Construction of 4-phase Golay Complementary Sequence Sets with Small Number of Constituent Sequences and Arbitrary Length

The construction of a Golay complementary sequence (GCS) set with small number of constituent sequences and flexible length has long been an open problem. Compared with the 2-phase GCS pairs, more possible lengths were founded in literatures by extending the phase and the number of con-stituent sequences. For both mathematical theory and engineering applications, it is desirable to keep the number of constituent sequences as small as possible. In this work, we construct a 4-phase GCS set of 2
³
+ [log2
$r$
sequences of arbitrary length
$n$
, where the 10
¹³
-base expansion of
$n$
has
$r$
nonzero digits. Specifically, the 4-phase GCS octets (sets of eight sequences) cover all the lengths no greater than 10
¹³
.