On the Asymptotic Existence of Hadamard Matrices
| Autor: | de Launey, Warwick |
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| Rok vydání: | 2010 |
| Předmět: | |
| Zdroj: | Journal of Combinatorial Theory, Series A 116 (2009) 1002-1008. |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1016/j.jcta.2009.01.001 |
| Popis: | It is conjectured that Hadamard matrices exist for all orders $4t$ ($t>0$). However, despite a sustained effort over more than five decades, the strongest overall existence results are asymptotic results of the form: for all odd natural numbers $k$, there is a Hadamard matrix of order $k2^{[a+b\log_2k]}$, where $a$ and $b$ are fixed non-negative constants. To prove the Hadamard Conjecture, it is sufficient to show that we may take $a=2$ and $b=0$. Since Seberry's ground-breaking result, which showed that we may take $a=0$ and $b=2$, there have been several improvements where $b$ has been by stages reduced to 3/8. In this paper, we show that for all $\epsilon>0$, the set of odd numbers $k$ for which there is a Hadamard matrix of order $k2^{2+[\epsilon\log_2k]}$ has positive density in the set of natural numbers. The proof adapts a number-theoretic argument of Erdos and Odlyzko to show that there are enough Paley Hadamard matrices to give the result. Comment: Keywords: Hadamard matrices, Asymptotic existence, Cocyclic Hadamard matrices, Relative difference sets, Riesel numbers, Extended Riemann hypothesis. (Received 2 August 2008, Available online 18 March 2009) |
| Databáze: | arXiv |
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