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We investigate binary sequence pairs with two-level correlation in terms of their corresponding cyclic difference pairs (CDPs). We define multipliers of a cyclic difference pair and present an existence theorem for multipliers, which could be applied to check the existence/nonexistence of certain hypothetical cyclic difference pairs. Then, we focus on the ideal case where all the out-of-phase correlation coefficients are zero. It is known that such an ideal binary sequence pair exists for length υ = 4*u* for every *u* ≥ 1. Using the techniques developed here on the theory of multipliers of a CDP and some exhaustive search, we are able to determine that, for lengths υ ≤ 30, (1) there does not exist "any other" *ideal*/ binary sequence pair and (2) every example in this range is equivalent to the one of length υ = 4*u* above. We conjecture that if there is a binary sequence pair with an ideal two-level correlation then its in-phase correlation must be 4. This implies so called the circulant Hadamard matrix conjecture.

- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E93-A No.11 pp.2266-2271

- Publication Date
- 2010/11/01

- Publicized

- Online ISSN
- 1745-1337

- DOI
- 10.1587/transfun.E93.A.2266

- Type of Manuscript
- Special Section PAPER (Special Section on Signal Design and its Application in Communications)

- Category
- Sequences

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Seok-Yong JIN, Hong-Yeop SONG, "Binary Sequence Pairs with Two-Level Correlation and Cyclic Difference Pairs" in IEICE TRANSACTIONS on Fundamentals,
vol. E93-A, no. 11, pp. 2266-2271, November 2010, doi: 10.1587/transfun.E93.A.2266.

Abstract: We investigate binary sequence pairs with two-level correlation in terms of their corresponding cyclic difference pairs (CDPs). We define multipliers of a cyclic difference pair and present an existence theorem for multipliers, which could be applied to check the existence/nonexistence of certain hypothetical cyclic difference pairs. Then, we focus on the ideal case where all the out-of-phase correlation coefficients are zero. It is known that such an ideal binary sequence pair exists for length υ = 4*u* for every *u* ≥ 1. Using the techniques developed here on the theory of multipliers of a CDP and some exhaustive search, we are able to determine that, for lengths υ ≤ 30, (1) there does not exist "any other" *ideal*/ binary sequence pair and (2) every example in this range is equivalent to the one of length υ = 4*u* above. We conjecture that if there is a binary sequence pair with an ideal two-level correlation then its in-phase correlation must be 4. This implies so called the circulant Hadamard matrix conjecture.

URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E93.A.2266/_p

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@ARTICLE{e93-a_11_2266,

author={Seok-Yong JIN, Hong-Yeop SONG, },

journal={IEICE TRANSACTIONS on Fundamentals},

title={Binary Sequence Pairs with Two-Level Correlation and Cyclic Difference Pairs},

year={2010},

volume={E93-A},

number={11},

pages={2266-2271},

abstract={We investigate binary sequence pairs with two-level correlation in terms of their corresponding cyclic difference pairs (CDPs). We define multipliers of a cyclic difference pair and present an existence theorem for multipliers, which could be applied to check the existence/nonexistence of certain hypothetical cyclic difference pairs. Then, we focus on the ideal case where all the out-of-phase correlation coefficients are zero. It is known that such an ideal binary sequence pair exists for length υ = 4*u* for every *u* ≥ 1. Using the techniques developed here on the theory of multipliers of a CDP and some exhaustive search, we are able to determine that, for lengths υ ≤ 30, (1) there does not exist "any other" *ideal*/ binary sequence pair and (2) every example in this range is equivalent to the one of length υ = 4*u* above. We conjecture that if there is a binary sequence pair with an ideal two-level correlation then its in-phase correlation must be 4. This implies so called the circulant Hadamard matrix conjecture.},

keywords={},

doi={10.1587/transfun.E93.A.2266},

ISSN={1745-1337},

month={November},}

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TY - JOUR

TI - Binary Sequence Pairs with Two-Level Correlation and Cyclic Difference Pairs

T2 - IEICE TRANSACTIONS on Fundamentals

SP - 2266

EP - 2271

AU - Seok-Yong JIN

AU - Hong-Yeop SONG

PY - 2010

DO - 10.1587/transfun.E93.A.2266

JO - IEICE TRANSACTIONS on Fundamentals

SN - 1745-1337

VL - E93-A

IS - 11

JA - IEICE TRANSACTIONS on Fundamentals

Y1 - November 2010

AB - We investigate binary sequence pairs with two-level correlation in terms of their corresponding cyclic difference pairs (CDPs). We define multipliers of a cyclic difference pair and present an existence theorem for multipliers, which could be applied to check the existence/nonexistence of certain hypothetical cyclic difference pairs. Then, we focus on the ideal case where all the out-of-phase correlation coefficients are zero. It is known that such an ideal binary sequence pair exists for length υ = 4*u* for every *u* ≥ 1. Using the techniques developed here on the theory of multipliers of a CDP and some exhaustive search, we are able to determine that, for lengths υ ≤ 30, (1) there does not exist "any other" *ideal*/ binary sequence pair and (2) every example in this range is equivalent to the one of length υ = 4*u* above. We conjecture that if there is a binary sequence pair with an ideal two-level correlation then its in-phase correlation must be 4. This implies so called the circulant Hadamard matrix conjecture.

ER -