Related papers: De Bruijn Cycles for Covering Codes

Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. In this paper we prove an upper bound on the minimal…

The well-known approach of Bose, Ray-Chaudhuri and Hocquenghem and its generalization by Hartmann and Tzeng are lower bounds on the minimum distance of simple-root cyclic codes. We generalize these two bounds to the case of repeated-root…

Covers being one of the most popular form of regularities in strings, have drawn much attention over time. In this paper, we focus on the problem of linear time inference of strings from cover arrays using the least sized alphabet possible.…

The nodes of the de Bruijn graph B(d,3) consist of all strings of length 3, taken from an alphabet of size d, with edges between words which are distinct substrings of a word of length 4. We give an inductive characterization of the maximum…

We introduce the notions of de Bruijn polyominoes and prismatic polyominoes, which generalize the notions of de Bruijn sequences and arrays. Given a small fixed polyomino $p$ and a set of colors $[n]$, a de Bruijn polyomino for $(p,n)$ is a…

This paper proves the existence of $t$-identifying codes on the class of undirected de Bruijn graphs with string length $n$ and alphabet size $d$, referred to as $\mathcal{B}(d,n)$. It is shown that $\mathcal{B}(d,n)$ is $t$-identifiable…

We use modular arithmetic to construct a de Bruijn array containing all fillings of an L (a 2 x 2 array with the upper right corner removed) with digits chosen from {0,...,k-1}.

The de Bruijn torus (or grid) problem looks to find an $n$-by-$m$ binary matrix in which every possible $j$-by-$k$ submatrix appears exactly once. The existence and construction of these binary matrices was determined in the 70's, with…

For positive integers $k,n$, a de Bruijn sequence $B(k,n)$ is a finite sequence of elements drawn from $k$ characters whose subwords of length $n$ are exactly the $k^n$ words of length $n$ on $k$ characters. This paper introduces the…

A $t$-covering array with entries from the alphabet ${\cal Q}=\{0,1,\ldots,q-1\}$ is a $k\times n$ stack, so that for any choice of $t$ (typically non-consecutive) columns, each of the $q^{t}$ possible $t$-letter words over ${\cal Q}$…

The paired de Bruijn graph is an extension of de Bruijn graph incorporating mate pair information for genome assembly proposed by Mevdedev et al. However, unlike in an ordinary de Bruijn graph, not every path or cycle in a paired de Bruijn…

This paper investigates the (quasi-)periodicity of a string when the string is edited. A string $C$ is called a cover (as known as a quasi-period) of a string $T$ if each character of $T$ lies within some occurrence of $C$. By definition, a…

Pseudo-random arrays and perfect maps are the two-dimensional analogs of M-sequences and de Bruijn sequences, respectively. We modify the definitions to be applied to codes. These codes are also the two-dimensional analogs of certain…

Let $K_q(n,r)$ denote the minimum size of a $q$-ary covering code of word length $n$ and covering radius $r$. In other words, $K_q(n,r)$ is the minimum size of a set of $q$-ary codewords of length $n$ such that the Hamming balls of radius…

Every binary De~Bruijn sequence of order n satisfies a recursion 0=x_n+x_0+g(x_{n-1}, ..., x_1). Given a function f on (n-1) bits, let N(f; r) be the number of functions generating a De Bruijn sequence of order n which are obtained by…

A Gray code is a listing structure for a set of combinatorial objects such that some consistent (usually minimal) change property is maintained throughout adjacent elements in the list. While Gray codes for m-ary strings have been…

We introduce subsequence covers (s-covers, in short), a new type of covers of a word. A word $C$ is an s-cover of a word $S$ if the occurrences of $C$ in $S$ as subsequences cover all the positions in $S$. The s-covers seem to be…

Repeat finding in strings has important applications in subfields such as computational biology. Surprisingly, all prior work on repeat finding did not consider the constraint on the locality of repeats. In this paper, we propose and study…

We propose a new camera-based method of robot identification, tracking and orientation estimation. The system utilises coloured lights mounted in a circle around each robot to create unique colour sequences that are observed by a camera.…

The goal of this paper is to solve Problem 481 from the list of research problems in the special issue of Discrete Mathematics dedicated to the Banff International Research Station workshop on "Generalizations of de Bruijn Cycles and Gray…