Related papers: On the $N$th $2$-adic complexity of binary sequenc…

Let $G$ be a finite abelian group, and $r$ be a multiple of its exponent. The generalized Erd\H{o}s-Ginzburg-Ziv constant $s_r(G)$ is the smallest integer $s$ such that every sequence of length $s$ over $G$ has a zero-sum subsequence of…

Suppose $G$ is a finite abelian group and $S=g_{1}\cdots g_{l}$ is a sequence of elements in $G$. For any element $g$ of $G$ and $A\subseteq\mathbb{Z}\backslash\left\{ 0\right\} $, let $N_{A,g}(S)$ denote the number of subsequences…

Let $G$ be a finite cyclic group of order $n \ge 2$. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot ... \cdot (n_lg)$ where $g\in G$ and $n_1,..., n_l \in [1,\ord(g)]$, and the index $\ind (S)$ of $S$ is defined as…

We define a superalgebra S2(N/2) as a Z2 graded algebra of dimension 2N+3, where N is a positive, odd integer. The even component is a three-dimensional abelian subalgebra, while the odd component is made up of two N-dimensional, mutually…

The $S$-adic conjecture claims that there exists a condition $C$ such that a sequence has a sub-linear complexity if and only if it is an $S$-adic sequence satisfying Condition $C$ for some finite set $S$ of morphisms. We present an…

Let $S=(s_1,s_2,...,s_m,...)$ be a linear recurring sequence with terms in $GF(q^n)$ and $T$ be a linear transformation of $GF(q^n)$ over $GF(q)$. Denote $T(S)=(T(s_1),T(s_2),...,T(s_m),...)$. In this paper, we first present counter…

This paper investigates the problem of active learning for binary label prediction on a graph. We introduce a simple and label-efficient algorithm called S2 for this task. At each step, S2 selects the vertex to be labeled based on the…

We prove that a sequence satisfying a certain symmetry property is $2$-regular in the sense of Allouche and Shallit, i.e., the $\mathbb{Z}$-module generated by its $2$-kernel is finitely generated. We apply this theorem to develop a general…

The $N$th linear complexity of a sequence is a measure of predictability. Any unpredictable sequence must have large $N$th linear complexity. However, in this paper we show that for $q$-automatic sequences over $\mathbb{F}_q$ the converse…

Let $G$ be an abelian group, let $S$ be a sequence of terms $s_1,s_2,...,s_{n}\in G$ not all contained in a coset of a proper subgroup of $G$, and let $W$ be a sequence of $n$ consecutive integers. Let $$W\odot S=\{w_1s_1+...+w_ns_n:\;w_i…

The Arnoux-Rauzy-Poincar\'e multidimensional continued fraction algorithm is obtained by combining the Arnoux-Rauzy and Poincar\'e algorithms. It is a generalized Euclidean algorithm. Its three-dimensional linear version consists in…

We study the complexity of multiplication in noncommutative group algebras which is closely related to the complexity of matrix multiplication. We characterize such semisimple group algebras of the minimal bilinear complexity and show…

The theory of asymptotic complexity provides an approach to characterizing the behavior of programs in terms of bounds on the number of computational steps executed or use of computational resources. We describe work using ACL2 to prove…

The generalized cyclotomic binary sequences $S=S(a, b, c)$ with period $n=pq$ have good autocorrelation property where $(a, b, c)\in \{0, 1\}^3$ and $p, q$ are distinct odd primes. For some cases, the sequences $S$ have ideal or optimal…

In 1952, Dirac proved the following theorem about long cycles in graphs with large minimum vertex degrees: Every $n$-vertex $2$-connected graph $G$ with minimum vertex degree $\delta\geq 2$ contains a cycle with at least $\min\{2\delta,n\}$…

A fast algorithm is presented for determining the linear complexity and the minimal polynomial of periodic sequences over GF(q) with period q n p m, where p is a prime, q is a prime and a primitive root modulo p2. The algorithm presented…

Let $p$ be an odd prime with $2$-adic expansion $\sum_{i=0}^kp_i\cdot2^i$. For a sequence $\underline{a}=(a(t))_{t\ge 0}$ over $\mathbb{F}_{p}$, each $a(t)$ belongs to $\{0,1,\ldots, p-1\}$ and has a unique $2$-adic expansion…

If $G$ is a finite Abelian group, define $s_{k}(G)$ to be the minimal $m$ such that a sequence of $m$ elements in $G$ always contains a $k$-element subsequence which sums to zero. Recently Bitz et al. proved that if $n = exp(G)$, then…

In [Ergodic Theory Dynam. System, 16 (1996) 663--682], S. Ferenczi proved that any minimal subshift with first difference of complexity bounded by 2 is $S$-adic with $\card S \leq 3^{27}$. In this paper, we improve this result by giving an…

Binary periodic sequences with good autocorrelation property have many applications in many aspects of communication. In past decades many series of such binary sequences have been constructed. In the application of cryptography, such…