exists a partition of Vn(q) into xi subspaces of dimension ni, 1 ≤ i ≤ k. Then x1, …, xk satisfy the Diophantine equation $\sum_{i=1}^{k}{(q^{n_i}-1)x_i}=q^n-1$. However, not every solution of the Diophantine equation corresponds to a partition of Vn(q). In this article, we show that there exists a partition of Vn(2) into x subspaces of dimension 3 and y subspaces of dimension 2 if and only if 7x + 3y = 2n − 1 and y ≠ 1. In doing so, we introduce techniques useful in constructing further partitions. We also show that partitions of Vn(q) induce uniformly resolvable designs on qn points. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 329–341, 2008

Vector quantization of image subbands: a survey

Subband and wavelet decompositions are powerful tools in image coding because of their decorrelating effects on image pixels, the concentration of energy in a few coefficients, their multirate/multiresolution framework, and their frequency splitting, which allows for efficient coding matched to the statistics of each frequency band and to the characteristics of the human visual system. Vector quantization (VQ) provides a means of converting the decomposed signal into bits in a manner that takes advantage of remaining inter and intraband correlation as well as of the more flexible partitions of higher dimensional vector spaces. Since 1988, a growing body of research has examined the use of VQ for subband/wavelet transform coefficients. We present a survey of these methods.

Sum and difference free partitions of vector spaces

Sum and difference free partitions of vector spaces

Partitions of vector spaces

Partitions of vector spaces