Near Butson Hadamard Matrices with Small off diagonal Entries

Kurt, Sibel
Yayla, Oğuz
For an integer m ≥ 2let ξm denote a primitive complex m − th root of unity. We call a v-periodic sequence a = (a1, a1, ..., av−1, ...) an m − ary sequence if a1, a1, ..., av−1 ∈ εm. (εm = 1, ξm, ξ2 m, ξ(m−1) m ). An almost m-ary sequence if a0 = 0anda0, a1, ..., av−1 ∈ εm.. For 1 ≤ t ≤ v1 the autocorrelation function Ca(t) is defined by Ca(t) = vX−1 i=0 aiai+t where a is the complex conjugate of a. An m − ary or almost m − ary sequence a of period v is called a perfect sequence (PS) if Ca(t) = 0 for all 1 ≤ t ≤ v − 1. Similarly, an almost m − ary sequence a of period v is called a nearly perfect sequence (NPS) of type γ ∈ {−1, +1} if Ca(t) = γ for all 1 ≤ t ≤ v − 1. This definition is extended to any γ ∈ IR ∩ Z[ξm] with ”small” absolute value with respect to n. Such sequences can be used in applications requiring a sequence with good correaliton properties.[1]. A NPS can be identified with a circulant near Butson-Hadamard matrice. A square matrix H of order v with entries in m is called a near Butson-Hadamard matrix BHγ(v, m) of type if HH T = (υ − γ)I + γJ for a γ ∈ IR ∩ Z[ξm]. A γ −BH(v, m) is called Butson-Hadamard matrix. Very recently, new properties of m − ary γ − BH matrices for γ ∈ Zare studied in Winterhof, Yayla, Ziegler [2].We study m − ary γ − BH matrices for γ /∈ Z, and look for new γ − BH examples and their existence conditions. In addition, we use the methods in Winterhof, Yayla, Ziegler[2] approve some nonexistence result for certain γ−BH matrices. We know that γ ∈ IR ∩ Z[ξm]. In this study,we consider, the case Z[ξm] \ Z. Our motivation is to obtain γ − BH matrices having | γ | as small as possible. So we obtain sequences that provides γ ∈ IR ∩ Z [ξm] and γ /∈ Z. For instance there exist γ − BH(3, 7), γ − BH(4, 7), γ − BH(5, 5), γ − BH(6, 5),γ − BH(7, 5), γ − BH(7, 7), γ − BH(8, 5), γ − BH(9, 5) for certain values of γ such that n γ /∈ Z. In particular γ − BH(5, 5) exist for γ ∈ {−ξ 3 5 − ξ 2 5 +2, 0, 5, ξ3 5 +ξ 2 5 +3} with absolute value γ ∈ {1.38, 0, 5, 3.61} respectively. For a concrete example a = (1, 1, −ξ 2 5 , 1, 1) has γ = −ξ 3 5 −ξ 2 5 + 2 with |γ| = 1.38. We also consider γ − BH(8, 5), it exist for γ ∈ {−ξ 3 5 − ξ 2 5 + 5, −ξ 3 5 − ξ 2 5 , 8, ξ3 5 + ξ 2 5 + 1, ξ3 5 + ξ 2 5 + 6} with absolute value γ ∈ {6.61, 1.61, 8, 0.61, 4.38} respectively.For a concrete example a = (1, 1, ξ2 5 , ξ3 5 , 1, ξ3 5 , ξ5, 1) has γ = −ξ 3 5 − ξ 2 5 + 2 with |γ| = 0.61. We obtained these examples by exhaustive computer search by MAGMA [3]. Moreover we present a method for excluding existence of γ-BH for certain dimensions in case Z[ξm] is not principal ideal domain that is an extension of a method presented in WYZ[2]. The authors are supported by the Scientific and Technological Research Council of Turkey (TUB¨ ˙ITAK) under Grant No. 116R001


Gardner's deformations of the Boussinesq equations
Karasu, Atalay (IOP Publishing, 2006-09-15)
Using the algebraic method of Gardner's deformations for completely integrable systems, we construct recurrence relations for densities of the Hamiltonians for the Boussinesq and the Kaup-Boussinesq equations. By extending the Magri schemes for these equations, we obtain new integrable systems adjoint with respect to the initial ones and describe their Hamiltonian structures and symmetry properties.
A filtration on equivariant Borel-Moore homology
Bingham, Aram; Can, Mahir Bilen; Ozan, Yıldıray (2019-07-04)
Let G / H be a homogeneous variety and let X be a G-equivariant embedding of G / H such that the number of G-orbits in X is finite. We show that the equivariant Borel-Moore homology of X has a filtration with associated graded module the direct sum of the equivariant Borel-Moore homologies of the G-orbits. If T is a maximal torus of G such that each G-orbit has a T-fixed point, then the equivariant filtration descends to give a filtration on the ordinary Borel-Moore homology of X. We apply our findings to c...
Value sets of folding polynomials over finite fields
Küçüksakallı, Ömer (2019-01-01)
Let k be a positive integer that is relatively prime to the order of the Weyl group of a semisimple complex Lie algebra g. We find the cardinality of the value sets of the folding polynomials P-g(k)(x) is an element of Z[x] of arbitrary rank n >= 1, over finite fields. We achieve this by using a characterization of their fixed points in terms of exponential sums.
Pamuk, Semra (2014-07-03)
Let G be a finite group and F be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative F-projective resolution for Z when F is the family of all subgroups HG with rkHrkG-1. We answer this question negatively by calculating the relative group cohomology FH*(G, ?(2)) where G = Z/2xZ/2 and F is the family of cyclic subgroups of G. To do this calculation we first observe that the relative group cohomology FH*(G, M) can be calc...
L Polynomials of the Curve yqn y xqh 1 over Fqm
Özbudak, Ferruh (null; 2014-09-28)
Let chi be a smooth, geometrically irreducible and projective curve over a finite field F-q of odd characteristic. The L-polynomial L-chi(t) of chi determines the number of rational points of chi not only over F-q but also over F-qs for any integer s >= 1. In this paper we determine L-polynomials of a class of such curves over F-q.
Citation Formats
S. Kurt and O. Yayla, “Near Butson Hadamard Matrices with Small off diagonal Entries,” presented at the Near Butson Hadamard Matrices with Small off diagonal Entries”, 3rd Istanbul Design Theory, Graph Theory and Combinatorics Workshop, 13 - 17 Haziran 2016, 2016, Accessed: 00, 2021. [Online]. Available: