Orthogonal designs and weighing matrices have many applications in areas such as coding theory, cryptography, wireless networking, and communication. In this paper, we first show that if positive integer 𝑘 cannot be written as the sum of three integer squares, then there does not exist any skew-symmetric weighing matrix of order 4𝑛 and weight 𝑘, where 𝑛 is an odd positive integer. Then we show that, for any square 𝑘, there is an integer 𝑁(𝑘) such that, for each 𝑛 ≥ 𝑁(𝑘), there is a symmetric weighing matrix of order 𝑛 and weight 𝑘. Moreover, we improve some of the asymptotic existence results for weighing matrices obtained by Eades, Geramita, and Seberry.

Some nonexistence and asymptotic existence results for weighing matrices / Ghaderpour, Ebrahim. - In: INTERNATIONAL JOURNAL OF COMBINATORICS. - ISSN 1687-9163. - 2016:(2016), pp. 1-6. [10.1155/2016/2162849]

Some nonexistence and asymptotic existence results for weighing matrices

Ebrahim Ghaderpour
Primo
2016

Abstract

Orthogonal designs and weighing matrices have many applications in areas such as coding theory, cryptography, wireless networking, and communication. In this paper, we first show that if positive integer 𝑘 cannot be written as the sum of three integer squares, then there does not exist any skew-symmetric weighing matrix of order 4𝑛 and weight 𝑘, where 𝑛 is an odd positive integer. Then we show that, for any square 𝑘, there is an integer 𝑁(𝑘) such that, for each 𝑛 ≥ 𝑁(𝑘), there is a symmetric weighing matrix of order 𝑛 and weight 𝑘. Moreover, we improve some of the asymptotic existence results for weighing matrices obtained by Eades, Geramita, and Seberry.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1655289
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