scholarly journals Lower Bounds for $q$-ary Codes with Large Covering Radius

10.37236/222 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Immanuel Halupczok ◽  
Jan-Christoph Schlage-Puchta
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Efficient Method ◽  
Winning Strategy ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Original Proof ◽  
Sphere Covering

Let $K_q(n,R)$ denote the minimal cardinality of a $q$-ary code of length $n$ and covering radius $R$. Recently the authors gave a new proof of a classical lower bound of Rodemich on $K_q(n,n-2)$ by the use of partition matrices and their transversals. In this paper we show that, in contrast to Rodemich's original proof, the method generalizes to lower-bound $K_q(n,n-k)$ for any $k>2$. The approach is best-understood in terms of a game where a winning strategy for one of the players implies the non-existence of a code. This proves to be by far the most efficient method presently known to lower-bound $K_q(n,R)$ for large $R$ (i.e. small $k$). One instance: the trivial sphere-covering bound $K_{12}(7,3)\geq 729$, the previously best bound $K_{12}(7,3)\geq 732$ and the new bound $K_{12}(7,3)\geq 878$.

scholarly journals On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

10.37236/969 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas ◽  
Jörn Quistorff
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Covering Radius ◽  
Minimal Cardinality ◽  
Finite Sets ◽  
Open Problems ◽  
Latin Rectangle ◽  
Mixed Codes ◽  
Special Case

Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

scholarly journals Lower Bounds for the Football Pool Problem for 7 and 8 Matches

10.37236/945 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Wolfgang Haas
Keyword(s):  
Lower Bounds ◽  
Covering Radius ◽  
Linear Inequalities ◽  
Minimal Cardinality ◽  
Covering Code ◽  
Ternary Code ◽  
Systems Of Linear Inequalities

Let $k_3(n)$ denote the minimal cardinality of a ternary code of length $n$ and covering radius one. In this paper we show $k_3(7)\ge 156$ and $k_3(8)\ge 402$ improving on the best previously known bounds $k_3(7)\ge 153$ and $k_3(8)\ge 398$. The proofs are founded on a recent technique of the author for dealing with systems of linear inequalities satisfied by the number of elements of a covering code, that lie in $k$-dimensional subspaces of F${}_3^n$.

scholarly journals On Third-Order Nonlinearity of Biquadratic Monomial Boolean Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Brajesh Kumar Singh
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Boolean Function ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Coding Theory ◽  
Block Ciphers ◽  
Covering Radius ◽  
Third Order ◽  
Complicated Problem

The rth-order nonlinearity of Boolean function plays a central role against several known attacks on stream and block ciphers. Because of the fact that its maximum equals the covering radius of the rth-order Reed-Muller code, it also plays an important role in coding theory. The computation of exact value or high lower bound on the rth-order nonlinearity of a Boolean function is very complicated problem, especially when r>1. This paper is concerned with the computation of the lower bounds for third-order nonlinearities of two classes of Boolean functions of the form Tr1nλxd for all x∈𝔽2n, λ∈𝔽2n*, where a d=2i+2j+2k+1, where i, j, and   k are integers such that i>j>k≥1 and n>2i, and b d=23ℓ+22ℓ+2ℓ+1, where ℓ is a positive integer such that gcdℓ,𝓃=1 and n>6.

A Lower Bound for Schur Numbers and Multicolor Ramsey Numbers

10.37236/1188 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Geoffrey Exoo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Ramsey Numbers

For $k \geq 5$, we establish new lower bounds on the Schur numbers $S(k)$ and on the k-color Ramsey numbers of $K_3$.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

scholarly journals On Computing Multilinear Polynomials Using Multi- r -ic Depth Four Circuits

2021 ◽  
Vol 13 (3) ◽  
pp. 1-21
Author(s):  
Suryajith Chillara
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Complexity Measure ◽  
Natural Question ◽  
Multilinear Polynomial ◽  
Arithmetic Circuit ◽  
Theory Of Computing ◽  
Small Constant ◽  
Multilinear Polynomials ◽  
The Individual

In this article, we are interested in understanding the complexity of computing multilinear polynomials using depth four circuits in which the polynomial computed at every node has a bound on the individual degree of r ≥ 1 with respect to all its variables (referred to as multi- r -ic circuits). The goal of this study is to make progress towards proving superpolynomial lower bounds for general depth four circuits computing multilinear polynomials, by proving better bounds as the value of r increases. Recently, Kayal, Saha and Tavenas (Theory of Computing, 2018) showed that any depth four arithmetic circuit of bounded individual degree r computing an explicit multilinear polynomial on n O (1) variables and degree d must have size at least ( n / r 1.1 ) Ω(√ d / r ) . This bound, however, deteriorates as the value of r increases. It is a natural question to ask if we can prove a bound that does not deteriorate as the value of r increases, or a bound that holds for a larger regime of r . In this article, we prove a lower bound that does not deteriorate with increasing values of r , albeit for a specific instance of d = d ( n ) but for a wider range of r . Formally, for all large enough integers n and a small constant η, we show that there exists an explicit polynomial on n O (1) variables and degree Θ (log 2 n ) such that any depth four circuit of bounded individual degree r ≤ n η must have size at least exp(Ω(log 2 n )). This improvement is obtained by suitably adapting the complexity measure of Kayal et al. (Theory of Computing, 2018). This adaptation of the measure is inspired by the complexity measure used by Kayal et al. (SIAM J. Computing, 2017).

