I. Introduction

M. J. Golay first proposed the Golay complementary pairs (GCPs) in 1951 while studying infrared multislit spectroscopy [1]. GCPs are a pair of sequences having zero aperiodic autocorrelation sums (AACSs) at each non-zero time shift. Because of its ideal correlation properties, GCPs have been found several applications in modern day communication systems [2] and radar [3]. However, the lengths of GCPs are very limited [4]. Specifically, binary GCPs are available only for lengths of the form , where are non-negative integers [4]. In 1972, Tseng and Liu extended the idea of GCPs to complementary sequence set (CSS), where each set contains multiple sequences, and have zero AACSs except at zero time shift [5]. In addition, the idea was further extended to a set of CSSs, which are mutually orthogonal to each other, known as mutually orthogonal complementary sequence sets (MOCSSs). A -MOCSS is a family of CSSs, where denotes the set size (i.e., the number of users), denotes the flock size (i.e., the number of sub-carriers) and denotes the sequence length [5]. In [6], the upper bound of the set size of the -MOCSS is given, i.e., . When the set size equals the flock size, MOCSS is called a complete complementary code (CCC). CCCs have been applied in multi-carrier code division multiple access (MC-CDMA) systems [7], MIMO channel estimation and suppressing the multiple access interference. However, a CCC has a limited number of users, which is upper bounded by its set size.