A multidimensional Costas array is the function graph of a bijection between two finite boxes of integer
lattice points for which all directed vector
differences between distinct points of the function
graph are different (Rubio and Torres 2023). Writing ![[a]={1,...,a}](/images/equations/MultidimensionalCostasArray/Inline1.svg)
, take ![X=[n_1]×...×[n_k]](/images/equations/MultidimensionalCostasArray/Inline2.svg)
and ![Y=[n_(k+1)]×...×[n_m]](/images/equations/MultidimensionalCostasArray/Inline3.svg)
, where every 
. The graph of a bijection 
is a subset of 
, and the order of the array is 
. The ordinary Costas array
corresponds to 
and 
.
The array is periodic Costas when every translated window of side lengths 
, ..., 
in its periodic extension also has distinct directed vector differences. Differences in a window are
ordinary integer differences, not differences reduced
modulo the side lengths.
Munagala (2026) gave the six-point array
 |
It is the function graph of a bijection ![[2]×[3]->[2]×[3]](/images/equations/MultidimensionalCostasArray/Inline12.svg)
and is periodic Costas. Its periods are 2, 3, 2, and 3, so 36 residue windows suffice
for verification. Each contains six points and 30 distinct directed vector
differences. This refutes the conjecture of Rubio and Torres (2023) that a periodic
Costas array with 
has order 
, while leaving their two- and three-dimensional results
intact.
Munagala (2026) reports that Claude assisted with the search, verification script, and Lean formalization. The 36 windows have been independently checked by integer arithmetic here, but the Lean formalization has not been replayed.
