The linear codes of t-designs held in the Reed-Muller and Simplex codes

Ding, Cunsheng; Tang, Chunming

doi:10.48550/arxiv.2008.09935

Cited by 1 publication

(2 citation statements)

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“…The linear code C q (D) over the prime field GF(q) of the design D is defined to be the linear subspace of GF(q) v spanned by the row vectors of the incidence matrix M D . Linear codes C q (D) of designs D have been extensively investigated (see, for example, [13], [16], [17], [18]).…”

Section: Combinatorial T-designs and Some Related Resultsmentioning

confidence: 99%

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An infinite family of antiprimitive cyclic codes supporting Steiner systems $S(3,8, 7^m+1)$

Xiang¹,

Tang²,

Liu³

2021

Preprint

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Coding theory and combinatorial t-designs have close connections and interesting interplay. One of the major approaches to the construction of combinatorial t-designs is the employment of error-correcting codes. As we all known, some t-designs have been constructed with this approach by using certain linear codes in recent years. However, only a few infinite families of cyclic codes holding an infinite family of 3-designs are reported in the literature. The objective of this paper is to study an infinite family of cyclic codes and determine their parameters. By the parameters of these codes and their dual, some infinite family of 3-designs are presented and their parameters are also explicitly determined. In particular, the complements of the supports of the minimum weight codewords in the studied cyclic code form a Steiner system. Furthermore, we show that the infinite family of cyclic codes admit 3-transitive automorphism groups.

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Section: Combinatorial T-designs and Some Related Resultsmentioning

confidence: 99%

“…It is known that linear codes and t-designs are closely related. A t-design can be induced to a linear code (see, for example, [12], [13]). Meanwhile, a linear code C may induce a t-design under certain conditions.…”

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confidence: 99%