scholarly journals Quadratic form Approach for the Number of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

2017 ◽  
Vol 9 (3) ◽  
pp. 8
Author(s):  
Yasanthi Kottegoda
Keyword(s):  
Quadratic Form ◽  
Finite Field ◽  
Characteristic Polynomial ◽  
Finite Fields ◽  
Quadratic Forms ◽  
Linear Recurring Sequences ◽  
Exact Number ◽  
Number Of Zeros ◽  
Homogeneous Linear ◽  
Form Approach

We consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on an irreducible characteristic polynomial of degree $n$ and order $m$. Let $t=(q^{n}-1)/ m$. We use quadratic forms over finite fields to give the exact number of occurrences of zeros of the sequence within its least period when $t$ has q-adic weight 2. Consequently we prove that the cardinality of the set of zeros for sequences from this category is equal to two.

The Distribution of Zeros of Quadratic Forms over Finite Fields

2015 ◽  
Vol 7 (2) ◽  
pp. 18
Author(s):  
Ali H. Hakami
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Quadratic Forms ◽  
Distribution Of Zeros ◽  
Linearly Independent ◽  
Set Of Points

Let $m$ be a positive integer with $m < p/2$ and $p$ is a prime. Let $\mathbb{F}_q$ be the finite field in $q = p^f$ elements, $Q({\mathbf{x}})$ be a nonsinqular quadratic form over $\mathbb{F}_q$ with $q$ odd, $V$ be the set of points in $\mathbb{F}_q^n$ satisfying the equation $Q({\mathbf{x}}) = 0$ in which the variables are restricted to a box of points of the type\[\mathcal{B}(m) = \left\{ {{\mathbf{x}} \in \mathbb{F}_q^n \left| {x_i  = \sum\limits_{j = 1}^f {x_{ij} \xi _j } ,\;\left| {x_{ij} } \right| < m,\;1 \leqslant i \leqslant n,\;1 \leqslant j \leqslant f} \right.} \right\},\]where $\xi _1 , \ldots ,\xi _f$ is a basis for $\mathbb{F}_q$ over $\mathbb{F}_p$ and $n > 2$ even. Set $\Delta  = \det Q$ such that $\chi \left( {( - 1)^{n/2} \Delta } \right) = 1.$ We shall motivate work of (Cochrane, 1986) to obtain lower bounds on $m,$ size of the box $\mathcal{B},$ so that $\mathcal{B} \cap V$ is nonempty. For this we show that the box $\mathcal{B}(m)$ contains a zero of $Q({\mathbf{x}})$ provided that $m \geqslant p^{1/2}.$ We also show that the box $\mathcal{B}(m)$ contains $n$ linearly independent zeros of $Q({\mathbf{x}})$ provided that $m \geqslant 2^{n/2} p^{1/2} .$

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

2017 ◽  
Vol 9 (2) ◽  
pp. 56
Author(s):  
Yasanthi Kottegoda ◽  
Robert Fitzgerald
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Lower Bounds ◽  
Upper Bound ◽  
Cyclic Codes ◽  
Quadratic Residue ◽  
Upper And Lower Bounds ◽  
Linear Recurring Sequences ◽  
Irreducible Cyclic Codes ◽  
Homogeneous Linear

Consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, 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 $a\in \mathbb{N}$.

On the Number of Zeros Over a Finite Field of Certain Symmetric Polynomials

1980 ◽  
Vol 23 (3) ◽  
pp. 327-332
Author(s):  
P. V. Ceccherini ◽  
J. W. P. Hirschfeld
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Polynomial Equations ◽  
Symmetric Polynomials ◽  
Number Of Solutions ◽  
Number Of Zeros

A variety of applications depend on the number of solutions of polynomial equations over finite fields. Here the usual situation is reversed and we show how to use geometrical methods to estimate the number of solutions of a non-homogeneous symmetric equation in three variables.

scholarly journals Pairs of Quadratic Forms over Finite Fields

10.37236/4072 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Alexander Pott ◽  
Kai-Uwe Schmidt ◽  
Yue Zhou
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Boolean Functions ◽  
Quadratic Forms ◽  
Symmetric Matrices ◽  
Explicit Expressions

Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a set of matrices over $\mathbb{F}_q$. The main results of this paper are explicit expressions for the number of pairs $(A,B)$ of matrices in $X$ such that $A$ has rank $r$, $B$ has rank $s$, and $A+B$ has rank $k$ in the cases that (i) $X$ is the set of alternating matrices over $\mathbb{F}_q$ and (ii) $X$ is the set of symmetric matrices over $\mathbb{F}_q$ for odd $q$. Our motivation to study these sets comes from their relationships to quadratic forms. As one application, we obtain the number of quadratic Boolean functions that are simultaneously bent and negabent, which solves a problem due to Parker and Pott.

