Finite Field Of Order Research Articles

Explicit incidence bounds over general finite fields

Let $\mathbb{F}_{q}$ be a finite field of order $q=p^k$ where $p$ is prime. Let $P$ and $L$ be sets of points and lines respectively in $\mathbb{F}_{q} \times \mathbb{F}_{q}$ with $|P|=|L|=n$. We establish the incidence bound $I(P,L) \leq \gamma n^{3/2 - 1/12838}$, where $\gamma$ is an absolute constant, so long as $P$ satisfies the conditions of being an `antifield'. We define this to mean that the projection of $P$ onto some coordinate axis has no more than half-dimensional interaction with large subfields of $\mathbb{F}_q$. In addition, we give examples of sets satisfying these conditions in the important cases $q=p^2$ and $q=p^4$.

On the equivalence of quadratic APN functions

Establishing the CCZ-equivalence of a pair of APN functions is generally quite difficult. In some cases, when seeking to show that a putative new infinite family of APN functions is CCZ inequivalent to an already known family, we rely on computer calculation for small values of n. In this paper we present a method to prove the inequivalence of quadratic APN functions with the Gold functions. Our main result is that a quadratic function is CCZ-equivalent to the APN Gold function $${x^{2^r+1}}$$ if and only if it is EA-equivalent to that Gold function. As an application of this result, we prove that a trinomial family of APN functions that exist on finite fields of order 2 n where n � 2 mod 4 are CCZ inequivalent to the Gold functions. The proof relies on some knowledge of the automorphism group of a code associated with such a function.

On the number of distinct values of a class of functions over a finite field

Several authors have recently shown that a planar function over a finite field of order q must have at least ( q + 1 ) / 2 distinct values. In this note this result is extended by weakening the hypothesis significantly and strengthening the conclusion. We also give an algorithm for determining whether a given bivariate polynomial ϕ ( X , Y ) can be written as f ( X + Y ) − f ( X ) − f ( Y ) for some polynomial f. Using the ideas of the algorithm, we then show a Dembowski–Ostrom polynomial is planar over a finite field of order q if and only if it yields exactly ( q + 1 ) / 2 distinct values under evaluation; that is, it meets the lower bound of the image size of a planar function.

The minimum rank problem over the finite field of order 2: Minimum rank 3

Our main result is a sharp bound for the number of vertices in a minimal forbidden subgraph for the graphs having minimum rank at most 3 over the finite field of order 2. We also list all 62 such minimal forbidden subgraphs. We conclude by exploring how some of these results over the finite field of order 2 extend to arbitrary fields and demonstrate that at least one third of the 62 are minimal forbidden subgraphs over an arbitrary field for the class of graphs having minimum rank at most 3 in that field.

On [formula omitted]-invariants of representations of [formula omitted

Let F q be a finite field of order q and whose characteristic is not equal to 2 or 3. Let G denote a split algebraic group of type D 4 over F q . It contains a split algebraic group G 2 over F q . In this paper we will state a conjecture on the dimensions of G 2 ( F q ) -invariant vectors on representations of G ( F q ) , and we will prove this conjecture in the cases where the representation of G ( F q ) is a principal series representation, a generic representation, or a unipotent representation.

Construction of pseudorandom binary sequences using additive characters over GF(2 k )

In a series of papers Mauduit and Sarkozy (partly with coauthors) studied finite pseudorandom binary sequences and they constructed sequences with strong pseudorandom properties. In these constructions fields with prime order were used. In this paper a new construction is presented, which is based on finite fields of order 2 k .

Generalized weighing matrices and self-orthogonal codes

It is shown that the row spaces of certain generalized weighing matrices, Bhaskar Rao designs, and generalized Hadamard matrices over a finite field of order q are hermitian self-orthogonal codes. Certain matrices of minimum rank yield optimal codes. In the special case when q = 4 , the codes are linked to quantum error-correcting codes, including some codes with optimal parameters.

Generalized H-codes and type II codes over GF(4)

The type II codes have been studied widely in applications since their appearance. With analysis of the algebraic structure of finite field of order 4 (i.e., GF(4)), some necessary and sufficient conditions that a generalized H-code (i.e., GH-code) is a type II code over GF(4) are given in this article, and an efficient and simple method to generate type II codes from GH-codes over GF(4) is shown. The conclusions further extend the coding theory of type II.

