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: | , |
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| Format: | Article |
| Language: | English |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/141391 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | 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. |
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