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<p>A recent conjecture by C. Carlet on the sum-freedom of the binary multiplicative inverse function can be stated as follows: for each pair of positive integers (<i>n</i>,&#xa0;<i>k</i>) with <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(3\le k\le n-3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>≤</mo> <mi>k</mi> <mo>≤</mo> <mi>n</mi> <mo>-</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, there is a <i>k</i>-dimensional <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb F_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-subspace <i>E</i> of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb F_{2^n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mi>n</mi> </msup> </msub> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\sum _{0\ne u\in E}1/u=0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <mn>0</mn> <mo>≠</mo> <mi>u</mi> <mo>∈</mo> <mi>E</mi> </mrow> </msub> <mn>1</mn> <mo stretchy="false">/</mo> <mi>u</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We confirm this conjecture when <i>n</i> is not a prime.</p>

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On a conjecture about the sum-freedom of the binary multiplicative inverse function

  • 期刊论文

  • 发表日期：

    2026年4月9日

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  • Xiang-dong Hou,
  • Shujun Zhao

摘要

A recent conjecture by C. Carlet on the sum-freedom of the binary multiplicative inverse function can be stated as follows: for each pair of positive integers (n, k) with \(3\le k\le n-3\) 3 ≤ k ≤ n - 3 , there is a k-dimensional \(\mathbb F_2\) F 2 -subspace E of \(\mathbb F_{2^n}\) F 2 n such that \(\sum _{0\ne u\in E}1/u=0\) ∑ 0 ≠ u ∈ E 1 / u = 0 . We confirm this conjecture when n is not a prime.