In this paper we study ultraflat sequences (Pn) of unimodular polynomials Pn ∈ Kn in general, not necessarily those produced by Kahane in his paper [Ka]. We examine how far is a sequence (Pn) of unimodular polynomials Pn ∈ Kn from being conjugate reciprocal. Our main results include the following. Theorem. Given a sequence (εn) of positive numbers tending to 0, assume that (Pn) is a (εn)-ultraf...

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by ∂D. Let Kn := ( pn : pn(z) = n X k=0 akz , ak ∈ C , |ak| = 1 ) . The class Kn is often called the collection of all (complex) unimodular polynomials of degree n. Given a sequence (εn) of positive numbers tending to 0, we say that a sequence (Pn) of unimodular polynomials Pn ∈ Kn i...

PAR-constrained sequences are widely used in communication systems and radars due to various practical needs; specifically, sequences are required to be unimodular or of low peak-to-average power ratio (PAR). For unimodular sequence design, plenty of efforts have been devoted to obtaining good correlation properties. Regarding channel estimation, however, sequences of such properties do not nec...

A Golay Complementary Sequence (CS) has Peak-to-Average-Power-Ratio (PAPR) ≤ 2.0 for its one-dimensional continuous Discrete Fourier Transform (DFT) spectrum. Davis and Jedwab showed that all known length 2m CS, (GDJ CS), originate from certain quadratic cosets of Reed-Muller (1,m). These can be generated using the Rudin-Shapiro construction. This paper shows that GDJ CS have PAPR ≤ 2.0 under a...

We are mainly concerned with the completablity of unimodular rows to an invertible matrix. We first relate this to stable range of the ring. Further, we consider rows generating an idempotent of the ring to obtain completability under special conditions. We try to obtain analogues of important theorems regarding unimodular rows in the case of these “idempotent rows”. By ring, we shall always me...

Let Pn(z) = ∑n k=0 ak,nz k ∈ C[z] be a sequence of unimodular polynomials (|ak,n| = 1 for all k, n) which is ultraflat in the sense of Kahane, i.e., lim n→∞ max |z|=1 ∣

A recursive construction is provided for sequence sets which possess good Hamming Distance and low Peak-to-Average Power Ratio (PAR) with respect to any Local Unitary Unimodular Transform (including all one and multi-dimensional Discrete Fourier Transforms).