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<p>We show that a “mate” <i>B</i> of a set <i>A</i> in a near-factorization (<i>A</i>,&#xa0;<i>B</i>) of a finite group <i>G</i> is unique. Further, we describe how to compute the mate <i>B</i> very efficiently using an explicit formula for <i>B</i>. We use this approach to give an alternate proof of a theorem of Wu, Yang and Feng, which states that a strong circular external difference family cannot have more than two sets. We prove some new structural properties of near-factorizations in certain classes of groups. Then, we examine all the noncyclic abelian groups of order less than 200 in a search for a possible nontrivial near-factorization. All of these possibilities are ruled out, either by theoretical criteria or by exhaustive computer searches. (In contrast, near-factorizations in cyclic or dihedral groups are known to exist by previous results.) We also look briefly at nontrivial near-factorizations of index <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\lambda &gt; 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>&gt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> in noncyclic abelian groups. Various examples are found with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\lambda = 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> by computer.</p>

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Uniqueness and explicit computation of mates in near-factorizations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月15日

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  • Donald L. Kreher,
  • William J. Martin,
  • Douglas R. Stinson

摘要

We show that a “mate” B of a set A in a near-factorization (A, B) of a finite group G is unique. Further, we describe how to compute the mate B very efficiently using an explicit formula for B. We use this approach to give an alternate proof of a theorem of Wu, Yang and Feng, which states that a strong circular external difference family cannot have more than two sets. We prove some new structural properties of near-factorizations in certain classes of groups. Then, we examine all the noncyclic abelian groups of order less than 200 in a search for a possible nontrivial near-factorization. All of these possibilities are ruled out, either by theoretical criteria or by exhaustive computer searches. (In contrast, near-factorizations in cyclic or dihedral groups are known to exist by previous results.) We also look briefly at nontrivial near-factorizations of index \(\lambda > 1\) λ > 1 in noncyclic abelian groups. Various examples are found with \(\lambda = 2\) λ = 2 by computer.