The weight distributions of cyclic codes with two zeros and zeta functions

Boston, Nigel; McGuire, Gary

doi:10.1016/j.jsc.2010.03.007

Cited by 39 publications

(48 citation statements)

References 4 publications

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“…There are a lot of references on the codes C (u,v,q,m) (see for example [2,20,28,29,[47][48][49][54][55][56]59,60]). It is obvious that C (u,v,q,m) cannot be a constant-weight code as its parity-check polynomial has two zeros.…”

Section: Three-weight Cyclic Codes Whose Duals Have Two Zerosmentioning

confidence: 99%

Three-weight cyclic codes and their weight distributions

Ding

et al. 2016

Discrete Mathematics

116

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a b s t r a c tCyclic codes have been an important topic of both mathematics and engineering for decades. They have been widely used in consumer electronics, data transmission technologies, broadcast systems, and computer applications as they have efficient encoding and decoding algorithms. The objective of this paper is to provide a survey of three-weight cyclic codes and their weight distributions. Information about the duals of these codes is also given when it is available.

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Section: Three-weight Cyclic Codes Whose Duals Have Two Zerosmentioning

confidence: 99%

Three-weight cyclic codes and their weight distributions

Ding

et al. 2016

Discrete Mathematics

116

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“…In general, we have m 1 = m 2 = m. Thus the corresponding cyclic code C is an [n, 2m] code. If F called the dual of primitive cyclic code with two zeros had been studied in [9,13,14,18,62,63,74]. In this subsection, we only consider cyclic codes whose weight distributions are determined by Gauss periods, so we do not describe the results on primitive cyclic codes here.…”

Section: Gauss Periods and Weight Distributionsmentioning

confidence: 99%

Recent progress on weight distributions of cyclic codes over finite fields

Dinh¹,

Li²,

Yue³

2015

JACODESMATH

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Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability. In this paper, we present the recent progress on the weight distributions of cyclic codes over finite fields, which had been determined by exponential sums. The cyclic codes with few weights which are very useful are discussed and their existence conditions are listed. Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions.

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“…For the weight distributions of the cyclic codes with two nonzeros α −(2 2·1 +1) and α −(2 1 +1) , there are the following results in [2]. For m = 7, the corresponding cyclic code C 1 has three nonzero weights and Using Matlab, it can be found that the cyclic code with nonzeros α −(2 2·3 +1) and α −(2 3 +1) has the same weight distribution as C 1 for m = 7, and C 2 for m = 11.…”

Section: Resultsmentioning

confidence: 99%

“…In this note we study the case k = 3. Boston and McGuire considered the weight distributions of binary cyclic codes with at most five nonzero weights [2]. For relevant studies of cyclic codes, please refer to [3]- [5], [14], [15], [18]- [20].…”

Section: Conjecturementioning

confidence: 99%

A Note on the Five Valued Conjectures of Johansen, Helleseth and Kholosha and Zeta Functions

2014

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For the complete five-valued cross-correlation distribution between two m-sequences s t and s dt of period 2 m − 1 that differ by the decimation d = (2 2k + 1)/(2 k + 1) where m is odd and gcd(k, m) = 1, Johansen and Helleseth expressed it in terms of some exponential sums. And two conjectures were presented about them. In this correspondence we study these conjectures for the particular case where k = 3, and the cases k = 1, 2 can also be analyzed in a similar process. When k > 3, the degrees of the relevant polynomials will become higher.

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The weight distributions of cyclic codes with two zeros and zeta functions

Cited by 39 publications

References 4 publications

Three-weight cyclic codes and their weight distributions

Three-weight cyclic codes and their weight distributions

Recent progress on weight distributions of cyclic codes over finite fields

A Note on the Five Valued Conjectures of Johansen, Helleseth and Kholosha and Zeta Functions

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