Quaternary Legendre pairs II

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2025-03-27 DOI:10.1016/j.disc.2025.114501

Ilias S. Kotsireas , Christoph Koutschan , Arne Winterhof

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引用次数: 0

Abstract

Quaternary Legendre pairs are pertinent to the construction of quaternary Hadamard matrices and have many applications, for example in coding theory and communications.

In contrast to binary Legendre pairs, quaternary ones can exist for even length ℓ as well. It is conjectured that there is a quaternary Legendre pair for any even ℓ. The smallest open case until now had been

ℓ = 28

, and

ℓ = 38

was the only length ℓ with

28 \leq ℓ \leq 60

resolved before. Here we provide constructions for

ℓ = 28, 30, 32

, and 34. In parallel and independently, Jedwab and Pender found a construction of quaternary Legendre pairs of length

ℓ = (q - 1) / 2

for any prime power

q \equiv 1 \mod 4

, which in particular covers

ℓ = 30

, 36, and 40, so that now

ℓ = 42

is the smallest unresolved case.

The main new idea of this paper is a way to separate the search for the subsequences along even and odd indices which substantially reduces the complexity of the search algorithm.

In addition, we use Galois theory for cyclotomic fields to derive conditions which improve the PSD test.

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来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.