1,551 research outputs found
Root-Hadamard transforms and complementary sequences
In this paper we define a new transform on (generalized) Boolean functions,
which generalizes the Walsh-Hadamard, nega-Hadamard, -Hadamard,
consta-Hadamard and all -transforms. We describe the behavior of what we
call the root- Hadamard transform for a generalized Boolean function in
terms of the binary components of . Further, we define a notion of
complementarity (in the spirit of the Golay sequences) with respect to this
transform and furthermore, we describe the complementarity of a generalized
Boolean set with respect to the binary components of the elements of that set.Comment: 19 page
New Sets of Optimal Odd-length Binary Z-Complementary Pairs
A pair of sequences is called a Z-complementary pair (ZCP) if it has zero
aperiodic autocorrelation sums (AACSs) for time-shifts within a certain region,
called zero correlation zone (ZCZ). Optimal odd-length binary ZCPs (OB-ZCPs)
display closest correlation properties to Golay complementary pairs (GCPs) in
that each OB-ZCP achieves maximum ZCZ of width (N+1)/2 (where N is the sequence
length) and every out-of-zone AACSs reaches the minimum magnitude value, i.e.
2. Till date, systematic constructions of optimal OB-ZCPs exist only for
lengths , where is a positive integer. In this
paper, we construct optimal OB-ZCPs of generic lengths (where are non-negative integers and
) from inserted versions of binary GCPs. The key leading to the
proposed constructions is several newly identified structure properties of
binary GCPs obtained from Turyn's method. This key also allows us to construct
OB-ZCPs with possible ZCZ widths of , and through proper
insertions of GCPs of lengths , respectively. Our proposed OB-ZCPs have applications in
communications and radar (as an alternative to GCPs)
Dynamical Algebraic Combinatorics, Asynchronous Cellular Automata, and Toggling Independent Sets
Though iterated maps and dynamical systems are not new to combinatorics, they have enjoyed a renewed prominence over the past decade through the elevation of the subfield that has become known as dynamical algebraic combinatorics. Some of the problems that have gained popularity can also be cast and analyzed as finite asynchronous cellular automata (CA). However, these two fields are fairly separate, and while there are some individuals who work in both, that is the exception rather than the norm. In this article, we will describe our ongoing work on toggling independent sets on graphs. This will be preceded by an overview of how this project arose from new combinatorial problems involving homomesy, toggling, and resonance. Though the techniques that we explore are directly applicable to ECA rule 1, many of them can be generalized to other cellular automata. Moreover, some of the ideas that we borrow from cellular automata can be adapted to problems in dynamical algebraic combinatorics. It is our hope that this article will inspire new problems in both fields and connections between them
Two-Dimensional Z-Complementary Array Quads with Low Column Sequence PMEPRs
In this paper, we first propose a new design strategy of 2D -complementary
array quads (2D-ZCAQs) with feasible array sizes. A 2D-ZCAQ consists of four
distinct unimodular arrays satisfying zero 2D auto-correlation sums for
non-trivial 2D time-shifts within certain zone. Then, we obtain the upper
bounds on the column sequence peak-to-mean envelope power ratio (PMEPR) of the
constructed 2D-ZCAQs by using specific auto-correlation properties of some seed
sequences. The constructed 2D-ZCAQs with bounded column sequence PMEPR can be
used as a potential alternative to 2D Golay complementary array sets for
practical applicationsComment: This work has been presented in 2023 IEEE International Symposium on
Information Theory (ISIT), Taipei, Taiwa
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