Related papers: On the $k$-error linear complexity of binary seque…
We consider two quantities that measure complexity of binary strings: $\mathit{KA}(x)$ is defined as the minus logarithm of continuous a priori probability on the binary tree, and $\mathit{KP}(x)$ denotes prefix complexity of a binary…
Generalized cyclotomic sequences of period pq have several desirable randomness properties if the two primes p and q are chosen properly. In particular,Ding deduced the exact formulas for the autocorrelation and the linear complexity of…
We consider the quantum complexities of the following three problems: searching an ordered list, sorting an un-ordered list, and deciding whether the numbers in a list are all distinct. Letting N be the number of elements in the input list,…
We consider the complexities of substitutive sequences over a binary alphabet. By studying various types of special words, we show that, knowing some initial values, its complexity can be completely formulated via a recurrence formula…
Motivated by the constructions of binary sequences by utilizing the cyclic elliptic function fields over the finite field $\mathbb{F}_{2^{n}}$ by Jin \textit{et al.} in [IEEE Trans. Inf. Theory 71(8), 2025], we extend the construction to…
We derive B\'ezout identities for the minimal polynomials of a finite sequence and use them to prove a theorem of Wang and Massey on binary sequences with a perfect linear complexity profile. We give a new proof of Rueppel's conjecture and…
Let $ \prod_{i=1}^d (X-\alpha_i Y) \in{\mathbb C}[X,Y]$ be a binary form and let $\epsilon_1,\dots,\epsilon_d$ be nonzero complex numbers. We consider the family of binary forms $ \prod_{i=1}^d (X-\alpha_i \epsilon_i^aY)$, $a\in {\mathbb…
New generalized cyclotomic binary sequences of period $p^2$ are proposed in this paper, where $p$ is an odd prime. The sequences are almost balanced and their linear complexity is determined. The result shows that the proposed sequences…
We study possible Scott sentence complexities of linear orderings using two approaches. First, we investigate the effect of the Friedman-Stanley embedding on Scott sentence complexity and show that it only preserves…
In recent years a variant of the resonance method was developed which allowed to obtain improved $\Omega$-results for the Riemann zeta function along vertical lines in the critical strip. In the present paper we show how this method can be…
We address the question of computing one selected term of an algebraic power series. In characteristic zero, the best algorithm currently known for computing the $N$th coefficient of an algebraic series uses differential equations and has…
A binary code Enc$:\{0,1\}^k \to \{0,1\}^n$ is $(0.5-\epsilon,L)$-list decodable if for all $w \in \{0,1\}^n$, the set List$(w)$ of all messages $m \in \{0,1\}^k$ such that the relative Hamming distance between Enc$(m)$ and $w$ is at most…
Binary sequences with optimal autocorrelation and large linear complexity have important applications in cryptography and communications. Very recently, a class of binary sequences of period $4p$ with optimal autocorrelation was proposed…
We study the computational complexity of decision problems in $k$-level linear programming (LP). Seminal work by Jeroslow establishes that determining whether the optimal objective value of a $k$-level LP is at least as good as a given…
We identify a binary sequence $\mathcal{S}=(s_n)_{n=0}^\infty$ with the $2$-adic integer $G_\mathcal{S}(2)=\sum\limits_{n=0}^\infty s_n2^n$. In the case that $G_\mathcal{S}(2)$ is algebraic over $\mathbb{Q}$ of degree $d\ge 2$, we prove…
Let $q$ be a prime power and $n, r$ integers such that $r\mid q^n-1$. An element of $\mathbb{F}_{q^n}$ of multiplicative order $(q^n-1)/r$ is called \emph{$r$-primitive}. For any odd prime power $q$, we show that there exists a…
A modified Dirichlet character $f$ is a completely multiplicative function such that for some Dirichlet character $\chi$, $f(p)=\chi(p)$ for all but a finite number of primes $p\in S$, and for those exceptional primes $p\in S$, $|f(p)|\leq…
We investigate when the sequence of binomial coefficients \binom{k}{i} modulo a prime p, for a fixed positive integer k, satisfies a linear recurrence relation of (positive) degree h in the finite range 0\le i\le k. In particular, we prove…
Let $q$ be an odd prime power and write \[ \theta_q := \frac{\phi(q-1)}{q-1}. \] If $\theta_q < \tfrac{1}{3}$, or if $\theta_q = \tfrac{1}{3}$ and $q \notin \{7,13,19,25,37\}$, then the finite field $\F$ contains a pair of consecutive…
A permutation $\pi$ contains a permutation $\sigma$ as a pattern if it contains a subsequence of length $|\sigma|$ whose elements are in the same relative order as in the permutation $\sigma$. This notion plays a major role in enumerative…