Handbook of Finite Fields

Mullen, Gary L.; Panario, Daniel

doi:10.1201/b15006

Cited by 288 publications

(206 citation statements)

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“…Our initial motivation arose from the following observation about the cosets of a multiplicative subgroup in a finite field (see [10] or [11] for background on finite fields). If q is a prime power, then the multiplicative group of the finite field GF (q) is cyclic: we denote the multiplicative group by GF (q) * .…”

Section: Motivation: Multiplicative Cosets In Finite Fieldsmentioning

confidence: 99%

Near-complete external difference families

Davis

Huczynska

Mullen

2016

Des. Codes Cryptogr.

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We introduce and explore near-complete external difference families, a partitioning of the nonidentity elements of a group so that each nonidentity element is expressible as a difference of elements from distinct subsets a fixed number of times. We show that the existence of such an object implies the existence of a near-resolvable design. We provide examples and general constructions of these objects, some of which lead to new parameter families of near-resolvable designs on a non-prime-power number of points. Our constructions employ cyclotomy, partial difference sets, and Galois Rings.AMS classification: 94C30 51E20 94A62 05B10

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Section: Motivation: Multiplicative Cosets In Finite Fieldsmentioning

confidence: 99%

Near-complete external difference families

Davis

Huczynska

Mullen

2016

Des. Codes Cryptogr.

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“…However, in the general case such an equitable distribution of zeros cannot be expected. Theorem 6.84 in Lidl and Niederreiter (1994) provides an estimate for the number of occurrences of zeros based on Gaussian sums and Mullen and Panario (2013) provides an improved bound. Table 1 gives some observations on the number of zeros of some linear recurring sequences over F 2 computed via MAPLE (Kottegoda, 2010, Appendix I-VIII) with the degrees and orders of their corresponding irreducible minimal polynomials.…”

Section: Then the Least Period Of The Sequence Is Equal To Ord(m(x))mentioning

confidence: 99%

The Cardinality of the Set of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

Kottegoda¹,

Fitzgerald²

2017

JMR

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Consider homogeneous linear recurring sequences over a finite field F q , based on the irreducible characteristic polynomial of degree d and order m. We give upper and lower bounds, and in some cases the exact values of the cardinality of the set of zeros of the sequences within its least period. We also prove that the cyclotomy bound introduced here is the best upper bound as it is reached in infinitely many cases. In addition, the exact number of occurrences of zeros is determined using the correlation with irreducible cyclic codes when (q d − 1)/m follows the quadratic residue conditions and also when it has the form q 2a − q a + 1 where a ∈ N.

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“…Из классифика-ции квадратичных форм [1,2] следует, что квадратичную форму можно линейным преобразованием аргументов привести к одному из следующих вариантов:…”

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On the Rank of Random Quadratic Form Over Finite Field

Черемушкин¹

2017

Applied Discrete Mathematics

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Изучаются свойства распределения ранга случайной квадратичной формы над конечным полем GF(q). Отдельно рассматриваются случаи чётной и нечётной характеристик поля. Доказаны асимптотические нижние оценки значения ранга для почти всех квадратичных форм f от n переменных вида rank(f ) n − 2 log q n + c + 1,Ключевые слова: конечное поле, симплектическая группа, квадратичная форма.

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Handbook of Finite Fields

Cited by 288 publications

References 0 publications

Near-complete external difference families

Near-complete external difference families

The Cardinality of the Set of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

On the Rank of Random Quadratic Form Over Finite Field

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