Non-existence of Some Nearly Perfect Sequences, Near Butson–Hadamard Matrices, and Near Conference Matrices

In this paper we study almost p-ary sequences and their autocorrelation coefficients. We first study the number ℓ of distinct out-of-phase autocorrelation coefficients for an almost p-ary sequence of period n + s with s consecutive zero-symbols. We prove an upper bound and a lower bound on ℓ. It is shown that ℓ can not be less than min{s, p, n}. In particular, it is shown that a nearly perfect sequence with at least two consecutive zero symbols does not exist. Next we define a new difference set, partial direct product difference set (PDPDS), and we prove the connection between an almost p-ary nearly perfect sequence of type (γ1, γ2) and period n + 2 with two consecutive zero-symbols and a cyclic (n + 2, p, n, n−γ 2 −2 p + γ2, 0, n−γ 1 −1 p + γ1, n−γ 2 −2 p , n−γ 1 −1 p ) PDPDS for arbitrary integers γ1 and γ2. Then we prove a necessary condition on γ2 for the existence of such sequences. In particular, we show that they don't exist for γ2 ≤ −3.

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“…Therefore we get that |T 0,j | = s j (s j − 1) for 0 ≤ j ≤ p − 1. Hence using (11), (12) and (13) we conclude that…”

Section: Autocorrelation Coefficientsmentioning

confidence: 89%

“…The following result on vanishing sums of roots of unity due to Lam and Leung [5], see also [11,Proposition 2.1].…”

Section: Preliminariesmentioning

confidence: 99%

Almost p-ary sequences

Özden

Yayla

2020

Cryptogr. Commun.

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“…Proof. We know by Theorem 3 in [21] that there exists m ∈ Z + such that n = mp. We will prove the minimum distance formula by induction on m. Let R i with i = 1, .…”

Section: Proposition 7 Let H K Be Prime Numbers Andmentioning

confidence: 99%

Power Hadamard matrices and Plotkin-optimal $p^k$-ary codes

Acar¹,

Saraç²,

Yayla³

2020

Preprint

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A power Hadamard matrix H(x) is a square matrix of dimension n with entries from Laurent polynomial ring, where f is some Laurent polynomial of degree greater than 0. In the first part of this work, some new results on power Hadamard matrices are studied, where we mainly extend the work of Craigen and Woodford. In the second part, codes obtained from Butson-Hadamard matrices are discussed and some bounds on the minimum distance of these codes are proved. In particular, we show that the code obtained from a Butson-Hadamard matrix meets the Plotkin bound under a non-homegeneous weight.

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Ideal Factorization Method and Its Applications

Kurt

Yayla

2019

Springer Proceedings in Mathematics &Amp; Statistics

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Non-existence of Some Nearly Perfect Sequences, Near Butson–Hadamard Matrices, and Near Conference Matrices

Cited by 3 publications

References 13 publications

Almost p-ary sequences

Almost p-ary sequences

Power Hadamard matrices and Plotkin-optimal $p^k$-ary codes

Ideal Factorization Method and Its Applications

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