A Direct Construction of Golay Complementary Pairs and Binary Complete Complementary Codes of Length Non-Power of Two

Abstract

Golay complementary pairs (GCPs) and complete complementary codes (CCCs) have found a wide range of practical applications in coding, signal processing and wireless communication due to their ideal correlation properties. The binary CCCs have special advantages in spread spectrum communication for their simple modulo-2 arithmetic operation, modulation, and correlation simplicity; however, they are limited in length. In this paper, we present a direct construction of GCPs, mutually orthogonal Golay complementary sets (MOGCSs) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -ary, including binary CCCs of non-power of two lengths to widen their application in the recent communication field. First, a generalised Boolean function (GBF) based truncation technique is used to construct GCPs of non-power of two lengths which has never been reported before. Then Golay complementary sets (GCSs) and MOGCSs of lengths of the form <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{m-1}+2^{m-3}$ </tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m \geq 5$ </tex-math></inline-formula> ) and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{m-1}+2^{m-2}+2^{m-4}$ </tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m \geq 6$ </tex-math></inline-formula> ) are generated by higher order GBFs. The later length of MOGCSs with direct construction is not available in the literature. Finally, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -ary sequences including binary CCCs with non-power of two lengths are constructed using the union of MOGCSs. There is no such direct construction of binary CCC of non-power of two lengths available in the literature. The column sequence peak to mean envelope power ratio (PMEPR) have been investigated for GCSs and CCCs respectively, and compared with existing works. The column sequence PMEPR of resultant CCCs is effectively upper bounded by 2, which is much lower than that of paraunitary based CCC construction. The proposed construction is also compared with existing works.