A class of two or three weights linear codes and their complete weight enumerators

Zheng, Dabin; Qing, Zhao; Wang, Xiaoqiang; Zhang, Yan

doi:10.1016/j.disc.2021.112355

Cited by 11 publications

(11 citation statements)

References 42 publications

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“…In fact, if q > 2, the complete weight enumerator contains much more information than the ordinary weight enumerator. For this reason, the determination of the complete weight enumerators of cyclic codes or linear codes over finite fields has received a great deal of attention in recent years (see for example [2], [5], [12], [13], [22], [24], [25], [26], and [27]). In this respect, it should be pointed out that, for a prime field IF p , several classes of three-weight linear codes and their complete weight enumerators were recently presented in [11], [23], [25], [26], and [27].…”

Section: Introductionmentioning

confidence: 99%

“…For this reason, the determination of the complete weight enumerators of cyclic codes or linear codes over finite fields has received a great deal of attention in recent years (see for example [2], [5], [12], [13], [22], [24], [25], [26], and [27]). In this respect, it should be pointed out that, for a prime field IF p , several classes of three-weight linear codes and their complete weight enumerators were recently presented in [11], [23], [25], [26], and [27]. In that context, we want to emphasize that the three-weight codes that we are going to present here are not only linear, but also cyclic, optimal and defined over any finite field IF q .…”

Section: Introductionmentioning

confidence: 99%

“…As an application of our enlarged class of optimal five-weight cyclic codes, we determine the complete weight enumerator of a subclass of the optimal three-weight cyclic codes (Theorem 3) that were studied in [20]. In this respect, it should be pointed out that, for a prime field IF p , several classes of three-weight linear codes and their complete weight enumerators were recently presented in [11], [23], [25], [26], and [27]. In that context, we want to emphasize that the three-weight codes presented here are not only linear, but also cyclic, optimal and defined over any finite field IF q .…”

mentioning

confidence: 99%

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The complete weight enumerator of a subclass of optimal three-weight cyclic codes

Vega¹,

Hernández²

2021

Preprint

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A class of optimal three-weight cyclic codes of dimension 3 over any finite field was presented by Vega [Finite Fields Appl., 42 (2016) 23-38]. Shortly thereafter, Heng and Yue [IEEE Trans. Inf. Theory, 62(8) (2016) 4501-4513] generalized this result by presenting several classes of cyclic codes with either optimal three weights or a few weights. On the other hand, a class of optimal five-weight cyclic codes of dimension 4 over a prime field was recently presented by Li, et al. [Adv. Math. Commun.,13(1) (2019) 137-156]. One of the purposes of this work is to present a more general description for these optimal five-weight cyclic codes, which gives place to an enlarged class of optimal five-weight cyclic codes of dimension 4 over any finite field. As an application of this enlarged class, we present the complete weight enumerator of a subclass of the optimal three-weight cyclic codes over any finite field that were studied by Vega [Finite Fields Appl., 42 (2016) 23-38]. In addition, we study the dual codes in this enlarged class of optimal five-weight cyclic codes, and show that they are cyclic codes of length q 2 − 1, dimension q 2 − 5, and minimum Hamming distance 4. In fact, through several examples, we see that those parameters are the best known parameters for linear codes.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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The complete weight enumerator of a subclass of optimal three-weight cyclic codes

Vega¹,

Hernández²

2021

Preprint

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“…For example, some minimal linear codes have been successfully employed in secret sharing and secure two-party computations. The recent progress on this kind of codes can be found in [3,17,18,23]. Let Tr q/p denote the trace function from F q to F p .…”

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confidence: 99%

Two-weight and three-weight linear codes constructed from Weil sums

Zhang¹,

Lü²,

Yang³

2022

MFC

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<p style='text-indent:20px;'>Linear codes with few weights are widely used in strongly regular graphs, secret sharing schemes, association schemes and authentication codes. In this paper, we construct several two-weight and three-weight linear codes over finite fields by choosing suitable different defining sets. We also give some examples and some of the codes are optimal or almost optimal. Their applications to secret sharing schemes are also investigated.</p>

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“…There are many works on the puncturing technique due to the defining-set construction. The reader is referred to, for example, [22,9,19,28,26] for information.…”

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confidence: 99%

The punctured codes of two classes of cyclic codes with few weights

Zheng

Wang

2022

AMC

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<p style='text-indent:20px;'>The puncturing technique is one of important tools for constructing new codes from old ones. In recent years, many classes of linear codes with interesting parameters have been obtained with this technique. Based on quadratic Gauss sums, the puncturing technique and cyclotomic classes, we investigate two classes of punctured codes of cyclic codes over finite fields. The weight distribution and the parameters of the duals of these codes are studied. The parameters of these codes are new.</p>

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A class of two or three weights linear codes and their complete weight enumerators

Cited by 11 publications

References 42 publications

The complete weight enumerator of a subclass of optimal three-weight cyclic codes

The complete weight enumerator of a subclass of optimal three-weight cyclic codes

Two-weight and three-weight linear codes constructed from Weil sums

The punctured codes of two classes of cyclic codes with few weights

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