In orthogonal frequency division multiplexing (OFDM), peak-to-mean envelope power ratio (PMEPR) plays a vital role in the system's performance. In binary Golay comple-mentary pair (GCP), PMEPR is less than or equal to 2. However, the binary GCPs are limited in length. This paper proposes a novel construction of a new family of binary complementary pairs, named an odd shift complementary pair (OSCP), having all possible lengths. The PMEPR depends on the function we choose. However, we have calculated the PMEPR of OSCP having a length of$2^{m}-1$, where the function is a second-order Boolean function and asymptotically bounded by 4. OSCP is the pair of two complex-valued sequences having their sum of aperiodic auto-correlation function (AACF) zero at every odd time shift.

One of the important applications of Golay complementary sets (GCSs) is the reduction of peak-to-mean en-velope power ratio (PMEPR) in orthogonal frequency division multiplexing (OFDM) systems. OFDM has played a major role in modern wireless systems such as long-term-evolution (LTE), 5th generation (5G) wireless standards, etc. This paper searches for systematic constructions of GCSs of arbitrary lengths and alphabet sizes. The proposed constructions are based on extended Boolean functions (EBFs). For the first time, we can generate codes of independent parameter choices.

This paper presents a direct construction of an optimal symmetrical Z-complementary code set (SZCCS) of prime power lengths using a multi-variable function (MVF). SZCCS is a natural extension of the Z-complementary code set (ZCCS), which has only front-end zero correlation zone (ZCZ) width. SZCCS has both front-end and tail-end ZCZ width. SZCCSs are used in developing optimal training sequences for broadband generalized spatial modulation systems over frequency-selective channels because they have ZCZ width on both the front and tail ends. The construction of optimal SZCCS with large set sizes and prime power lengths is presented for the first time in this paper. Furthermore, it is worth noting that several existing works on ZCCS and SZCCS can be viewed as special cases of the proposed construction.

Multiple sets of multiple zero correlation zone (MZCZ) sequence sets are suitable for a quasi-synchronous code division multiple access (QS-CDMA) system in multi-user, multi-cluster and multi-cell environments to eliminate the interferences. In this paper, we present a two-folded direct construction of asymptotically optimal multiple sets of MZCZ sequence sets with inter-set zero-cross correlation zone (ZCCZ) from multivariable functions (MVFs). Each MZCZ sequence set forms a ZCZ sequence set after taking the union of elements in the MZCZ sequence set. Therefore, multiple sets of MZCZ sequence sets will form a new MZCZ sequence set, which is referred to as fold one. Further, this newly formed MZCZ sequence set again generates a ZCZ sequence set in the same manner, which is referred to as fold two. Also, for the first time in the literature, we are presenting multiple sets of MZCZ sequence sets. In this paper, many new lengths of MZCZ sequence sets are covered. Finally, the construction is compared with the existing state-of-the-art.

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$q$-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$2^{m-1}+2^{m-3}$($m \geq 5$) and$2^{m-1}+2^{m-2}+2^{m-4}$($m \geq 6$) are generated by higher order GBFs. The later length of MOGCSs with direct construction is not available in the literature. Finally,$q$-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.

Teramoto et al. [ IEICE Trans. Inform. Syst., 89-D (2006), pp. 2348--2356] defined a measure called the gap ratio that measures the uniformity of a finite point set sampled from $\cal S$, a bounded subset of $\mathbb{R}^2$. This definition of uniformity measure can be generalized over all metric spaces by appealing to covering and packing radius. We consider discrete spaces like graph and set of points in the Euclidean space and continuous spaces like the unit square and path connected spaces. The definition of the gap ratio needs only a metric unlike discrepancy, a widely used uniformity measure, that depends on the notion of a range space and its volume. We show some interesting connections of the gap ratio to Delaunay triangulation and packing and covering. Asano [ Inform. Process. Lett., 109 (2008), pp. 57--60] opined that the discrete version of the uniformity problem makes it amenable to pose combinatorial optimization related questions. The major focus of this work is on finding lower bounds and solving optimization related questions about selecting uniform point samples from metric spaces using the gap ratio. In deducing lower bounds on the gap ratio, we exploit its relation to packing and covering. We have been able to show existence of point configurations with certain cardinality obtained using farthest point insertion and characterized using a recurrence to achieve the lower bounds deduced. Apart from the lower bounds, we prove hardness and approximation hardness results. We show that a general approximation algorithm framework gives different approximation ratios for different metric spaces based on the lower bound we deduce. Apart from the above, we show existence of coresets for sampling uniform points from the Euclidean space---for both the static and the streaming case. This leads to a $( 1+\epsilon)$-approximation algorithm for uniform sampling from the Euclidean space.