Near Butson Hadamard Matrices with Small off diagonal Entries

2016-06-17
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
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: https://hdl.handle.net/11511/81276.