scholarly journals Lower bounds for the state complexity of probabilistic languages and the language of prime numbers

2020 ◽  
Vol 30 (1) ◽  
pp. 175-192
Author(s):  
NathanaËl Fijalkow
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Prime Numbers ◽  
Regular Languages ◽  
Automata Theory ◽  
Seminal Paper ◽  
Probabilistic Automata ◽  
State Complexity ◽  
Probabilistic Languages ◽  
Alternating Automata

Abstract This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. Motivated by the seminal paper of Rabin from 1963 introducing probabilistic automata, we study the (deterministic) state complexity of probabilistic languages and prove that probabilistic languages can have arbitrarily high deterministic state complexity. We then look at alternating automata as introduced by Chandra, Kozen and Stockmeyer: such machines run independent computations on the word and gather their answers through boolean combinations. We devise a lower bound technique relying on boundedly generated lattices of languages, and give two applications of this technique. The first is a hierarchy theorem, stating that there are languages of arbitrarily high polynomial alternating state complexity, and the second is a linear lower bound on the alternating state complexity of the prime numbers written in binary. This second result strengthens a result of Hartmanis and Shank from 1968, which implies an exponentially worse lower bound for the same model.

scholarly journals Encoding Two-Dimensional Range Top-k Queries

Algorithmica ◽  
2021 ◽  
Author(s):  
Seungbum Jo ◽  
Rahul Lingala ◽  
Srinivasa Rao Satti
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Total Order ◽  
Two Dimensional ◽  
Information Theoretic ◽  
Cartesian Tree ◽  
Dimensional Range

AbstractWe consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering $${\text{Top-}}{k}$$ Top- k queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For an $$m \times n$$ m × n array, with $$m \le n$$ m ≤ n , we first propose an encoding for answering 1-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, whose query range is restricted to $$[1 \dots m][1 \dots a]$$ [ 1 ⋯ m ] [ 1 ⋯ a ] , for $$1 \le a \le n$$ 1 ≤ a ≤ n . Next, we propose an encoding for answering for the general (4-sided) $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries that takes $$(m\lg {{(k+1)n \atopwithdelims ()n}}+2nm(m-1)+o(n))$$ ( m lg ( k + 1 ) n n + 2 n m ( m - 1 ) + o ( n ) ) bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial $$O(nm\lg {n})$$ O ( n m lg n ) -bit encoding, our encoding takes less space when $$m = o(\lg {n})$$ m = o ( lg n ) . In addition to the upper bound results for the encodings, we also give lower bounds on encodings for answering 1 and 4-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, which show that our upper bound results are almost optimal.

Lower Bounds to the Eigenvalues in One-Dimensional Problems by a Shift in the Weight Function

1970 ◽  
Vol 37 (2) ◽  
pp. 267-270 ◽  
Author(s):  
D. Pnueli
Keyword(s):  
Lower Bound ◽  
Weight Function ◽  
Lower Bounds ◽  
Variational Formulation ◽  
Ritz Method ◽  
The Other ◽  
One Dimensional ◽  
Systematic Shift ◽  
The One ◽  
Detailed Computation

A method is presented to obtain both upper and lower bound to eigenvalues when a variational formulation of the problem exists. The method consists of a systematic shift in the weight function. A detailed procedure is offered for one-dimensional problems, which makes improvement of the bounds possible, and which involves the same order of detailed computation as the Rayleigh-Ritz method. The main contribution of this method is that it yields the “other bound;” i.e., the one which cannot be obtained by the Rayleigh-Ritz method.

Lower Bounds For Induced Forests in Cubic Graphs

1987 ◽  
Vol 30 (2) ◽  
pp. 193-199 ◽  
Author(s):  
J. A. Bondy ◽  
Glenn Hopkins ◽  
William Staton
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Cubic Graphs ◽  
Cubic Graph

AbstractIf G is a connected cubic graph with ρ vertices, ρ > 4, then G has a vertex-induced forest containing at least (5ρ - 2)/8 vertices. In case G is triangle-free, the lower bound is improved to (2ρ — l)/3. Examples are given to show that no such lower bound is possible for vertex-induced trees.

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