Constant Amplitude (CA), Zero Auto Correlation (ZAC) sequences (or CAZAC sequences, aka perfect sequences) have numerous applications. We generalize the CAZAC notion to what we term as CASAC by permitting small autocorrelations (SAC). We extend Björck’s classification result of two-valued CAZAC sequences by providing a complete classification of all almost 2-valued (i.e., two-valued except for the first position which uses a third value) CASAC sequences. While Björck’s original work dealt only with primes p, we extend his ideas to any abelian group of order $v\equiv 1\pmod {4}$ , as opposed to restricting just to the prime fields GF(p). Björck sequences have better ambiguity function than Zadoff-Chu sequences, making them suitable for radar and communications applications in the presence of high Doppler shifts. In fact, the discrete narrow band ambiguity function has an optimal bound in case of Björck sequences (as opposed to Gauss sequences). A one-parameter infinite family of CASAC we construct would have applications in Multiple-Input Multiple-Output (MIMO) areas. Toward MIMO applications, we introduce a performance measure we term as cross merit factor to study cross correlation behavior, generalizing the well-known notion of Golay Merit Factor (GMF).

We investigate how Legendre $G$-array pairs are related to several different perfect binary $G$-array families. In particular we study the relations between Legendre $G$-array pairs, Sidelnikov-Lempel-Cohn-Eastman $\mathbb{Z}_{q-1}$-arrays, Yamada-Pott $G$-array pairs, Ding-Helleseth-Martinsen $\mathbb{Z}_{2}\times \mathbb{Z}_p^{m}$-arrays, Yamada $\mathbb{Z}_{(q-1)/2}$-arrays, Szekeres $\mathbb{Z}^m_{p}$-array pairs, Paley $\mathbb{Z}^m_{p}$-array pairs, and Baumert $\mathbb{Z}^{m_1}_{p_1}\times \mathbb{Z}^{m_2}_{p_2}$-array pairs. Our work also solves one of the two open problems posed in Ding~[J. Combin. Des. 16 (2008), 164-171]. Moreover, we provide several computer search based existence and non-existence results regarding Legendre $\mathbb{Z}_n$-array pairs. Finally, by using cyclotomic cosets, we provide a previously unknown Legendre $\mathbb{Z}_{57}$-array pair.

We propose using periodic binary sequences with optimal correlation energy (CE) to generate near E(s(2))-optimal supersaturated designs (SSDs) and near D-optimal 2-symbol fractional factorial designs for the all main effects and the intercept model. We derive a lower bound for the CE of odd length periodic sequences and provide previously unknown odd length periodic sequences with optimal CE up to length 43.

We employ algebraic methods to provide some new constructions of what we call optimal high-energy ternary sequences (a sequence with entries in {0,1,-1} with a single zero, having optimal correlation properties). Our motivation for these constructions stems from their usefulness in several areas related to communication and radar systems.

In the context of radar waveforms, there are many references to “Doppler Tolerance” in the literature, but a formal, complete, precise, and reasonable definition has not been forthcoming. We attempt to fill this void in this paper. We revisit existing definitions and demonstrate, that they are either too restrictive for any practical use, incomplete, or imprecise. Our definition uses the ambiguity function as its main ingredient. We emphasize that the Doppler tolerance is a 3D function. The first parameter is a spatial variable which relates to the measures of connectedness of possible disjoint ambiguity function peaks. The second parameter is the time delay at which the Doppler tolerance is itself specified, and the third parameter is similar to a threshold and is related to height of the ambiguity function used in measuring the Doppler tolerance. As a byproduct of our definition, we analytically conclude that for small time bandwidth products the linear frequency modulated (LFM) waveform is only as Doppler tolerant as an unmodulated rectangular pulse. We therefore bust the well known myth that “(all) chirps are Doppler tolerant”.

A weighing matrix is a square matrix whose entries are 1, 0 or -1, such that the matrix times its transpose is some integer multiple of the identity matrix. We examine the case where these matrices are said to be developed by an abelian group. Through a combination of extending previous results and by giving explicit constructions we will answer the question of existence for 318 such matrices of order and weight both below 100. At the end, we are left with 98 open cases out of a possible 1,022. Further, some of the new results provide insight into the existence of matrices with larger weights and orders.