On Generalizations of Maiorana-McFarland and $\mathcal{PS}_{ap}$ Functions
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910085388697600 |
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| author | Alkan, Sezel Anbar, Nurdagül Avrantini, Athina Etxabarri-Alberdi, Erroxe Kalaycı, Tekgül Toesca, Beatrice |
| author_facet | Alkan, Sezel Anbar, Nurdagül Avrantini, Athina Etxabarri-Alberdi, Erroxe Kalaycı, Tekgül Toesca, Beatrice |
| contents | We study generalizations of two classical primary constructions of Boolean bent functions, namely the Maiorana-McFarland ($MM$) class and the (Desarguesian) partial spread ($\mathcal{PS}_{ap}$) class. The construction of bent functions lying outside the completed $MM$ class has attracted considerable attention in recent years. In this direction, we construct families of generalized Maiorana--McFarland bent functions that are not equivalent to any function in the classical $MM$ or $\mathcal{PS}_{ap}$ classes, and hence lie outside their completed classes. As a second contribution, we investigate the decomposition of generalized $\mathcal{PS}_{ap}$ functions. We prove that when the degree is sufficiently small relative to the size of the underlying finite field, such functions do not, in general, admit a decomposition into bent or semibent functions. Consequently, they cannot be obtained from known secondary constructions based on concatenation. Finally, we present a secondary construction of Boolean bent functions arising from the concatenation of components of vectorial generalized $\mathcal{PS}_{ap}$ functions. Our constructions and proofs rely on classical results concerning second-order derivatives of bent functions and their duals. In addition, we employ methods from the theory of algebraic curves and their function fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_28485 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Generalizations of Maiorana-McFarland and $\mathcal{PS}_{ap}$ Functions Alkan, Sezel Anbar, Nurdagül Avrantini, Athina Etxabarri-Alberdi, Erroxe Kalaycı, Tekgül Toesca, Beatrice Combinatorics Number Theory 11T06, 94A60, 14H05 We study generalizations of two classical primary constructions of Boolean bent functions, namely the Maiorana-McFarland ($MM$) class and the (Desarguesian) partial spread ($\mathcal{PS}_{ap}$) class. The construction of bent functions lying outside the completed $MM$ class has attracted considerable attention in recent years. In this direction, we construct families of generalized Maiorana--McFarland bent functions that are not equivalent to any function in the classical $MM$ or $\mathcal{PS}_{ap}$ classes, and hence lie outside their completed classes. As a second contribution, we investigate the decomposition of generalized $\mathcal{PS}_{ap}$ functions. We prove that when the degree is sufficiently small relative to the size of the underlying finite field, such functions do not, in general, admit a decomposition into bent or semibent functions. Consequently, they cannot be obtained from known secondary constructions based on concatenation. Finally, we present a secondary construction of Boolean bent functions arising from the concatenation of components of vectorial generalized $\mathcal{PS}_{ap}$ functions. Our constructions and proofs rely on classical results concerning second-order derivatives of bent functions and their duals. In addition, we employ methods from the theory of algebraic curves and their function fields. |
| title | On Generalizations of Maiorana-McFarland and $\mathcal{PS}_{ap}$ Functions |
| topic | Combinatorics Number Theory 11T06, 94A60, 14H05 |
| url | https://arxiv.org/abs/2603.28485 |