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.

scholarly journals Three new lengths for cyclic Legendre pairs

2021 ◽  
pp. 2-7
Author(s):  
Nikolay Balonin ◽  
Dragomir Dokovic
Keyword(s):  
Autocorrelation Function ◽  
Cyclic Group ◽  
Hadamard Matrices ◽  
Related Structure ◽  
Practical Relevance ◽  
Ring Of Integers ◽  
Video Information ◽  
On Line ◽  
Periodic Autocorrelation ◽  
Periodic Autocorrelation Function

Introduction: It is conjectured that the cyclic Legendre pairs of odd lengths >1 always exist. Such a pair consists of two functions a, b: G→Z, whose values are +1 or −1, and whose periodic autocorrelation function adds up to the constant value −2 (except at the origin). Here G is a finite cyclic group and Z is the ring of integers. These conditions are fundamental and the closely related structure of Hadamard matrices having a two circulant core and double border is incompletely described in literature, which makes its study especially relevant. Purpose: To describe the two-border two-circulant-core construction for Legendre pairs having three new lengths. Results: To construct new Legendre pairs we use the subsets X={x∈G: a(x)=–1} and Y={x∈G: b(x)=–1} of G. There are 20 odd integers v less than 200 for which the existence of Legendre pairs of length v is undecided. The smallest among them is v=77. We have constructed Legendre pairs of lengths 91, 93 and 123 reducing thereby the number of undecided cases to 17. In the last section of the paper we list some new examples of cyclic Legendre pairs for lengths v≤123. Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, and compression and masking of video information. Programs for search of Hadamard matrices and a library of constructed matrices are used in the mathematical network “mathscinet.ru” together with executable on-line algorithms

scholarly journals Symmetric Hadamard Matrices of Orders 268, 412, 436 and 604

2018 ◽  
pp. 2-8 ◽  
Author(s):  
N. A. Balonin ◽  
D. Z. Ðokovic'
Keyword(s):  
Hash Function ◽  
Hadamard Matrices ◽  
Orbit Method ◽  
Practical Relevance ◽  
Video Information ◽  
The Past ◽  
Difference Families ◽  
Feasible Sets ◽  
On Line ◽  
First Time

Purpose.To investigate more fully, than what was done in the past, certain families of symmetric Hadamard matrices of small orders by using the so called propus construction.Methods.Orbit method for the search of three cyclic blocks to construct Hadamard matrices of propus type. This method speeds up the classical search of required sequences by distributing them into different bins using a hash-function.Results. Our main result is that we have constructed, for the first time, symmetric Hadamard matrices of order 268, 412, 436 and 604. The necessary difference families are constructed by restricting the search to the families which admit a nontrivial multiplier. A wide collection of new symmetric Hadamard matrices was obtained and tabulated, according to the feasible sets of parameters.Practical relevance.Hadamard matrices are used extensively in the problems of error-free coding, compression and masking of video information. Programs for search of symmetric Hadamard matrices and a library of constructed matrices are used in the mathematical network “Internet” together with executable on-line algorithms. 

scholarly journals Hadamard matrices from Goethals — Seidel difference families with a repeated block

2019 ◽  
pp. 2-9 ◽  
Author(s):  
Leonid Abuzin ◽  
Nikolai Unknown ◽  
Dragomir Ðoković ◽  
Ilias Kotsireas
Keyword(s):  
Additional Condition ◽  
Cyclic Subgroup ◽  
Finite Abelian Group ◽  
Hadamard Matrices ◽  
Difference Family ◽  
Main Ingredient ◽  
Difference Families ◽  
Symmetry Properties ◽  
On Line ◽  
Repeated Block

Purpose: To construct Hadamard matrices by using Goethals — Seidel difference families having a repeated block, generalizingthe so called propus construction. In particular we construct the first examples of symmetric Hadamard matrices of order 236.Methods: The main ingredient of the propus construction is a difference family in a finite abelian group of order v consisting offour blocks (X1, X2, X3, X4) where X1 is symmetric and X2 X3. The parameters (v; k1, k2, k3, k4; λ) of such family must satisfythe additional condition ki  λ  v. We modify this construction by imposing different symmetry conditions on some of theblocks and construct many examples of Hadamard matrices of this kind. In this paper we work with the cyclic group Zv of order v.For larger values of v we build the blocks Xi by using the orbits of a suitable small cyclic subgroup of the automorphism groupof Zv. Results: We continue the systematic search for symmetric Hadamard matrices of order 4v by using the propus construction.Such searches were carried out previously for odd v  51. We extend it to cover the case v53. Moreover we construct thefirst examples of symmetric Hadamard matrices of order 236. A wide collection of symmetric and skew-symmetric Hadamardmatrices was obtained and the corresponding difference families tabulated by using the symmetry properties of their blocks.Practical relevance: Hadamard matrices are used extensively in the problems of error-free coding, compression and masking ofvideo information. Programs for search of symmetric Hadamard matrices and a library of constructed matrices are used in themathematical network Internet together with executable on line algorithms.

