摘要
Golay在1961年的論文中定義一維的互補序列，1992年Dymond首先介紹了二進位±1值的格雷陣列對的產生方法。2008年Fiedler、Jedwab、Parker介紹了PSK多維格雷陣列的產生方法。該文所提出的方法是合併一維格雷序列對產生出高維格雷陣列對，但無法由一維的格雷序列直接產生二維的格雷矩陣，若需要二維格雷矩陣，必須經由合併指標降低維度才能達成。 本論文將焦點放在二維格雷矩陣的的產生方法與結構探討，藉由兩種方法來產生格雷矩陣。方法一由目標二維格雷矩陣的row與column數目，選擇兩組格雷序列對，再將其元素組合成二維格雷矩陣，此方法優點在於使用一維格雷序列僅需單一步驟，即可產生二維的格雷矩陣。第二種產生格雷矩陣的方法，由目標二維格雷矩陣的總元素數（row×column 數），選擇足夠數量的長度1的trivial pairs，產生高維度的格雷陣列對。若所需的二維格雷矩陣的元素數量為2^m 個，則需要m+1對trivial pairs。產生出的高維度格雷陣列的尺寸為2^(m)，再乘常數變化，則可以產生出更多的高維格雷陣列，將這些高維格雷陣列的指標合併為兩個，也就是將維度由高維降至2維。此方法可以產生出較方法一更多的二維格雷矩陣。 本論文使用上述兩種方法來產生 、QPSK二維格雷矩陣，方法一使用64對長度2的QPSK格雷序列對與512對長度4的QPSK格雷序列對，產生出2112個4×4、QPSK二維格雷矩陣；方法二使用5對trivial pairs，產生出6272個4×4、QPSK二維格雷矩陣，其中包含了所有方法一所產生的矩陣。
並列摘要
The construction of one-dimensional complementary sequences was proposed by Golay in 1961, and the construction of binary Golay array pairs was introduced by Dymond in 1992. In 2008, Fiedler, Jedwab, and Parker introduced the construction of PSK multi-dimensional Golay complementary arrays. Each multi-dimensional Golay array pair was constructed from one-dimensional Golay sequence pairs. However, two-dimensional Golay array pairs cannot be directly constructed by one dimensional Golay sequence pairs in their method. In this paper we focus on the construction of two-dimensional Golay matrices from one-dimensional Golay sequence pairs. Two construction methods are discussed. The first method uses the numbers of the rows and columns of the desired two-dimensional Golay matrix pairs to choose two one-dimensional Golay sequence pairs, and combine their elements form a two-dimensional Golay matrix pair. The advantage of the first method is that there is only one step to construct two-dimensional Golay matrix from one-dimensional Golay sequence pairs. The second method finds the total number of the elements(row×column ) of the desired two-dimensional Golay matrix, 2^m , and select m+1 trivial sequence pairs to construct a basic multi-dimensional Golay array pair of size 2^(m) . By taking “affine offsets”, a set of Golay array pairs can be generated from the basic pair. Combining the indexes of these multi-dimensional Golay arrays to two indexes, decreases the array dimension to two. The second method constructs more two-dimensional Golay matrix pairs compare the first method. We apply these two methods to the construction of 4×4, QPSK two-dimensional Golay matrices. The first method uses 64 QPSK Golay sequence pairs of length 2 and 512 QPSK Golay sequence pairs of length 4 to construct 2112 matrices. The second method uses 5 trivial pairs to construct 6272 4×4, QPSK two-dimensional Golay matrices. All the matrices constructed by the first method are included in the matrices constructed by the second method.