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Erschienen in:
01.03.2014
Perfect arrays of unbounded sizes over the basic quaternions
Erschienen in: Cryptography and Communications | Ausgabe 1/2014
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Abstract
We show the existence of perfect arrays, of unbounded sizes, over the basic quaternions {1, − 1,i, − i,j, − j,k, − k}. We translate the algorithm of Arasu and de Launey, to inflate perfect arrays over the four roots of unity, from a polynomial, into a simple matrix approach. Then, we modify this algorithm to inflate perfect arrays over the basic quaternions {1, − 1,i, − i,j, − j,k, − k}. We show that all modified Lee Sequences (in the sense of Barrera Acevedo and Hall, Lect Notes Comput Sci 159–167, 2012) of length m = p + 1 ≡ 2 (mod 4), where p is a prime number, can be folded into a perfect two-dimensional array (with only one occurrence of the element j) of size \(2\times \frac{m}{2}\), with \(GCD(2,\frac{m}{2})=1\). Then, each of these arrays can be inflated into perfect arrays of sizes \(2p\times \frac{m}{2}p\) (previously unknown sizes), with a random appearance of all the elements 1, − 1,i, − i,j, − j,k, − k.