Perfect arrays of unbounded sizes over the basic quaternions

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• 作者：Santiago Barrera Acevedo (1)
  Nathan Jolly (1)
  
• 关键词：Perfect arrays over the basic quaternions ; Lee sequences ; Perfect autocorrelation ; Perfect arrays ; Quaternions ; 94A99
• 刊名：Cryptography and Communications
• 出版年：2014
• 出版时间：March 2014
• 年：2014
• 卷：6
• 期：1
• 页码：47-57
• 全文大小：314 KB
• 作者单位：Santiago Barrera Acevedo (1)
  Nathan Jolly (1)
  
  1. Monash University, Clatyon, VIC, Australia
  
• ISSN：1936-2455

文摘

We show the existence of perfect arrays, of unbounded sizes, over the basic quaternions {1,??-,i,??-em class="a-plus-plus">i,j,??-em class="a-plus-plus">j,k,??-em class="a-plus-plus">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,??-,i,??-em class="a-plus-plus">i,j,??-em class="a-plus-plus">j,k,??-em class="a-plus-plus">k}. We show that all modified Lee Sequences (in the sense of Barrera Acevedo and Hall, Lect Notes Comput Sci 159-67, 2012) of length m--em class="a-plus-plus">p--?≡- (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,??-,i,??-em class="a-plus-plus">i,j,??-em class="a-plus-plus">j,k,??-em class="a-plus-plus">k.