scholarly journals Applying Balanced Generalized Weighing Matrices to Construct Block Designs

10.37236/1556 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Yury J. Ionin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Block Designs ◽  
Weighing Matrices ◽  
Symmetric Designs

Balanced generalized weighing matrices are applied for constructing a family of symmetric designs with parameters $(1+qr(r^{m+1}-1)/(r-1),r^{m},r^{m-1}(r-1)/q)$, where $m$ is any positive integer and $q$ and $r=(q^{d}-1)/(q-1)$ are prime powers, and a family of non-embeddable quasi-residual $2-((r+1)(r^{m+1}-1)/(r-1),r^{m}(r+1)/2,r^{m}(r-1)/2)$ designs, where $m$ is any positive integer and $r=2^{d}-1$, $3\cdot 2^{d}-1$ or $5\cdot 2^{d}-1$ is a prime power, $r\geq 11$.

scholarly journals New symmetric designs from regular Hadamard matrices

10.37236/1339 ◽  
1997 ◽  
Vol 5 (1) ◽  
Author(s):  
Yury J. Ionin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Hadamard Matrices ◽  
Weighing Matrices ◽  
Symmetric Designs ◽  
Parameter Values

For every positive integer $m$, we construct a symmetric $(v,k,\lambda )$-design with parameters $v={{h((2h-1)^{2m}-1)}\over{h-1}}$, $k=h(2h-1)^{2m-1}$, and $\lambda =h(h-1)(2h-1)^{2m-2}$, where $h=\pm 3\cdot 2^d$ and $|2h-1|$ is a prime power. For $m\geq 2$ and $d\geq 1$, these parameter values were previously undecided. The tools used in the construction are balanced generalized weighing matrices and regular Hadamard matrices of order $9\cdot 4^d$.

scholarly journals On the Twin Designs with the Ionin–type Parameters

10.37236/1479 ◽  
1999 ◽  
Vol 7 (1) ◽  
Author(s):  
H. Kharaghani
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Symmetric Designs ◽  
New Class

Let $4n^2$ be the order of a Bush-type Hadamard matrix with $q=(2n-1)^2$ a prime power. It is shown that there is a weighing matrix $$ W(4(q^m+q^{m-1}+\cdots+q+1)n^2,4q^mn^2) $$ which includes two symmetric designs with the Ionin–type parameters $$ \nu=4(q^m+q^{m-1}+\cdots+q+1)n^2,\;\;\; \kappa=q^m(2n^2-n), \;\;\; \lambda=q^m(n^2-n) $$ for every positive integer $m$. Noting that Bush–type Hadamard matrices of order $16n^2$ exist for all $n$ for which an Hadamard matrix of order $4n$ exist, this provides a new class of symmetric designs.

Sign changes of Fourier coefficients of cusp forms supported on prime power indices

2014 ◽  
Vol 10 (08) ◽  
pp. 1921-1927 ◽  
Author(s):  
Winfried Kohnen ◽  
Yves Martin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Power Indices ◽  
Hecke Operator ◽  
Sign Changes ◽  
Integral Weight ◽  
Hecke Eigenform ◽  
Almost All

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL2(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(pjn))n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.

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].

An Existence Theorem for Room Squares*

1969 ◽  
Vol 12 (4) ◽  
pp. 493-497 ◽  
Author(s):  
R. C. Mullin ◽  
E. Nemeth
Keyword(s):  
Positive Integer ◽  
Existence Theorem ◽  
Prime Power ◽  
Unordered Pair ◽  
Room Squares ◽  
Square Array

It is shown that if v is an odd prime power, other than a prime of the form 22n + 1, then there exists a Room square of order v + 1.A room square of order 2n, where n is a positive integer, is an arrangement of 2n objects in a square array of 2 side 2n - 1, such that each of the (2n - 1)2 cells of the array is either-empty or contains exactly two distinct objects; each of the 2n objects appears exactly once in each row and column; and each (unordered) pair of objects occurs in exactly one cell.

scholarly journals A congruence involving harmonic sums modulo pαqβ

2017 ◽  
Vol 13 (05) ◽  
pp. 1083-1094 ◽  
Author(s):  
Tianxin Cai ◽  
Zhongyan Shen ◽  
Lirui Jia
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Prime Factors ◽  
Necessary And Sufficient ◽  
Positive Integers

In 2014, Wang and Cai established the following harmonic congruence for any odd prime [Formula: see text] and positive integer [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] denote the set of positive integers which are prime to [Formula: see text]. In this paper, we obtain an unexpected congruence for distinct odd primes [Formula: see text], [Formula: see text] and positive integers [Formula: see text], [Formula: see text] and the necessary and sufficient condition for [Formula: see text] Finally, we raise a conjecture that for [Formula: see text] and odd prime power [Formula: see text], [Formula: see text], [Formula: see text] However, we fail to prove it even for [Formula: see text] with three distinct prime factors.

ON THE CLASSIFICATION OF GROUPS OF PRIME-POWER ORDER BY COCLASS: THE 3-GROUPS OF COCLASS 2

2013 ◽  
Vol 23 (05) ◽  
pp. 1243-1288 ◽  
Author(s):  
BETTINA EICK ◽  
C. R. LEEDHAM-GREEN ◽  
M. F. NEWMAN ◽  
E. A. O'BRIEN
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Prime Power Order ◽  
Computational Tools ◽  
Significant Step ◽  
New Phenomena ◽  
Full Classification

In this paper we take a significant step forward in the classification of 3-groups of coclass 2. Several new phenomena arise. Theoretical and computational tools have been developed to deal with them. We identify and are able to classify an important subset of the 3-groups of coclass 2. With this classification and further extensive computations, it is possible to predict the full classification. On the basis of the work here and earlier work on the p-groups of coclass 1, we formulate another general coclass conjecture. It implies that, given a prime p and a positive integer r, a finite computation suffices to determine the p-groups of coclass r.

scholarly journals Nonexistence Results for Hadamard-like Matrices

10.37236/1842 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Justin D. Christian ◽  
Bryan L. Shader
Keyword(s):  
Positive Integer ◽  
Weighing Matrices

The class of square $(0,1,-1)$-matrices whose rows are nonzero and mutually orthogonal is studied. This class generalizes the classes of Hadamard and Weighing matrices. We prove that if there exists an $n$ by $n$ $(0,1,-1)$-matrix whose rows are nonzero, mutually orthogonal and whose first row has no zeros, then $n$ is not of the form $p^k$, $2p^k$ or $3p$ where $p$ is an odd prime, and $k$ is a positive integer.

scholarly journals MÖBIUS–FROBENIUS MAPS ON IRREDUCIBLE POLYNOMIALS

2020 ◽  
pp. 1-12
Author(s):  
F. E. BROCHERO MARTÍNEZ ◽  
DANIELA OLIVEIRA ◽  
LUCAS REIS
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Fixed Points ◽  
Prime Power ◽  
Irreducible Polynomials ◽  
Natural Action

Abstract Let n be a positive integer and let $\mathbb{F} _{q^n}$ be the finite field with $q^n$ elements, where q is a prime power. We introduce a natural action of the projective semilinear group on the set of monic irreducible polynomials over the finite field $\mathbb{F} _{q^n}$ . Our main results provide information on the characterisation and number of fixed points.

Primitive symmetric designs with prime power number of points

2009 ◽  
pp. n/a-n/a
Author(s):  
Snježana Braić ◽  
Anka Golemac ◽  
Joško Mandić ◽  
Tanja Vučičić
Keyword(s):  
Prime Power ◽  
Power Number ◽  
Symmetric Designs
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