GOLOMB RULERS AND DIFFERENCE SETS FOR SUCCINCT QUANTUM AUTOMATA

2003 ◽  
Vol 14 (05) ◽  
pp. 871-888 ◽  
Author(s):  
ALBERTO BERTONI ◽  
CARLO MEREGHETTI ◽  
BEATRICE PALANO
Keyword(s):  
Finite Automaton ◽  
Prime Power ◽  
Difference Sets ◽  
Minimal Size ◽  
Quantum Automata ◽  
Size Number ◽  
Minimal Difference ◽  
Golomb Rulers ◽  
Efficient Construction ◽  
Stochastic Event

Given a function p : N → [0,1] of period n, we study the minimal size (number of states) of a one-way quantum finite automaton (Iqfa) inducing the stochastic event ap + b, for real constants a>0, b≥0, a+b≤1. First of all, we relate the estimation of the minimal size to the problem of finding a minimal difference cover for a suitable subset of Zn. Then, by observing that the cardinality of a difference cover Δ for a set A ⊆ Zn, must satisfy [Formula: see text], we investigate the class of sets A admitting difference covers of cardinality exactly [Formula: see text]. We relate this problem with the efficient construction of Golomb rulers and difference sets. We design an algorithm which outputs each of the Golomb rulers (if any) of a given set in pseudo-polynomial time. As a consequence, we obtain an efficient algorithm that construct minimal difference covers for a non-trivial class of sets. Moreover, by using projective geometry arguments, we give an algorithm that, for any n=q2+q+1 with q prime power, constructs difference sets for Zn in quadratic time.

On a combinatorial search problem

2002 ◽  
Vol 12 (4) ◽  
Author(s):  
E. V. Debrev
Keyword(s):  
Complete Graph ◽  
Basic Research ◽  
Difference Sets ◽  
Search Problem ◽  
Combinatorial Search ◽  
Research Grants ◽  
Federal Program ◽  
Minimal Difference ◽  
Hamiltonian Circuits ◽  
Program Integration

AbstractIn this paper, we consider the problem of searching for undirected Hamiltonian circuits in the complete graph on n vertices with the use of unconditional edge tests. We prove that the minimal test contains exactly n(n - 3)/2 - [n/3] + 1 edges. We propose an explicit characterisation of all minimal difference sets of edges.This research was supported by the Russian Foundation for Basic Research, grants 02-01-00985 and 00-15-96103, by the Program ‘Universities of Russia,’ and the Federal Program ‘Integration.’

Quantum Automata with Open Time Evolution

2012 ◽  
pp. 74-89
Author(s):  
Mika Hirvensalo
Keyword(s):  
Time Evolution ◽  
Finite Automaton ◽  
Finite Automata ◽  
Quantum Evolution ◽  
Closure Properties ◽  
Probabilistic Measure ◽  
Quantum Automata ◽  
Open Quantum ◽  
Quantum Measure ◽  
Open Time

In this paper, a model for finite automaton with an open quantum evolution is introduced, and its basic properties are studied. It is shown that the (fuzzy) languages accepted by open evolution quantum automata obey various closure properties. More importantly, it is shown that major other models of finite automata, including probabilistic, measure once quantum, measure many quantum, and Latvian quantum automata can be simulated by the open quantum evolution automata without increasing the number of the states.

scholarly journals Some classes of Hadamard matrices with constant diagonal

1972 ◽  
Vol 7 (2) ◽  
pp. 233-249 ◽  
Author(s):  
Jennifer Wallis ◽  
Albert Leon Whiteman
Keyword(s):  
Prime Power ◽  
Abelian Groups ◽  
Difference Sets ◽  
Hadamard Matrices ◽  
Incidence Matrices

The concepts of circulant and backcirculant matrices are generalized to obtain incidence matrices of subsets of finite additive abelian groups. These results are then used to show the existence of skew-Hadamard matrices of order 8(4f+1) when f is odd and 8f + 1 is a prime power. This shows the existence of skew-Hadamard matrices of orders 296, 592, 1184, 1640, 2280, 2368 which were previously unknown.A construction is given for regular symmetric Hadamard matrices with constant diagonal of order 4(2m + 1)2 when a symmetric conference matrix of order 4m + 2 exists and there are Szekeres difference sets, X and Y, of size m satisfying x є X ⇒ −xє X, y є Y ⇒ −y єY.