Sum-product estimates via directed expanders

Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain constructions of expanders, we characterize all polynomials $P$ for which the following holds \vskip2mm \centerline{\it If $|A+A|$ is small (compared to $|A|$), then $|P(A)|$ is large.} \vskip2mm \noindent The case $P=x_1x_2$ corresponds to the well-known sum-product problem.

Commutative presemifields and semifields

Strong conditions are derived for when two commutative presemifields are isotopic. It is then shown that any commutative presemifield of odd order can be described by a planar Dembowski–Ostrom polynomial and conversely, any planar Dembowski–Ostrom polynomial describes a commutative presemifield of odd order. These results allow a classification of all planar functions which describe presemifields isotopic to a finite field and of all planar functions which describe presemifields isotopic to Albert's commutative twisted fields. A classification of all planar Dembowski–Ostrom polynomials over any finite field of order p 3 , p an odd prime, is therefore obtained. The general theory developed in the article is then used to show the class of planar polynomials X 10 + a X 6 − a 2 X 2 with a ≠ 0 describes precisely two new commutative presemifields of order 3 e for each odd e ⩾ 5 .

On the independence number of the Erdős‐Rényi and projective norm graphs and a related hypergraph

AbstractThe Erdős‐Rényi and Projective Norm graphs are algebraically defined graphs that have proved useful in supplying constructions in extremal graph theory and Ramsey theory. Their eigenvalues have been computed and this yields an upper bound on their independence number. Here we show that in many cases, this upper bound is sharp in the order of magnitude. Our result for the Erdős‐Rényi graph has the following reformulation: the maximum size of a family of mutually non‐orthogonal lines in a vector space of dimension three over the finite field of order q is of order q3/2. We also prove that every subset of vertices of size greater than q2/2 + q3/2 + O(q) in the Erdős‐Rényi graph contains a triangle. This shows that an old construction of Parsons is asymptotically sharp. Several related results and open problems are provided. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 113–127, 2007

Counting characters of upper triangular groups

Working over a finite field of order q, let U n ( q ) be the group of upper triangular n × n matrices with all diagonal entries equal to 1. It is known that all complex irreducible characters of U n ( q ) have degrees that are powers of q and that all powers q e occur for 0 ⩽ e ⩽ μ ( n ) , where the upper bound μ ( n ) is defined by the formulas μ ( 2 m ) = m ( m − 1 ) and μ ( 2 m + 1 ) = m 2 . It has been conjectured that the number of irreducible characters having degree q e is some polynomial in q with integer coefficients (depending on n and e). In this paper, we construct explicit polynomials for the cases where e is one of μ ( n ) , μ ( n ) − 1 and 1. Also, a list of polynomials is presented that appear to give the correct character counts for n ⩽ 9 . Several related results concerning pattern groups are also included.

An area-efficient bit-serial integer and GF(2 n) multiplier

This paper presents the design of a new multiplier architecture for normal integer multiplication of positive and negative numbers as well as for multiplication in finite fields of order 2 n . It has been developed to increase the performance of algorithms for cryptographic and signal processing applications on implementations of the Instruction Systolic Array (ISA) parallel computer model [M. Kunde, H.W. Lang, M. Schimmler, H. Schmeck, H. Schröder, Parallel Computing 7 (1988) 25–39, H.W. Lang, Integration, the VLSI Journal 4 (1986) 65–74]. The multiplier operates least significant bit (LSB)-first for integer multiplication and most significant bit ( )-first for finite field multiplication. It is a modular bit-serial design, which on the one hand can be efficiently implemented in hardware and on the other hand has the advantage that it can handle operands of arbitrary length.

The affinity of a permutation of a finite vector space

For a permutation f of an n-dimensional vector space V over a finite field of order q we let k -affinity ( f ) denote the number of k-flats X of V such that f ( X ) is also a k-flat. By k -spectrum ( n , q ) we mean the set of integers k -affinity ( f ) , where f runs through all permutations of V. The problem of the complete determination of k -spectrum ( n , q ) seems very difficult except for small or special values of the parameters. However, we are able to determine ( n - 1 ) -spectrum ( n , 2 ) and establish that 0 ∈ k -spectrum ( n , q ) in the following cases: (i) q ≥ 3 and 1 ≤ k ≤ n - 1 ; (ii) q = 2 , 3 ≤ k ≤ n - 1 ; (iii) q = 2 , k = 2 , n ≥ 3 odd. For 1 ≤ k ≤ n - 1 and ( q , k ) ≠ ( 2 , 1 ) , the maximum of k -affinity ( f ) is obtained when f is any semiaffine mapping. We conjecture that the next to largest value of k -affinity ( f ) occurs when f is a transposition, and we are able to prove it when q = 2 , k = 2 , n ≥ 3 and when q ≥ 3 , k = 1 , n ≥ 2 .