The Zeta Function of a Pair of Quadratic Forms

2001 ◽  
Vol 44 (2) ◽  
pp. 242-256
Author(s):  
Laura Mann Schueller
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Function Field ◽  
Quadratic Forms ◽  
Arbitrary Characteristic ◽  
Number Of Zeros ◽  
Even Order ◽  
Hyperelliptic Function

AbstractThe zeta function of a nonsingular pair of quadratic forms defined over a finite field, k, of arbitrary characteristic is calculated. A. Weil made this computation when char k ≠ 2. When the pair has even order, a relationship between the number of zeros of the pair and the number of places of degree one in an appropriate hyperelliptic function field is established.

scholarly journals The existence of primitive normal elements of quadratic forms over finite fields

2020 ◽  
pp. 2250068
Author(s):  
Himangshu Hazarika ◽  
Dhiren Kumar Basnet ◽  
Stephen D. Cohen
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Quadratic Forms ◽  
Primitive Element ◽  
Sufficient Condition ◽  
Extension Field ◽  
Normal Element

For [Formula: see text] ([Formula: see text]), denote by [Formula: see text] the finite field of order [Formula: see text] and for a positive integer [Formula: see text], let [Formula: see text] be its extension field of degree [Formula: see text]. We establish a sufficient condition for existence of a primitive normal element [Formula: see text] such that [Formula: see text] is a primitive element, where [Formula: see text], with [Formula: see text] satisfying [Formula: see text] in [Formula: see text].

Linear Recurring Sequences and Explicit Factors of x2nd−1 in 𝔽q[x]

2020 ◽  
Vol 27 (03) ◽  
pp. 563-574
Author(s):  
Manjit Singh
Keyword(s):  
Field Theory ◽  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Second Order ◽  
Order Linear ◽  
Linear Recurring Sequences

Let 𝔽q be a finite field of odd characteristic containing q elements, and n be a positive integer. An important problem in finite field theory is to factorize xn − 1 into the product of irreducible factors over a finite field. Beyond the realm of theoretical needs, the availability of coefficients of irreducible factors over finite fields is also very important for applications. In this paper, we introduce second order linear recurring sequences in 𝔽q and reformulate the explicit factorization of [Formula: see text] over 𝔽q in such a way that the coefficients of its irreducible factors can be determined from these sequences when d is an odd divisor of q + 1.

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

2007 ◽  
Vol 03 (04) ◽  
pp. 541-556 ◽  
Author(s):  
WAI KIU CHAN ◽  
A. G. EARNEST ◽  
MARIA INES ICAZA ◽  
JI YOUNG KIM
Keyword(s):  
Quadratic Form ◽  
Number Field ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Integral Quadratic Form ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Totally Positive ◽  
Finiteness Result

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

scholarly journals On small zeros of quadratic forms over finite fields

1989 ◽  
Vol 31 (3) ◽  
pp. 272-284 ◽  
Author(s):  
Wang Yuan
Keyword(s):  
Finite Fields ◽  
Quadratic Forms

scholarly journals REPRESENTATIONS OF ALTERNATIVE CLIFFORD ALGEBRAS OF QUADRATIC FORMS

2014 ◽  
Vol 57 (3) ◽  
pp. 579-590 ◽  
Author(s):  
STACY MARIE MUSGRAVE
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Algebraic Structure ◽  
Quadratic Forms ◽  
Clifford Algebras ◽  
Future Research ◽  
Octonion Algebras

AbstractThis work defines a new algebraic structure, to be called an alternative Clifford algebra associated to a given quadratic form. I explored its representations, particularly concentrating on connections to the well-understood octonion algebras. I finished by suggesting directions for future research.

Sign in / Sign up

Export Citation Format

Share Document