Related papers: De Bruijn Sequences with Minimum Discrepancy

We show that the de Bruijn-Erd\H{o}s condition for the error term in their improvement of Fekete's Lemma is not only sufficient but also necessary in the following strong sense. Suppose that given a sequence $0\leq f(1)\leq f(2)\leq…

In the trace reconstruction problem our goal is to learn an unknown string $x\in \{0,1\}^n$ given independent traces of $x$. A trace is obtained by independently deleting each bit of $x$ with some probability $\delta$ and concatenating the…

An $n$-bit Gray code is a sequence of all $n$-bit strings such that consecutive strings differ in a single bit. It is well-known that given $\alpha,\beta\in\{0,1\}^n$, an $n$-bit Gray code between $\alpha$ and $\beta$ exists iff the Hamming…

The problem of approximate string matching is important in many different areas such as computational biology, text processing and pattern recognition. A great effort has been made to design efficient algorithms addressing several variants…

We introduce the notion of de Bruijn entropy of an Eulerian quiver and show how the corresponding relative entropy can be applied to practical string similarity problems. This approach explicitly links the combinatorial and…

Let us call a sequence of numbers heapable if they can be sequentially inserted to form a binary tree with the heap property, where each insertion subsequent to the first occurs at a leaf of the tree, i.e. below a previously placed number.…

In this short note, we show a simple characterization of integers that reach records for a sequence described by adding binary strings to runs of 1's and 0's in a binary representation. In particular, we show that this set does not depend…

We put forward new general criteria to design successor rules that generate binary de Bruijn sequences. Prior fast algorithms based on successor rules in the literature are then shown to be special instances. We implemented the criteria to…

The problem of correcting deletions has received significant attention, partly because of the prevalence of these errors in DNA data storage. In this paper, we study the problem of correcting a consecutive burst of at most $t$ deletions in…

Minimizing the discrepancy of a set system is a fundamental problem in combinatorics. One of the cornerstones in this area is the celebrated six standard deviations result of Spencer (AMS 1985): In any system of n sets in a universe of size…

Correlation measure of order $k$ is an important measure of randomness in binary sequences. This measure tries to look for dependence between several shifted version of a sequence. We study the relation between the correlation measure of…

We propose a construction of de Bruijn sequences by the cycle joining method from linear feedback shift registers (LFSRs) with arbitrary characteristic polynomial $f(x)$. We study in detail the cycle structure of the set $\Omega(f(x))$ that…

Negative avoiding sequences of span $n$ are periodic sequences of elements from $\mathbb{Z}_k$ for some $k$ with the property that no $n$-tuple occurs more than once in a period and if an $n$-tuple does occur then its negative does not.…

We first give a construction of binary $t_1$-deletion-$t_2$-insertion-burst correcting codes with redundancy at most $\log(n)+(t_1-t_2-1)\log\log(n)+O(1)$, where $t_1\ge 2t_2$. Then we give an improved construction of binary codes capable…

When considering the number of subtrees of trees, the extremal structures which maximize this number among binary trees and trees with a given maximum degree lead to some interesting facts that correlate to other graphical indices in…

The problem of assembling DNA fragments starting from imperfect strings given by a sequencer, classified as NP hard when trying to get perfect answers, has a huge importance in several fields, because of its relation with the possibility of…

We consider a binary erasure version of the n-channel multiple descriptions problem with symmetric descriptions, i.e., the rates of the n descriptions are the same and the distortion constraint depends only on the number of messages…

M. B. Levin used Sobol-Faure low discrepancy sequences with Pascal matrices modulo $2$ to construct, for each integer $b$, a real number $x$ such that the first $N$ terms of the sequence $(b^n x \mod 1)_{n\geq 1}$ have discrepancy $O((\log…

We study the problems of finding a shortest synchronizing word and its length for a given prefix code. This is done in two different settings: when the code is defined by an arbitrary decoder recognizing its star and when the code is…

Nonlinear complexity is an important measure for assessing the randomness of sequences. In this paper we investigate how circular shifts affect the nonlinear complexities of finite-length binary sequences and then reveal a more explicit…