Perfect sequences over the quaternions and (4n, 2, 4n, 2n)-relative difference sets in Cn × Q 8

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Abstract

Perfect sequences over general quaternions were introduced in 2009 by Kuznetsov. The existence of perfect sequences of increasing lengths over the basic quaternions Q8 = {±1, ±i, ±j, ±k} was established in 2012 by Barrera Acevedo and Hall. The aim of this paper is to prove a 1–1 correspondence between perfect sequences of length n over Q8 ∪ qQ8 with q = (1 + i + j + k)/2, and (4n, 2, 4n, 2n)-relative difference sets in Cn × Q8 with forbidden subgroup C2; here Cm is a cyclic group of order m. We show that if n = pa + 1 for a prime p and integer a ≥ 0 with n ≡ 2 mod 4, then there exists a (4n, 2, 4n, 2n)-relative different set in Cn × Q8 with forbidden subgroup C2. Lastly, we show that every perfect sequence of length n over Q8 ∪ qQ8 yields a Hadamard matrix of order 4n (and a quaternionic Hadamard matrix of order n over Q8 ∪ qQ8).

Original languageEnglish
Pages (from-to)357-368
Number of pages12
JournalCryptography and Communications: discrete structures, Boolean functions and sequences
Volume10
Issue number2
DOIs
Publication statusPublished - 1 Mar 2018

Keywords

  • Hadamard matrices
  • Perfect sequences
  • Quaternions
  • Relative difference sets

Cite this

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title = "Perfect sequences over the quaternions and (4n, 2, 4n, 2n)-relative difference sets in Cn × Q 8",
abstract = "Perfect sequences over general quaternions were introduced in 2009 by Kuznetsov. The existence of perfect sequences of increasing lengths over the basic quaternions Q8 = {±1, ±i, ±j, ±k} was established in 2012 by Barrera Acevedo and Hall. The aim of this paper is to prove a 1–1 correspondence between perfect sequences of length n over Q8 ∪ qQ8 with q = (1 + i + j + k)/2, and (4n, 2, 4n, 2n)-relative difference sets in Cn × Q8 with forbidden subgroup C2; here Cm is a cyclic group of order m. We show that if n = pa + 1 for a prime p and integer a ≥ 0 with n ≡ 2 mod 4, then there exists a (4n, 2, 4n, 2n)-relative different set in Cn × Q8 with forbidden subgroup C2. Lastly, we show that every perfect sequence of length n over Q8 ∪ qQ8 yields a Hadamard matrix of order 4n (and a quaternionic Hadamard matrix of order n over Q8 ∪ qQ8).",
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AU - Barrera Acevedo, Santiago

AU - Dietrich, Heiko

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N2 - Perfect sequences over general quaternions were introduced in 2009 by Kuznetsov. The existence of perfect sequences of increasing lengths over the basic quaternions Q8 = {±1, ±i, ±j, ±k} was established in 2012 by Barrera Acevedo and Hall. The aim of this paper is to prove a 1–1 correspondence between perfect sequences of length n over Q8 ∪ qQ8 with q = (1 + i + j + k)/2, and (4n, 2, 4n, 2n)-relative difference sets in Cn × Q8 with forbidden subgroup C2; here Cm is a cyclic group of order m. We show that if n = pa + 1 for a prime p and integer a ≥ 0 with n ≡ 2 mod 4, then there exists a (4n, 2, 4n, 2n)-relative different set in Cn × Q8 with forbidden subgroup C2. Lastly, we show that every perfect sequence of length n over Q8 ∪ qQ8 yields a Hadamard matrix of order 4n (and a quaternionic Hadamard matrix of order n over Q8 ∪ qQ8).

AB - Perfect sequences over general quaternions were introduced in 2009 by Kuznetsov. The existence of perfect sequences of increasing lengths over the basic quaternions Q8 = {±1, ±i, ±j, ±k} was established in 2012 by Barrera Acevedo and Hall. The aim of this paper is to prove a 1–1 correspondence between perfect sequences of length n over Q8 ∪ qQ8 with q = (1 + i + j + k)/2, and (4n, 2, 4n, 2n)-relative difference sets in Cn × Q8 with forbidden subgroup C2; here Cm is a cyclic group of order m. We show that if n = pa + 1 for a prime p and integer a ≥ 0 with n ≡ 2 mod 4, then there exists a (4n, 2, 4n, 2n)-relative different set in Cn × Q8 with forbidden subgroup C2. Lastly, we show that every perfect sequence of length n over Q8 ∪ qQ8 yields a Hadamard matrix of order 4n (and a quaternionic Hadamard matrix of order n over Q8 ∪ qQ8).

KW - Hadamard matrices

KW - Perfect sequences

KW - Quaternions

KW - Relative difference sets

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