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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.

New Constructions of Extended Sonar Sequences From Sidon Sets