Abstract
A <inline-formula> <tex-math notation="LaTeX">$(m,n)$ </tex-math></inline-formula> sonar sequence is an <inline-formula> <tex-math notation="LaTeX">$m\times n$ </tex-math></inline-formula> array with exactly one dot in each column and where all lines connecting two dots in the array are distinct as vectors. These arrays are known to have many applications such as sonar and radar detection and these are studied as a particular case of Golomb rectangles or two-dimensional Sidon sets. The main open problem for sonar sequences is: for fixed <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>, find the largest <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> for which there is an <inline-formula> <tex-math notation="LaTeX">$(m,n)$ </tex-math></inline-formula> sonar sequence, these sequences are called the best sonar sequences. The extended sonar sequences are generalizations of sonar sequences where each column has at most one dot, the motivation to study these arrays are the best results obtained when applied to radar and sonar detection. In this paper, we give the best sonar sequences with <inline-formula> <tex-math notation="LaTeX">$m\leq 100$ </tex-math></inline-formula> obtained from an exhaustive computational search based on the Caicedo, Ruiz and Trujillo constructions and we present new constructions of extended sonar sequences that use Sidon sets.
Highlights
I N this paper Z, Z+, Zn and Fq denote the set of integers, positive integers, congruence classes modulo n and the finite field with q elements, respectively
We present new constructions of extended sonar sequences that use Sidon sets, these constructions can be considered as generalizations of those obtained by Caicedo, Ruiz and Trujillo in [24]
In this work we obtain two new constructions of extended sonar sequences using special one-dimensional Sidon sets, this contribution is a result for both the sonar detection and the theory of two-dimensional Sidon sets or Golomb rectangles
Summary
I N this paper Z, Z+, Zn and Fq denote the set of integers, positive integers, congruence classes modulo n and the finite field with q elements, respectively. For complete information of Costas arrays see [12] Sonar sequences are another class of Golomb rectangles, they were mentioned in [13]- [15], where m ≤ n and each column has exactly one dot. Our motivation for considering these sequences come from new applications to study related concepts, for example, the ESS are used to obtain new constructions of resolvable Golomb rulers in [20], for study the Costas extended in [21], in the search of two-dimensional patterns with distinct differences and multiple target sonar [22], [23]. We give the parameters of the best sonar sequences with m ≤ 100 obtained from an exhaustive computational search based on the Caicedo, Ruiz and Trujillo constructions [24].
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