Multipartite nearly orthogonal sets over finite fields
Rajko Nenadov, Lander Verlinde

TL;DR
This paper generalizes recent bounds on nearly orthogonal sets over finite fields, introducing a multipartite container lemma to establish lower bounds on the size of sets with strong orthogonality properties.
Contribution
It extends previous results by proving a lower bound for the size of sets with a stronger orthogonality property using a new multipartite asymmetric container lemma.
Findings
Established a lower bound on the size of nearly orthogonal sets over finite fields.
Proved a new multipartite asymmetric container lemma with a simplified proof.
Generalized previous results to sets with multiple orthogonality conditions.
Abstract
For a field and integers and , a set is called -nearly orthogonal if all vectors in are non-self-orthogonal and every vectors in contain pairwise orthogonal vectors. Recently, Haviv, Mattheus, Milojevi\'{c} and Wigderson have improved the lower bound on nearly orthogonal sets over finite fields, using counting arguments and a hypergraph container lemma. They showed that for every prime and an integer , there is a constant such that for every field of characteristic and for all integers , contains a -nearly orthogonal set of size . This nearly matches an upper bound coming from Ramsey theory. Moreover, they proved the same lower bound for the size of a largest set …
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Taxonomy
TopicsOptimization and Packing Problems
