Infinite families of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml43" display="inline" overflow="scroll" altimg="si33.gif"><mml:mn>2</mml:mn></mml:math>-designs and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml44" display="inline" overflow="scroll" altimg="si44.gif"><mml:mn>3</mml:mn></mml:math>-designs from linear codes

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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.

S (2, 3, 2^{m} - 1)

from the minimum codewords in the binary Hamming codes.…”

Section: Discussionmentioning

confidence: 99%

“…We inform the reader that an infinite family of conjectured Steiner systems

S (2, 4, false(3^{m} - 1 false) / 2)

for odd m was presented in . It would be good if more infinite families of Steiner systems from error correcting codes could be discovered.…”

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 84%

“…These results were proved by the Assmus–Mattson Theorem, and the designs of those codes were covered in .…”

Section: A Type Of Affine‐invariant Codes and Their Designsmentioning

confidence: 94%

See 2 more Smart Citations

An Infinite Family of Steiner Systems from Cyclic Codes

Ding

2017

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“…Conjecture was confirmed by Magma for

n \in {5, 7}

. If Conjecture is true, then

C_{2} ({double-struckKA}_{n, 1})

holds three

3

‐designs, and

C_{2} {({KA}_{n, 1})}^{⊥}

holds exponentially many

3

‐designs (see for detail).…”

Section: Two Constructions Of 3‐designs From Apn Functionsmentioning

confidence: 93%

Infinite families of 3‐designs from APN functions

Tang

2019

Combinatorial t‐designs have nice applications in coding theory, finite geometries, and several engineering areas. A classical method for constructing t‐designs is by the action of a permutation group that is t‐transitive or t‐homogeneous on a point set. This approach produces t‐designs, but may not yield (t+1)‐designs. The objective of this paper is to study how to obtain 3‐designs with 2‐transitive permutation groups. The incidence structure formed by the orbits of a base block under the action of the general affine groups, which are 2‐transitive, is considered. A characterization of such incidence structure to be a 3‐design is presented, and a sufficient condition for the stabilizer of a base block to be trivial is given. With these general results, infinite families of 3‐designs are constructed by employing almost perfect nonlinear functions. Some 3‐designs presented in this paper give rise to self‐dual binary codes or linear codes with optimal or best parameters known. Several conjectures on 3‐designs and binary codes are also presented.

Infinite families of 2‐designs from a class of cyclic codes

Wang

Fan

2019

Combinatorial t-designs have wide applications in coding theory, cryptography, communications and statistics. It is well known that the supports of all codewords with a fixed weight in a code may give a t-design. In this paper, we first determine the weight distribution of a class of linear codes derived from the dual of extended cyclic code with two non-zeros. We then obtain infinite families of 2-designs and explicitly compute their parameters from the supports of all the codewords with a fixed weight in the codes. By simple counting argument, we obtain exponentially many 2-designs.Combinatorial t-designs have very important applications in coding theory, cryptography, communications and statistics. There has been an interplay between codes and t-designs for both directions. On one hand, a linear code over any finite field can be derived from the incidence matrix of a t-design and much progress has been made and documented in [1,6,19,20]. On the other hand, linear and nonlinear codes might both hold t-designs. Till now, 4-designs and 5-designs with fixed parameters were only derived from binary and ternary Golay codes. The largest t for which the infinite families of t-designs could be obtained from linear codes is t = 3. In 2017, Ding and Li [8] obtained infinite families of 2-designs from p-ary Hamming codes, ternary projective cyclic codes, binary codes with two zeros and their duals as well as the infinite families of 3-designs from the extended codes of these codes and RM codes. Afterwards, infinite families of 2-designs and 3-designs were constructed from a class of binary linear codes with five weights [9]. For other constructions of t-designs, for example, we refer to [2,5,17,18].The rest of this paper is organized as follows. In Section 2, we introduce some notation, together with some preliminary results on affine-invariant codes and 2designs, which will be used in subsequent sections. In Section 3, we present the weight distributions of a class of linear codes derived from the dual of the extended cyclic codes with two non-zeros. By proving that the derived codes are affine-invariant, we then obtain infinite families of 2-designs and explicitly determine their parameters. The proof of the main results are given in Section 4. Section 5 concludes the paper.