Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On de Bruijn Covering Sequences and Arrays

Yeow Meng Chee, Tuvi Etzion, Hoang Ta, Van Khu Vu

TL;DR

This work advances two-dimensional de Bruijn covering structures by linking (m,n,R)-de Bruijn covering arrays (dBCAs) to one-dimensional $(mn,R)$-dBCS, and by providing both probabilistic and constructive methods to minimize area. It introduces folding techniques to convert dBCSs into dBCAs, and develops several new sequence-construction schemes based on cyclic covering codes, primitive polynomials, self-dual sequences, and interleaving, achieving shorter dBCSs and smaller-area dBCAs. A probabilistic upper bound on the minimal dBCA area is established, showing existence of near-optimal arrays, while the paper also demonstrates concrete, efficient constructions for small-parameter cases. Collectively, the results offer practical guidance for designing compact two-dimensional covering structures with applications in pattern design and related coding-theoretic tasks.

Abstract

An $(m,n,R)$-de Bruijn covering array (dBCA) is a doubly periodic $M \times N$ array over an alphabet of size $q$ such that the set of all its $m \times n$ windows form a covering code with radius $R$. An upper bound of the smallest array area of an $(m,n,R)$-dBCA is provided using a probabilistic technique which is similar to the one that was used for an upper bound on the length of a de Bruijn covering sequence. A folding technique to construct a dBCA from a de Bruijn covering sequence or de Bruijn covering sequences code is presented. Several new constructions that yield shorter de Bruijn covering sequences and $(m,n,R)$-dBCAs with smaller areas are also provided. These constructions are mainly based on sequences derived from cyclic codes, self-dual sequences, primitive polynomials, an interleaving technique, folding, and mutual shifts of sequences with the same covering radius. Finally, constructions of de Bruijn covering sequences codes are also discussed.

On de Bruijn Covering Sequences and Arrays

TL;DR

This work advances two-dimensional de Bruijn covering structures by linking (m,n,R)-de Bruijn covering arrays (dBCAs) to one-dimensional -dBCS, and by providing both probabilistic and constructive methods to minimize area. It introduces folding techniques to convert dBCSs into dBCAs, and develops several new sequence-construction schemes based on cyclic covering codes, primitive polynomials, self-dual sequences, and interleaving, achieving shorter dBCSs and smaller-area dBCAs. A probabilistic upper bound on the minimal dBCA area is established, showing existence of near-optimal arrays, while the paper also demonstrates concrete, efficient constructions for small-parameter cases. Collectively, the results offer practical guidance for designing compact two-dimensional covering structures with applications in pattern design and related coding-theoretic tasks.

Abstract

An -de Bruijn covering array (dBCA) is a doubly periodic array over an alphabet of size such that the set of all its windows form a covering code with radius . An upper bound of the smallest array area of an -dBCA is provided using a probabilistic technique which is similar to the one that was used for an upper bound on the length of a de Bruijn covering sequence. A folding technique to construct a dBCA from a de Bruijn covering sequence or de Bruijn covering sequences code is presented. Several new constructions that yield shorter de Bruijn covering sequences and -dBCAs with smaller areas are also provided. These constructions are mainly based on sequences derived from cyclic codes, self-dual sequences, primitive polynomials, an interleaving technique, folding, and mutual shifts of sequences with the same covering radius. Finally, constructions of de Bruijn covering sequences codes are also discussed.
Paper Structure (5 sections, 8 theorems, 35 equations, 2 tables)

This paper contains 5 sections, 8 theorems, 35 equations, 2 tables.

Table of Contents

  1. Introduction
  2. A probabilistic construction
  3. Folding of a dBCS into a dBCA
  4. Constructions for dBCSs and dBCSCs
  5. Small Area dBCAs with small parameters

Key Result

Theorem 1

Let $m,n$ be nonnegative integer numbers. For any $M\geq m$, there exists an $M \times N$$(m,n,R)$-dBCA such that $MN = \mathcal{O} \left( \frac{q^{mn}}{V_q(mn,R)} \cdot (\log m + \log n) \right)$, for fixed $q$ and $R$.

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 2
  • proof : Proof of Theorem \ref{['thm:prob_bound']}
  • Lemma 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • proof
  • Corollary 7
  • Lemma 8
  • ...and 5 more