Recently, (linear) codes over and quasi-cyclic (QC) codes (over fields) have been shown to yield useful results in coding theory. Combining these two ideas we study -QC codes and obtain new binary codes using the usual Gray map. Among the new codes, the lift of the famous Golay code to produces a new binary code, a (92 2 28)-code, which is the best among all binary codes (linear or nonlinear). ...

‎The sequences of the form ${E_{mb}g_{n}}_{m‎, ‎ninmathbb{Z}}$,‎ ‎where $E_{mb}$ is the modulation operator‎, ‎$b>0$ and $g_{n}$ is the‎ ‎window function in $L^{2}(mathbb{R})$‎, ‎construct Fourier-like‎ ‎systems‎. ‎We try to consider some sufficient conditions on the window‎ ‎functions of Fourier-like systems‎, ‎to make a frame and find a dual‎ ‎frame with the same structure‎. ‎We also extend t...

For a pair $(f,g)$ of sequences of length $\ell$ whose terms are in $\{-1,1\}$, Pursley and Sarwate established a lower bound on a combined measure of crosscorrelation and autocorrelation for $f$ and $g$. They showed that the sum of the mean square crosscorrelation between $f$ and $g$ and the geometric mean of $f$'s mean square autocorrelation and $g$'s mean square autocorrelation must be at le...

Golay sequences have been used extensively for constructing base sequences, Yang numbers, T-sequences, Hadamard matrices, SBIBDs and Hadamard matrices with maximum possible sums. The possibility of obtaining new Golay sequences is diminishing and only non-existence results are appearing nowadays. We introduce block Golay sequences. It turns out that every result on Golay sequences could be exte...

In this paper we show some very interesting properties of weak Chebyshev subspaces and use them to simplify Pinkus's characterization of Asubspaces of C[a, b]. As a consequence we obtain that if the metric projection PG from C[a, b] onto a finite-dimensional subspace G has a continuous selection and elements of G have no common zeros on (a, b), then G is an /4-subspace.

This thesis consists of two separate topics involved with Golay complementary sequences. In the first part, we present results of an experimental investigation where the distribution of peak-to-average power ratio (PAPR) in downlink MC-CDMA systems is modeled by the generalized extreme value (GEV) distribution. Two orthogonal sets of sequences, WalshHadamard and Golay complementary sequences, a...

A Chebyshev curve C(a, b, c, φ) has a parametrization of the form x(t) = Ta(t); y(t) = Tb(t); z(t) = Tc(t + φ), where a, b, c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ ∈ R. When C(a, b, c, φ) is nonsingular, it defines a polynomial knot. We determine all possible knot diagrams when φ varies. Let a, b, c be integers, a is odd, (a, b) = 1, we show that one can list all pos...

Apart from the ordinary and the periodic Golay pairs, we define also the negaperiodic Golay pairs. (They occurred first, under a different name, in a paper of Ito.) If a Hadamard matrix is also a Toeplitz matrix, we show that it must be either cyclic or negacyclic. We investigate the construction of Hadamard (and weighing matrices) from two negacyclic blocks (2N-type). The Hadamard matrices of ...

In this paper, based on binary Golay complementary sequences, we propose some methods to construct Golay complementary sequences of length 2 for integer n, over the M-QAM constellation and 2M -Q-PAM constellations ,where M = 2 for integer m . A method to judge whether a sequence constructed using the new general offset pairs over the QAM constellation is Golay complementary sequence is proposed...