A triple-error-correcting cyclic code from the Gold and Kasami–Welch APN power functions

Zeng, Xiangyong; Shan, Jinyong; Hu, Lei

doi:10.1016/j.ffa.2011.06.005

Cited by 45 publications

(17 citation statements)

References 19 publications

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“…For the duals of primitive cyclic codes with few quadratic type zeros, their weight distributions have been intensively studied, for example see [1,4,13,7,12,23,26,25,27,28] and references therein. For the duals of primitive cyclic codes with few non-quadratic type zeros, the reader is referred to [3,6,10,16,18,22,24] and references therein. For the non-primitive cyclic codes with few non-zeros, their weight distributions have been discussed in [2,8,9,15,[19][20][21] for instance.…”

Section: Introductionmentioning

confidence: 99%

The weight enumerators of several classes of p-ary cyclic codes

Zheng

Wang

et al. 2015

Discrete Mathematics

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a b s t r a c tLet p be an odd prime, and m and k be two positive integers with m ≥ 3. Let h ±1 (x) and h ±t (x) be the minimal polynomials of ±α −1 and ±α −t over F p , respectively, where α is a primitive element of F p m . Let C 1,−1,±t , C ±1,t,−t and C 1,−1,t,−t be the cyclic codes over F prespectively. This paper determines the weight distributions of the cyclic codes C 1,−1,±t , C ±1,t,−t and C 1,−1,t,−t for the parameter t satisfying some congruence equations.

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Section: Introductionmentioning

confidence: 99%

The weight enumerators of several classes of p-ary cyclic codes

Zheng

Wang

et al. 2015

Discrete Mathematics

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“…It is shown in [10] (see also [3]) that S ∈ {0, ±2 (n+1)/2 , ±2 (n+3)/2 }. This completes the proof of Theorem 6 except for some examples to show that each value can actually occur, so the theoretical restrictions cannot be improved in general.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Further results on the number of rational points of hyperelliptic supersingular curves in characteristic 2

McGuire

Yılmaz

2015

Des. Codes Cryptogr.

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Supersingular elliptic curves have played an important role in the development of elliptic curve crytography. Scott Vanstone introduced the first author to elliptic curve cryptography, a subject that continues to be a rich source of interesting problems, results and applications. One important fact about supersingular elliptic curves is that the number of rational points is tightly constrained to a small number of possible values. In this paper we present some similar results for curves of higher genus. We also present an application to the problem of determining abelian varieties that occur as jacobians.

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“…RELATED WORKS In [9], the authors determine the dimension, the minimum distance and the weight enumerators for BCH codes under some conditions and for well-defined cases; in an other work [10], the authors gave a study of dimension for three type of BCH codes over a finite field of order q (GF(q)). In [11], the authors propose a study of the minimum distance of a binary cyclic code of length n=2 m -1 and the weight divisibility of its dual code. Based on directed graphs, the authors of [12] have developed combinatorial algorithms for computing parameters of extensions of BCH codes.…”

Section: Introductionmentioning

confidence: 99%

A Fast Method to Estimate Partial Weights Enumerators by Hash Techniques and Automorphism Group

Alaoui¹,

Nouh²,

Marzak³

2017

ijacsa

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Abstract-BCH codes have high error correcting capability which allows classing them as good cyclic error correcting codes. This important characteristic is very useful in communication and data storage systems. Actually after almost 60 years passed from their discovery, their weights enumerators and therefore their analytical performances are known only for the lengths less than or equal to 127 and only for some codes of length as 255. The Partial Weights Enumerator (PWE) algorithm permits to obtain a partial weights enumerators for linear codes, it is based on the Multiple Impulse Method combined with a Monte Carlo Method; its main inconveniece is the relatively long run time. In this paper we present an improvement of PWE by integration of Hash techniques and a part of Automorphism Group (PWEHA) to accelerate it. The chosen approach applies to two levels. The first is to expand the sample which contains codewords of the same weight from a given codeword, this is done by adding a part of the Automorphism Group. The second level is to simplify the search in the sample by the use of hash techniques. PWEHA has allowed us to considerably reduce the run time of the PWE algorithm, for example that of PWEHA is reduced at more than 3900% for the BCH (127,71,19) code. This method is validated and it is used to approximate a partial weights enumerators of some BCH codes of unknown weights enumerators.

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A triple-error-correcting cyclic code from the Gold and Kasami–Welch APN power functions

Cited by 45 publications

References 19 publications

The weight enumerators of several classes of p-ary cyclic codes

The weight enumerators of several classes of p-ary cyclic codes

Further results on the number of rational points of hyperelliptic supersingular curves in characteristic 2

A Fast Method to Estimate Partial Weights Enumerators by Hash Techniques and Automorphism Group

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