Nonexistence results on generalized bent functions Zqm→Zq with odd m and q ≡ 2 (mod 4)

Nonexistence results on generalized bent functions Zqm→Zq with odd m and q ≡ 2 (mod 4)

Let p be an odd prime, let a be a positive integer, let m be an odd positive integer, and suppose that a generalized bent function from Z2pam to Z2pa exists. We show that this implies m≠1, p≤22m+2m+1, and ordp(2)≤2m−1. We obtain further necessary conditions and prove that p=7 if m=3 and p∈{7,23,31,7...

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Main Authors: Leung, Ka Hin, Schmidt, Bernhard
Other Authors: School of Physical and Mathematical Sciences
Format: Article
Language:English
Published: 2020
Subjects:
Science::Mathematics
Weil Numbers
Minimal Aliases
Online Access:https://hdl.handle.net/10356/141391
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Institution: Nanyang Technological University
Language: English
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spelling sg-ntu-dr.10356-1413912020-06-08T04:47:44Z Nonexistence results on generalized bent functions Zqm→Zq with odd m and q ≡ 2 (mod 4) Leung, Ka Hin Schmidt, Bernhard School of Physical and Mathematical Sciences Science::Mathematics Weil Numbers Minimal Aliases Let p be an odd prime, let a be a positive integer, let m be an odd positive integer, and suppose that a generalized bent function from Z2pam to Z2pa exists. We show that this implies m≠1, p≤22m+2m+1, and ordp(2)≤2m−1. We obtain further necessary conditions and prove that p=7 if m=3 and p∈{7,23,31,73,89} if m=5. Our results are based on new tools for the investigation of cyclotomic integers of prescribed complex modulus, including “minimal aliases” invariant under automorphisms, and bounds on the ℓ2-norms of their coefficient vectors. These methods have further applications, for instance, to relative difference sets, circulant Butson matrices, and other kinds of bent functions. MOE (Min. of Education, S’pore) 2020-06-08T04:47:44Z 2020-06-08T04:47:44Z 2019 Journal Article Leung, K. H., & Schmidt, B. (2019). Nonexistence results on generalized bent functions Zqm→Zq with odd m and q ≡ 2 (mod 4). Journal of Combinatorial Theory. Series A, 163, 1-33. doi:10.1016/j.jcta.2018.11.007 0097-3165 https://hdl.handle.net/10356/141391 10.1016/j.jcta.2018.11.007 2-s2.0-85056758225 163 1 33 en Journal of Combinatorial Theory. Series A © 2018 Elsevier Inc. All rights reserved.
institution Nanyang Technological University
building NTU Library
country Singapore
collection DR-NTU
language English
topic Science::Mathematics
Weil Numbers
Minimal Aliases
spellingShingle Science::Mathematics
Weil Numbers
Minimal Aliases
Leung, Ka Hin
Schmidt, Bernhard
Nonexistence results on generalized bent functions Zqm→Zq with odd m and q ≡ 2 (mod 4)
description Let p be an odd prime, let a be a positive integer, let m be an odd positive integer, and suppose that a generalized bent function from Z2pam to Z2pa exists. We show that this implies m≠1, p≤22m+2m+1, and ordp(2)≤2m−1. We obtain further necessary conditions and prove that p=7 if m=3 and p∈{7,23,31,73,89} if m=5. Our results are based on new tools for the investigation of cyclotomic integers of prescribed complex modulus, including “minimal aliases” invariant under automorphisms, and bounds on the ℓ2-norms of their coefficient vectors. These methods have further applications, for instance, to relative difference sets, circulant Butson matrices, and other kinds of bent functions.
author2 School of Physical and Mathematical Sciences
author_facet School of Physical and Mathematical Sciences
Leung, Ka Hin
Schmidt, Bernhard
format Article
author Leung, Ka Hin
Schmidt, Bernhard
author_sort Leung, Ka Hin
title Nonexistence results on generalized bent functions Zqm→Zq with odd m and q ≡ 2 (mod 4)
title_short Nonexistence results on generalized bent functions Zqm→Zq with odd m and q ≡ 2 (mod 4)
title_full Nonexistence results on generalized bent functions Zqm→Zq with odd m and q ≡ 2 (mod 4)
title_fullStr Nonexistence results on generalized bent functions Zqm→Zq with odd m and q ≡ 2 (mod 4)
title_full_unstemmed Nonexistence results on generalized bent functions Zqm→Zq with odd m and q ≡ 2 (mod 4)
title_sort nonexistence results on generalized bent functions zqm→zq with odd m and q ≡ 2 (mod 4)
publishDate 2020
url https://hdl.handle.net/10356/141391
_version_ 1681056715135516672

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