Steiner systems $S(2,4,v)$ - a survey

Reid, Colin D.; Rosa, Alexander

doi:10.37236/39

Cited by 30 publications

(40 citation statements)

References 113 publications

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“…Системой Штейнера S(t, k, v) называется пара (V, B), где V множество из v элементов, а B семейство k-элементных подмножеств V , называемых блоками, таких, что любое t-элементное подмножество V лежит ровно в одном блоке. С основными результатами по системам Штейнера с указанными параметрами можно ознакомиться, например, в [4].…”

Section: радиус покрытия R(c H )unclassified

On the covering radius of the linear codes generated by the affine geometries over GF(4)

Kovalenko¹

2014

Applied Discrete Mathematics

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Рассматриваются линейные коды, порождённые аффинными геометриями над полем из четырёх элементов. Для данных кодов приводятся некоторые свойства и вычисляется точное значение радиуса покрытия равное 4.Ключевые слова: линейные коды, конечные аффинные геометрии, радиус покрытия, покрывающие коды.

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Section: радиус покрытия R(c H )unclassified

On the covering radius of the linear codes generated by the affine geometries over GF(4)

Kovalenko¹

2014

Applied Discrete Mathematics

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“…Motivated by this fact, we obtain infinite families of 2-designs by using quadratic functions over finite fields and determine their parameters explicitly. For other constructions of t-designs, see [6,9,10,14,15] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Combinatorial t-designs from quadratic functions

Xiang

Ling

Wang

2019

Des. Codes Cryptogr.

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Combinatorial t-designs have been an interesting topic in combinatorics for decades. It was recently reported that the image sets of a fixed size of certain special polynomials may constitute a t-design. Till now only a small amount of work on constructing t-designs from special polynomials has been done, and it is in general hard to determine their parameters. In this paper, we investigate this idea further by using quadratic functions over finite fields, thereby obtain infinite families of 2-designs, and explicitly determine their parameters. The obtained designs cover some earlier 2-designs as special cases. Furthermore, we confirmed Conjecture 3 in Tang (arXiv: 1903.07375, 2019).

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“…According to the surveys [9,22], the following is a list of infinite families of Steiner systems S (2, 4, v): r S(2, 4, 4 n ), n ≥ 2 (affine geometries). r S(2, 4, 3 n + · · · + 3 + 1), n ≥ 2 (projective geometries).…”

Section: Introductionmentioning

confidence: 99%

An Infinite Family of Steiner Systems from Cyclic Codes

Ding

2017

J of Combinatorial Designs

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Steiner systems are a fascinating topic of combinatorics. The most studied Steiner systems are S(2, 3, v) (Steiner triple systems), S(3, 4, v) (Steiner quadruple systems), and S(2, 4, v). There are a few infinite families of Steiner systems S(2, 4, v) in the literature. The objective of this paper is to present an infinite family of Steiner systems S(2, 4, 2 m ) for all m ≡ 2 (mod 4) ≥ 6 from cyclic codes. This may be the first coding-theoretic construction of an infinite family of Steiner systems S (2, 4, v). As a by-product, many infinite families of 2-designs are also reported in this paper.

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Steiner systems $S(2,4,v)$ - a survey

Abstract: We survey the basic properties and results on Steiner systems $S(2, 4, v)$, as well as open problems.

Cited by 30 publications

References 113 publications

On the covering radius of the linear codes generated by the affine geometries over GF(4)

On the covering radius of the linear codes generated by the affine geometries over GF(4)

Combinatorial t-designs from quadratic functions

An Infinite Family of Steiner Systems from Cyclic Codes

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