Geodesic
Multidimensional threshold matrices and extremal matrices of order $2$
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1052-1063

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The paper is devoted to multidimensional $(0,1)$-matrices extremal with respect to containing a polydiagonal (a fractional generalization of a diagonal). Every extremal matrix is a threshold matrix, i.e., an entry belongs to its support whenever a weighted sum of incident hyperplanes exceeds a given threshold. Firstly, we prove that nonequivalent threshold matrices have different distributions of ones in hyperplanes. Next, we establish that extremal matrices of order $2$ are exactly selfdual threshold Boolean functions. Using this fact, we find the asymptotics of the number of extremal matrices of order $2$ and provide counterexamples to several conjectures on extremal matrices. Finally, we describe extremal matrices of order $2$ with a small diversity of hyperplanes.
Keywords: extremal matrix, threshold matrix, selfdual Boolean function.
Mots-clés : multidimensional matrix 
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     author = {A. A. Taranenko},
     title = {Multidimensional threshold matrices and extremal matrices of order $2$},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1052--1063},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a36/}
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A. A. Taranenko. Multidimensional threshold matrices and extremal matrices of order $2$. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 20 (2023) no. 2, pp. 1052-1063. http://geodesic.mathdoc.fr/item/SEMR_2023_20_2_a36/