On the minimum distance of cyclic codes

Lint, JH van; Wilson, R.

doi:10.1109/tit.1986.1057134

Cited by 161 publications

(106 citation statements)

References 8 publications

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“…and the generator polynomials G and H have binary coefficients. From Theorem 9 in [vLW86], the minimum distance of these codes over F 2 m is the same as that of their subfield subcodes over F 2 . As a consequence, we apply the Schaub algorithm on their binary subfield subcodes, since these one have much less cyclic subcodes.…”

Section: Schaub Algorithm and Algebraic Cryptanalysismentioning

confidence: 94%

On the Triple-Error-Correcting Cyclic Codes with Zero Set {1, 2 i + 1, 2 j + 1}

Herbert

Sarkar

2011

Cryptography and Coding

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Abstract. We consider a class of 3-error-correcting cyclic codes of length 2 m −1 over the two-element field F2. The generator polynomial of a code of this class has zeroes α, α 2 i +1 and α 2 j +1 , where α is a primitive element of the field F2m . In short, {1, 2 i + 1, 2 j + 1} refers to the zero set of these codes. Kasami in 1971 and Bracken and Helleseth in 2009, showed that cyclic codes with zeroes {1, 2 ℓ + 1, 2 3ℓ + 1} and {1, 2 ℓ + 1, 2 2ℓ + 1} respectively are 3-error correcting, where gcd(ℓ, m) = 1. We present a sufficient condition so that the zero set {1, 2 ℓ + 1, 2 pℓ + 1}, gcd(ℓ, m) = 1 gives a 3-error-correcting cyclic code. The question for p > 3 is open. In addition, we determine all the 3-error-correcting cyclic codes in the class {1, 2 i + 1, 2 j + 1} for m < 20. We investigate their weight distribution via their duals and observe that they have the same weight distribution as 3-error-correcting BCH codes for m < 14. Further our experiment shows that these codes are not equivalent to the 3-error-correcting BCH code in general. We also study the Schaub algorithm which determines a lower bound of the minimum distance of a cyclic code. We introduce a pruning strategy to improve the Schaub algorithm. Finally we study the cryptographic property of a Boolean function, called spectral immunity which is directly related to the minimum distance of cyclic codes over F2m . We apply the improved Schaub algorithm in order to find a lower bound of the spectral immunity of a Boolean function related to the zero set {1, 2 i + 1, 2 j + 1}.

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Section: Schaub Algorithm and Algebraic Cryptanalysismentioning

confidence: 94%

On the Triple-Error-Correcting Cyclic Codes with Zero Set {1, 2 i + 1, 2 j + 1}

Herbert

Sarkar

2011

Cryptography and Coding

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“…Cyclic codes have been studied in a series of papers and a lot of progress have been accomplished (see, for example [1], [6], [7], [11] and [13]). The Whiteman generalized cyclotomy was introduced by…”

Section: Introductionmentioning

confidence: 99%

Cyclic codes from the second class two-prime Whiteman’s generalized cyclotomic sequence with order 6

Kewat

Kumari

2016

Cryptogr. Commun.

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“…We do not consider bounds which have an exponential computational cost in the code length ( [AL96]), as for example the Van Lint-Wilson bound ([vLW86]) or the Massey-Schaub bound ( [Sch88]). …”

Section: Introductionmentioning

confidence: 99%

A New Bound for Cyclic Codes Beating the Roos Bound

Piva

Sala

2013

Algebraic Informatics

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We use the algebraic structure of cyclic codes and some properties of the discrete Fourier transform to give a reformulation of several classical bounds for the distance of cyclic codes, by extending techniques of linear algebra. We propose a bound, whose computational complexity is polynomial bounded, which is a generalization of the Hartmann-Tzeng bound and the Betti-Sala bound. In the majority of computed cases, our bound is the tightest among all known polynomial-time bounds, including the Roos bound.

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On the minimum distance of cyclic codes

Cited by 161 publications

References 8 publications

On the Triple-Error-Correcting Cyclic Codes with Zero Set {1, 2 i + 1, 2 j + 1}

On the Triple-Error-Correcting Cyclic Codes with Zero Set {1, 2 i + 1, 2 j + 1}

Cyclic codes from the second class two-prime Whiteman’s generalized cyclotomic sequence with order 6

A New Bound for Cyclic Codes Beating the Roos Bound

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