Date

2016-06-17

Author

Kurt, Sibel
Yayla, Oğuz

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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

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https://hdl.handle.net/11511/81276

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Graduate School of Applied Mathematics, Conference / Seminar