Formally dual subsets of cyclic groups of prime power order

Schüler, Robert

doi:10.1007/s13366-017-0337-7

Cited by 9 publications

(9 citation statements)

References 3 publications

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“…This conjecture has been verified when N is a prime power, by Schüler [21] for p odd or when N an even power of 2, and by Xia, Park, and Cohn [27] in the remaining cases, as well as when N is square-free.…”

Section: Introductionmentioning

confidence: 79%

“…Now we are ready to apply the methods already introduced; we will begin by providing two different proofs for the prime power case, one using the field descent method and one using the polynomial method. We emphasize that the polynomial method comprises of similar arguments as in [21], albeit with a different language. Theorem 6.1.…”

Section: The Prime Power Case Revisitedmentioning

confidence: 99%

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Formal duality in finite cyclic groups

Malikiosis

2017

Preprint

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The notion of formal duality in finite Abelian groups appeared recently in relation to spherical designs, tight sphere packings, and energy minimizing configurations in Euclidean spaces. For finite cyclic groups it is conjectured that there are no primitive formally dual pairs besides the trivial one and the TITO configuration. This conjecture has been verified for cyclic groups of prime power order, as well as of square-free order. In this paper, we will confirm the conjecture for other classes of cyclic groups, namely almost all cyclic groups of order a product of two prime powers, with finitely many exceptions for each pair of primes, or whose order N satisfies p || N , where p a prime satisfying the so-called self-conjugacy property with respect to N . For the above proofs, various tools were needed: the field descent method, used chiefly for the circulant Hadamard conjecture, the techniques of Coven & Meyerowitz for sets that tile Z or ZN by translations, dubbed herein as the polynomial method, as well as basic number theory of cyclotomic fields, especially the splitting of primes in a given cyclotomic extension.

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Section: Introductionmentioning

confidence: 79%

Section: The Prime Power Case Revisitedmentioning

confidence: 99%

Formal duality in finite cyclic groups

Malikiosis

2017

Preprint

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“…We recall that when N is a prime power, Conjecture 1 has been proved in [20] and [24]. The author of [24] essentially showed that a primitive formally dual set in the cyclic group Z 2 2k with k ≥ 1 must have rank three.…”

Section: Primitive Formally Dual Sets With Small Rankmentioning

confidence: 99%

Formal duality in finite abelian groups

Pott

Schüler

2019

Journal of Combinatorial Theory, Series A

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Inspired by an experimental study of energy-minimizing periodic configurations in Euclidean space, Cohn, Kumar and Schürmann proposed the concept of formal duality between a pair of periodic configurations, which indicates an unexpected symmetry possessed by the energy-minimizing periodic configurations. Later on, Cohn, Kumar, Reiher and Schürmann translated the formal duality between a pair of periodic configurations into the formal duality of a pair of subsets in a finite abelian group. This insight suggests to study the combinatorial counterpart of formal duality, which is a configuration named formally dual pair. In this paper, we initiate a systematic investigation on formally dual pairs in finite abelian groups, which involves basic concepts, constructions, characterizations and nonexistence results. In contrast to the belief that primitive formally dual pairs are very rare in cyclic groups, we construct three families of primitive formally dual pairs in noncyclic groups. These constructions enlighten us to propose the concept of even sets, which reveals more structural information about formally dual pairs and leads to a characterization of rank three primitive formally dual pairs. Finally, we derive some nonexistence results about primitive formally dual pairs, which are in favor of the main conjecture that except two small examples, no primitive formally dual pair exists in cyclic groups.

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“…Motivated by this conjecture, some follow-up works studied formally dual pairs in cyclic groups. Specifically, this conjecture was proved for cyclic groups of prime power order, where Schüler confirmed the odd prime power case [10] and Xia confirmed the even prime power case [11]. Malikiosis showed that the conjecture holds true in many cases when the order of the cyclic group is a product of two prime powers [7].…”

Section: Introductionmentioning

confidence: 96%

A direct construction of primitive formally dual pairs having subsets with unequal sizes

Pott

2019

Cryptogr. Commun.

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The concept of formal duality was proposed by Cohn, Kumar and Schürmann, which reflects a remarkable symmetry among energy-minimizing periodic configurations. This formal duality was later translated into a purely combinatorial property by Cohn, Kumar, Reiher and Schürmann, where the corresponding combinatorial objects were called formally dual pairs. So far, except the results presented in [5], we have little information about primitive formally dual pairs having subsets with unequal sizes. In this paper, we propose a direct construction of primitive formally dual pairs having subsets with unequal sizes in Z 2 × Z 2m 4 , where m ≥ 1. This construction recovers an infinite family obtained in [5], which was derived by employing a recursive approach. Although the resulting infinite family was known before, the idea of the direct construction is new and provides more insights which were not known from the recursive approach.

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Formally dual subsets of cyclic groups of prime power order

Cited by 9 publications

References 3 publications

Formal duality in finite cyclic groups

Formal duality in finite cyclic groups

Formal duality in finite abelian groups

A direct construction of primitive formally dual pairs having subsets with unequal sizes

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