Combinatorial t-designs from quadratic functions

Xiang, Can; Ling, Xin; Wang, Qi

doi:10.1007/s10623-019-00696-9

Cited by 14 publications

(4 citation statements)

References 22 publications

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“…In the literature, a lot of progress has been made in constructing designs from the codewords of a fixed weight in some linear codes (for example, see [1,17,19,20,27,28,26,7,11,29,25]). There are two general methods, which specify certain sufficient conditions for constructing t-designs from linear codes.…”

Section: Preliminariesmentioning

confidence: 99%

Steiner systems $ S(2,4, 2^m) $ supported by a family of extended cyclic codes

Wang

2022

AMC

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<p style='text-indent:20px;'>In [C. Ding, An infinite family of Steiner systems <inline-formula><tex-math id="M2">\begin{document}$ S(2, 4, 2^m) $\end{document}</tex-math></inline-formula> from cyclic codes, <i>J. Combin. Des.</i> 26 (2018), no.3, 126–144], Ding constructed a family of Steiner systems <inline-formula><tex-math id="M3">\begin{document}$ S(2, 4, 2^m) $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M4">\begin{document}$ m \equiv 2 \pmod{4} \ge 6 $\end{document}</tex-math></inline-formula> from a family of extended cyclic codes. The objective of this paper is to present a family of Steiner systems <inline-formula><tex-math id="M5">\begin{document}$ S(2, 4, 2^m) $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M6">\begin{document}$ m \equiv 0 \pmod{4} \ge 4 $\end{document}</tex-math></inline-formula> supported by this family of extended cyclic codes. The main result of this paper complements the previous work of Ding, and the results in the two papers will show that there exists a binary extended cyclic code that can support a Steiner system <inline-formula><tex-math id="M7">\begin{document}$ S(2, 4, 2^m) $\end{document}</tex-math></inline-formula> for all even <inline-formula><tex-math id="M8">\begin{document}$ m \geq 4 $\end{document}</tex-math></inline-formula>. Furthermore, this paper also determines the parameters of other <inline-formula><tex-math id="M9">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-designs supported by this family of extended cyclic codes.</p>

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Section: Preliminariesmentioning

confidence: 99%

Steiner systems $ S(2,4, 2^m) $ supported by a family of extended cyclic codes

Wang

2022

AMC

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“…and Stab B is the stabilizer of B for this action. For some recent works on t-designs from group actions, we refer the reader to [25,31].…”

Section: Definition 1 Given a Set X And A Group G A Left Action Of G...mentioning

confidence: 99%

On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and their Consequences

Liu¹,

Ding²,

Mesnager³

et al. 2021

Preprint

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The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field GF(q) with length q +1, which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some q-ary BCH codes with length q + 1, and their duals are determined in this paper. The dual codes of the narrowsense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of PGL(2, p m )-invariant codes over GF(p h ) is completed. As an application of this result, the p-ranks of all incidence structures invariant under the projective general linear group PGL(2, p m ) are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in ⋆ C.

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“…Recall that a group G acting on a set X is t-transitive (resp., t-homogeneous) if for any two ordered t-tuples (ν, k, λ) design admitting G as an automorphism group for some λ. For some recent works on t-designs from group actions, we refer the reader to [19,24].…”

Section: Group Actions and T-designsmentioning

confidence: 99%

The Projective General Linear Group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ and Linear Codes of Length $2^m+1$

Ding

Tang²,

Tonchev

2020

Preprint

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The projective general linear group PGL 2 (GF(2 m )) acts as a 3-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over GF(2 h ) that are invariant under PGL 2 (GF(2 m )) are trivial codes: the repetition code, the whole space GF(2 h ) 2 m +1 , and their dual codes. As an application of this result, the 2-ranks of the (0,1)-incidence matrices of all 3-(q + 1, k, λ) designs that are invariant under PGL 2 (GF(2 m )) are determined. The second objective is to present two infinite families of cyclic codes over GF(2 m ) such that the set of the supports of all codewords of any fixed nonzero weight is invariant under PGL 2 (GF(2 m )), therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters [q +1, q −3, 4] q , where q = 2 m , and m ≥ 4 is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3-(q + 1, 4, 2) design. A code from the second family has parameters [q + 1, 4, q − 4] q , q = 2 m , m ≥ 4 even, and the minimum weight codewords support a 3-(q+1, q−4, (q−4)(q−5)(q−6)/60) design, whose complementary 3-(q + 1, 5, 1) design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over GF(q) that can support a 3-(q +1, q −4, (q −4)(q −5)(q −6)/60) design is proved, and it is shown that the designs supported by the codewords of minimum weight in the codes from the second family of codes meet this bound.

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Combinatorial t-designs from quadratic functions

Cited by 14 publications

References 22 publications

Steiner systems $ S(2,4, 2^m) $ supported by a family of extended cyclic codes

Steiner systems $ S(2,4, 2^m) $ supported by a family of extended cyclic codes

On Infinite Families of Narrow-Sense Antiprimitive BCH Codes Admitting 3-Transitive Automorphism Groups and their Consequences

The Projective General Linear Group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ and Linear Codes of Length $2^m+1$

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