The combinatorics of binary arrays
This paper is a summary of a survey on (periodical binary) \(r\times s\) arrays, that is on infinite matrices \(A=\{a_{ij}\}\) (\(a_{ij}\in \{0,1\}\); \(i,j\) integers) such that \(a_{ij}=a_{i+r,s}=a_{i,j+s}\), for all \(i\) and \(j\). As generalisation of de Bruijn sequences is treated the possibility to have an \(r\times s\) array with \(rs=2^{mn}\) in which all \(m\times n\) subarrays are different. Certain autocorrelation functions for \(A\) give problems equivalent to questions on (relative or not) difference sets and planar functions and so lead for example to the projective plane conjecture. Other analogous functions lead for example to SONAR arrays, the Goulomb conjecture and Barker sequences. Obviously the paper can only point out some links and some (updated) references.
scientific article; zbMATH DE number 1057371
scientific article