Prime Rings Whose Units Satisfy a Group Identity. II

Let R be a prime ring and 𝒰(R) its group of units. We prove that if 𝒰(R) satisfies a group identity and 𝒰(R) generates R,then either R is a domain or R is isomorphic to the algebra of n × n matrices over a finite field of order d. Moreover the integers n and d depend only on the group identity satisfed by 𝒰(R). This result has been recently proved by C. H. Liu and T. K. Lee (Liu,C. H.; Lee,T. K. Group identities and prime rings generated by units. Comm. Algebra (to appear)) and here we present a new different proof.

A Polynomial Representation of the Diffie-Hellman Mapping

Let Fq be the finite field of order q and γ be an element of Fq of order d. The construction of an explicit polynomial \(\) of degree \(\) with the property \(\) for \(\) is described. In particular the exact degree and sparsity of f are determined.

Special prime numbers and discrete logs in finite prime fields

A set $A$ of primes $p$ involving numbers such as $ab^t+c$, where $|a|,|b|,|c|=O(1)$ and $t\to \infty$, is defined. An algorithm for computing discrete logs in the finite field of order $p$ with $p\in A$ is suggested. Its heuristic expected running time is $L_p[\frac 13;(\frac {32}{9})^{1/3}]$ for $(\frac {32}{9})^{1/3}=1.526\dotsb$, where $L_p[\alpha ;\beta ]=\exp ((\beta +o(1))\ln ^\alpha p(\ln \ln p)^{1-\alpha })$ as $p\to \infty$, $0<\alpha <1$, and $0<\beta$. At present, the most efficient algorithm for computing discrete logs in the finite field of order $p$ for general $p$ is Schirokauer’s adaptation of the Number Field Sieve. Its heuristic expected running time is $L_p[\frac 13;(\frac {64}{9})^{1/3}]$ for $(\frac {64}{9})^{1/3}=1.9229\dotsb$. Using $p\in A$ rather than general $p$ does not enhance the performance of Schirokauer’s algorithm. The definition of the set $A$ and the algorithm suggested in this paper are based on a more general congruence than that of the Number Field Sieve. The congruence is related to the resultant of integer polynomials. We also give a number of useful identities for resultants that allow us to specify this congruence for some $p$.

On Fusion in Unipotent Blocks

The context of this note is as follows. One considers a connected reductive group G and a Frobenius endomorphism F: G → G defining G over a finite field of order q. One denotes by GF the associated (finite) group of fixed points. Let l be a prime not dividing q. We are interested in the l-blocks of the finite group GF. Such a block is called unipotent if there is a unipotent character (see, for instance, [6, Definition 12.1]) among its representations in characteristic zero. Roughly speaking, it is believed that the study of arbitrary blocks of GF might be reduced to unipotent blocks (see [2, Théorème 2.3], [5, Remark 3.6]). In view of certain conjectures about blocks (see, for instance, [9]), it would be interesting to further reduce the study of unipotent blocks to the study of principal blocks (blocks containing the trivial character). Our Theorem 7 is a step in that direction: we show that the local structure of any unipotent block of GF is very close to that of a principal block of a group of related type (notion of ‘control of fusion’, see [13, §49]). 1991 Mathematics Subject Classification 20Cxx.

A quadratic form tr( AX2) and its application

We give the complete classification of quadratic forms tr( AX 2) obtained by taking the trace of n × n matrices AX 2 for n = 2,3, where A ϵ M n ( F q ), X is a square matrix of n 2 variables, and F q is a finite field of order q and of characteristic different from 2. As an application of the classification we can compute the number of solutions of a matrix equation X 2 1 + X 2 2 + … + X 2 m = B for a 2 × 2 matrix B over F q .

Irreducible polynomials and linear recurring arrays

For α ϵ F q the finite field of order q and β ϵ F q (α.), let F q ( α, β) = F q ( γ). We obtain an explicit formula for the minimal polynomial h gg ( x) of γ in terms of the greatest common divisor of two polynomials which are closely related to the minimal polynomials f α ( x) of α and gβ( x) of β. We also give an application of this result to linear recurring arrays.