Correction: The classifications of o-monomials and of 2-to-1 binomials are equivalent
- Open Access
- 15.05.2025
- Correction
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Correction to: Designs, Codes and Cryptography (2025) 93:961–970 https://doi.org/10.1007/s10623-024-01463-1
In the original publication of the article [1], there were misprints in the statement of \(F_7(x)\) in the Theorem 4.1. The proof given in [1] is true and yields the polynomial \(F_7(x)\). The corrected Theorem 4.1 is given below.
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Theorem 1.1
(Theorem 4.1. of [1]) The following polynomials define 2-to-1 maps on \(\mathbb F_{2^{2m+1}}\) for any \(a \in \mathbb F_{2^{2m+1}}^*\):
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\(F_1(x)=x^{2^{m+1}+2}+x^{2^{m+1}}+x^2+ax\),
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\(F_2(x)=x^{2^{m+1}+2}+ax^{2^{m+1}+1}+(a+1)x^{2^{m+1}}+x^2+ax\),
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\(F_3(x)=x^{2^{n}-2}+x^{2^n-2^{m+1}}+x^{2^n-2^{m+1}-2}+ax\),
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\(F_4(x)=x^6+x^4+ax^3+(a+1)x^2+ax\),
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\(F_5(x)=ax^6+x^5+x^3+x\),
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\(F_6(x)=x^{2^{m+1}+2^m}+x^{2^{m+1}}+x^{2^m}+ax^3+ax^2+ax\),
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\(F_7(x)=x^{16}+a^4x^{12}+x^8+a^2x^6+x^4+ax^3\) if \(m \not \equiv 1 \pmod 3\).
Acknowledgements
We thank Mike Zieve for pointing out the misprint.
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