scholarly journals On the realizable classes of the square root of the inverse different in the unitary class group

2017 ◽  
Vol 13 (04) ◽  
pp. 913-932 ◽  
Author(s):  
Sin Yi Cindy Tsang
Keyword(s):  
Abelian Group ◽  
Number Field ◽  
Bilinear Form ◽  
Class Group ◽  
Finite Abelian Group ◽  
Symmetric Bilinear Form ◽  
Square Root ◽  
Ring Of Integers ◽  
Trace Map ◽  
Odd Order

Let [Formula: see text] be a number field with ring of integers [Formula: see text] and let [Formula: see text] be a finite abelian group of odd order. Given a [Formula: see text]-Galois [Formula: see text]-algebra [Formula: see text], write [Formula: see text] for its trace map and [Formula: see text] for its square root of the inverse different, where [Formula: see text] exists by Hilbert’s formula since [Formula: see text] has odd order. The pair [Formula: see text] is locally [Formula: see text]-isometric to [Formula: see text] whenever [Formula: see text] is weakly ramified, in which case it defines a class in the unitary class group [Formula: see text] of [Formula: see text]. Here [Formula: see text] denotes the canonical symmetric bilinear form on [Formula: see text] defined by [Formula: see text] for all [Formula: see text]. We will study the set of all such classes and show that a subset of them forms a subgroup of [Formula: see text].

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.

A Stickelberger Condition on Cyclic Galois Extensions

1982 ◽  
Vol 34 (3) ◽  
pp. 686-690 ◽  
Author(s):  
L. N. Childs
Keyword(s):  
Abelian Group ◽  
Algebraic Number ◽  
Galois Extension ◽  
Class Group ◽  
Finite Abelian Group ◽  
Algebraic Number Field ◽  
Primitive Elements ◽  
Galois Extensions ◽  
Ring Of Integers ◽  
Isomorphism Classes

LetRbe a commutative ring,Ca finite abelian group,Sa Galois extension ofRwith groupC, in the sense of [1]. ViewingSas anRC-module defines the Picard invariant map [4] from the Harrison group Gal (R,C) of isomorphism classes of Galois extensions ofRwith groupCto CI (RC), the class group ofRC. The image of the Picard invariant map is known to be contained in the subgrouphCl (RC) of primitive elements of CI (RC) (for definition see below). Characterizing the image of the Picard invariant map has been of some interest, for the image describes the extent of failure of Galois extensions to have normal bases.LetRbe the ring of integers of an algebraic number fieldK.

An arithmetic characterization of algebraic number fields with a given class group

1983 ◽  
Vol 94 (1) ◽  
pp. 23-28 ◽  
Author(s):  
David E. Rush
Keyword(s):  
Abelian Group ◽  
Algebraic Number ◽  
Class Group ◽  
Finite Abelian Group ◽  
Number Fields ◽  
Unique Factorization ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Ring Of Integers

Let R be the ring of integers of a number field K with class group G. It is classical that R is a unique factorization domain if and only if G is the trivial group; and the finite abelian group G is generally considered as a measure of the failure of unique factorization in R. The first arithmetic description of rings of integers with non-trivial class groups was given in 1960 by L. Carlitz (1). He proved that G is a group of order ≤ two if and only if any two factorizations of an element of R into irreducible elements have the same number of factors. In ((6), p. 469, problem 32) W. Narkiewicz asked for an arithmetic characterization of algebraic number fields K with class numbers ≠ 1, 2. This problem was solved for certain class groups with orders ≤ 9 in (2), and for the case that G is cyclic or a product of k copies of a group of prime order in (5). In this note we solve Narkiewicz's problem in general by giving arithmetical characterizations of a ring of integers whose class group G is any given finite abelian group.

Existence of simple BIBDs from a prime power difference family with minimum index

2019 ◽  
Vol 18 (09) ◽  
pp. 1950166
Author(s):  
Hsin-Min Sun
Keyword(s):  
Simple Formula ◽  
Prime Power ◽  
Block Designs ◽  
Incomplete Block ◽  
Difference Family ◽  
Balanced Incomplete Block Designs ◽  
Extensive Analysis ◽  
Power Difference ◽  
Balanced Incomplete Block ◽  
Minimum Index

We show that under certain technical conditions that simple [Formula: see text] balanced incomplete block designs (BIBDs) exist for all allowable values of [Formula: see text], where [Formula: see text] is an odd prime power. Our primary technique is to argue for the existence of difference families in finite fields, in the flavor of Wilson [J. Number Theory 4 (1972) 17–47]. We provide an extensive analysis in the cases, where [Formula: see text] and also for [Formula: see text].

A characteristic property of elementary 2-groups of rank six

2011 ◽  
Vol 48 (3) ◽  
pp. 354-370
Author(s):  
Sándor Szabó
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Characteristic Property ◽  
Identity Element ◽  
Finite Abelian Group

Consider a finite abelian group G which is a direct product of its subsets A and B both containing the identity element e. If the non-periodicity of A and B forces that neither A nor B can span the whole G, then G must be an elementary 2-group of rank six.

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