The prime power conjecture is true for $n < 2{,}000{,}000$

10.37236/1186 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Daniel M. Gordon
Keyword(s):  
Prime Power ◽  
Necessary Conditions ◽  
Difference Sets ◽  
Cyclic Difference Sets

The Prime Power Conjecture (PPC) states that abelian planar difference sets of order $n$ exist only for $n$ a prime power. Evans and Mann verified this for cyclic difference sets for $n \leq 1600$. In this paper we verify the PPC for $n \leq 2{,}000{,}000$, using many necessary conditions on the group of multipliers.

scholarly journals Perfect difference sets

1964 ◽  
Vol 6 (4) ◽  
pp. 177-184 ◽  
Author(s):  
H. Halberstam ◽  
R. R. Laxton
Keyword(s):  
Prime Power ◽  
Difference Sets ◽  
Difference Set ◽  
Comprehensive Survey ◽  
Open Question

If the set K of r+1 distinct integers k0, k1 …, kr has the property that the (r+1)r differences ki–kj (0≦i, j≦r, i≠j) are distinct modulo r2+r+1, K is called a perfect difference set modr2+r+1. The existence of perfect difference sets seems intuitively improbable, at any rate for large r, but in 1938 J. Singer [1] proved that, whenever r is a prime power, say r = pn, a perfect difference set mod p2n+pn+1 exists. Since the appearance of Singer's paper several authors have succeeded in showing that for many kinds of number r perfect difference sets mod r2+r+1 do not exist; but it remains an open question whether perfect difference sets exist only when r is a prime power (for a comprehensive survey see [2]).

Quantum Automata with Open Time Evolution

2010 ◽  
Vol 1 (1) ◽  
pp. 70-85 ◽  
Author(s):  
Mika Hirvensalo
Keyword(s):  
Time Evolution ◽  
Finite Automaton ◽  
Finite Automata ◽  
Quantum Evolution ◽  
Closure Properties ◽  
Probabilistic Measure ◽  
Quantum Automata ◽  
Open Quantum ◽  
Quantum Measure ◽  
Open Time

In this paper, a model for finite automaton with an open quantum evolution is introduced, and its basic properties are studied. It is shown that the (fuzzy) languages accepted by open evolution quantum automata obey various closure properties. More importantly, it is shown that major other models of finite automata, including probabilistic, measure once quantum, measure many quantum, and Latvian quantum automata can be simulated by the open quantum evolution automata without increasing the number of the states.

Some Theorems on Difference Sets

1952 ◽  
Vol 4 ◽  
pp. 222-226 ◽  
Author(s):  
Henry B. Mann
Keyword(s):  
Prime Power ◽  
Sufficient Conditions ◽  
Difference Sets ◽  
Difference Set ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

A set a1 …, ak of different residues mod v is called a difference set (v, k, λ) (v>k > λ) if the congruence ai — aj ≡ d (mod v) has exactly λ solutions for d ≢ 0 (mod v). Singer [4] has demonstrated the existence of a difference set (v, k, 1) if k — 1 is a prime power, and difference sets for λ > 1 have been constructed by various authors; but necessary and sufficient conditions for the existence of a (v, k, λ) are not known. It has not been possible so far to find a difference set with λ = 1 if k — 1 is not a prime power and it has therefore been conjectured that no such difference set exists.

Conference matrices from Legendre C-pairs

2020 ◽  
pp. 2-10
Author(s):  
Nikolay Balonin ◽  
Dragomir Ðoković
Keyword(s):  
Prime Power ◽  
Identity Element ◽  
Finite Abelian Group ◽  
Difference Sets ◽  
Practical Relevance ◽  
Difference Family ◽  
Ring Of Integers ◽  
Video Information ◽  
On Line ◽  
Universal Structure

Introduction: There are just a few known methods for the construction of symmetric C-matrices, due to the lack of a universal structure for them. This obstruction is fundamental, in addition, the structure of C-matrices with a double border is incompletely described in literature, which makes its study especially relevant. The purpose: To describe the two-border two-circulant construction in detail with the proposal of the concept of C-pairs Legendre. Results: The paper deals with C-matrices of order n=2v+2 with two borders and extends the so called generalized Legendre pairs, v odd, to a wider class of Legendre C-pairs with even and odd v, defined on a finite abelian group G of order v. Such a pair consists of two functions a, b: G→Z, whose values are +1 or −1 except that a(e)=0, where e is the identity element of G and Z is  the ring of integers. To characterize the Legendre C-pairs we use the subsets X={xÎG: a(x)=–1} and Y={xÎG: b(x)=–1} of G. We show that a(x−1)=(−1)v a(x) for all x. For odd v we show that X and Y form a difference family, which is not true for even v. These difference families are precisely the so called Szekeres difference sets, used originally for the construction of skew-Hadamard matrices. We introduce the subclass of the special Legendre C-pairs and prove that they exist whenever 2v+1 is a prime power. In the last two sections of the paper we list examples of special cyclic Legendre C-pairs for lengths v<70. Practical relevance: C-matrices are used extensively in the problems of error-free coding, compression and masking of video information. Programs for search of conference matrices and a library of constructed matrices are used in the mathematical network “mathscinet.ru” together with executable on-line algorithms.

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