QE rings in characteristic pn

1983 ◽  
Vol 48 (1) ◽  
pp. 140-162 ◽  
Author(s):  
Chantal Berline ◽  
Gregory Cherlin
Keyword(s):  
Finite Field ◽  
Jacobson Radical ◽  
Prime Power ◽  
Galois Field ◽  
Power Characteristic ◽  
Witt Ring ◽  
Boolean Power ◽  
Three Components

AbstractWe show that all QE rings of prime power characteristic are constructed in a straightforward way out of three components: a filtered Boolean power of a finite field, a nilpotent Jacobson radical, and the ring Zp. or the Witt ring W2(F4) (which is the characteristic four analogue of the Galois field with four elements).

Projective Geometries that are Disjoint Unions of Caps

1980 ◽  
Vol 32 (6) ◽  
pp. 1299-1305 ◽  
Author(s):  
Barbu C. Kestenband
Keyword(s):  
Finite Field ◽  
Projective Geometry ◽  
Prime Power ◽  
Disjoint Union ◽  
Hermitian Matrices ◽  
Incidence Relation ◽  
Projective Geometries ◽  
Image Position ◽  
Natural Way ◽  
Square Matrix

We show that any PG(2n, q2) is a disjoint union of (q2n+1 − 1)/ (q − 1) caps, each cap consisting of (q2n+1 + 1)/(q + 1) points. Furthermore, these caps constitute the “large points” of a PG(2n, q), with the incidence relation defined in a natural way.A square matrix H = (hij) over the finite field GF(q2), q a prime power, is said to be Hermitian if hijq = hij for all i, j [1, p. 1161]. In particular, hii ∈ GF(q). If if is Hermitian, so is p(H), where p(x) is any polynomial with coefficients in GF(q).Given a Desarguesian Projective Geometry PG(2n, q2), n > 0, we denote its points by column vectors:All Hermitian matrices in this paper will be 2n + 1 by 2n + 1, n > 0.

CONSTRUCTING THE GROUP PRESERVING A SYSTEM OF FORMS

2008 ◽  
Vol 18 (02) ◽  
pp. 227-241 ◽  
Author(s):  
PETER A. BROOKSBANK ◽  
E. A. O'BRIEN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Linear Group ◽  
Jacobson Radical ◽  
Matrix Algebra ◽  
General Linear Group ◽  
Efficient Algorithms ◽  
Sesquilinear Forms ◽  
Group Of Units ◽  
Practical Algorithm

We present a practical algorithm to construct the subgroup of the general linear group that preserves a system of bilinear or sesquilinear forms on a vector space defined over a finite field. Components include efficient algorithms to construct the Jacobson radical and the group of units of a matrix algebra.

Zeros of random polynomials over finite fields

1992 ◽  
Vol 111 (2) ◽  
pp. 193-197 ◽  
Author(s):  
R. W. K. Odoni
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Prime Power ◽  
Random Polynomials ◽  
Polynomials Over Finite Fields ◽  
Image Position

Let be the finite field with q elements (q a prime power), let r 1 and let X1, , Xr be independent indeterminates over . We choose an arbitrary and a d 1 and consider

scholarly journals A characterisation of PSL2(Zþλ) and PGL2(Zþλ)

1968 ◽  
Vol 8 (3) ◽  
pp. 523-543 ◽  
Author(s):  
G. E. Wall
Keyword(s):  
Finite Field ◽  
Prime Power ◽  
Cyclic Subgroup ◽  
Residue Class ◽  
Linear Groups ◽  
Cyclic Subgroups ◽  
Even Order ◽  
Residue Class Ring ◽  
Projective Linear Groups ◽  
Class Ring

Let Fq denote the finite field with q elements, Zm the residue class ring Z/mZ. It is known that the projective linear groups G = PSL2(Fq) and PGL2(Fq) (q a prime-power ≥ 4) are characterised among finite insoluble groups by the property that, if two cyclic subgroups of G of even order intersect non-trivially, they generate a cyclic subgroup (cf. Brauer, Suzuki, Wall [2], Gorenstein, Walter [3]). In this paper, we give a similar characterisation of the groups G = PSL2 (Zþt+1) and PGL2 (Zþt+1) (p a prime ≥ 5, t ≥ 1).

Involutory Matrices Over Finite Local Rings

1972 ◽  
Vol 24 (3) ◽  
pp. 369-378 ◽  
Author(s):  
B. R. McDonald
Keyword(s):  
Prime Power ◽  
Quotient Ring ◽  
Careful Analysis ◽  
Commutative Rings ◽  
Local Rings ◽  
Power Characteristic ◽  
Special Cases ◽  
Algebraic Cryptography ◽  
Characteristic 2 ◽  
Finite Commutative Rings

A square matrix A over a commutative ring R is said to be involutory if A2 = I (identity matrix). It has been recognized for some time that involutory matrices have important applications in algebraic cryptography and the special cases where R is either a finite field or a quotient ring of the rational integers have been extensively researched. However, there has been no detailed attempt to extend the theory to all finite commutative rings. In this paper we illustrate in detail the theory of involutory matrices over finite commutative rings with 1 having odd characteristic. The method is a careful analysis of finite local rings of odd prime power characteristic. The techniques might be also used in the examination of involutory matrices over local rings of characteristic 2λ; however, as illustrated by finite fields of characteristic 2 and Z/2λZ (Z the rational integers), the arguments are basically different. The reader will note the methods are not limited to only questions on involutory matrices.

scholarly journals Coefficient subrings of certain local rings with prime-power characteristic

1995 ◽  
Vol 18 (3) ◽  
pp. 451-462 ◽  
Author(s):  
Takao Sumiyama
Keyword(s):  
Local Ring ◽  
Prime Power ◽  
Galois Ring ◽  
Local Rings ◽  
Power Characteristic ◽  
Commutative Field

IfRis a local ring whose radicalJ(R)is nilpotent andR/J(R)is a commutative field which is algebraic overGF(p), thenRhas at least one subringSsuch thatS=∪i=1∞Si, where eachSi, is isomorphic to a Galois ring andS/J(S)is naturally isomorphic toR/J(R). Such subrings ofRare mutually isomorphic, but not necessarily conjugate inR.

The construction of LDPC codes based on the subspaces of singular linear space over finite field

2016 ◽  
Vol 08 (04) ◽  
pp. 1650073
Author(s):  
Congcong Wang ◽  
Yingying Zhang ◽  
Zhuoqun Li ◽  
Xiaona Zhang ◽  
You Gao
Keyword(s):  
Finite Field ◽  
Linear Space ◽  
Prime Power ◽  
Ldpc Codes

Let [Formula: see text] be a finite field with [Formula: see text] elements, where [Formula: see text] is a prime power. [Formula: see text] denotes the [Formula: see text]-dimensional row linear space over [Formula: see text]. In this paper, we construct a series of LDPC codes based on the subspaces of singular linear space over [Formula: see text], and calculate their parameters.

scholarly journals Distinct Triangle Areas in a Planar Point Set over Finite Fields

10.37236/700 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Prime Power ◽  
Finite Plane ◽  
Expected Number ◽  
Planar Point ◽  
Point Set ◽  
Distinct Triangle

Let $\mathcal{P}$ be a set of $n$ points in the finite plane $\mathbb{F}_q^2$ over the finite field $\mathbb{F}_q$ of $q$ elements, where $q$ is an odd prime power. For any $s \in \mathbb{F}_q$, denote by $A (\mathcal{P}; s)$ the number of ordered triangles whose vertices in $\mathcal{P}$ having area $s$. We show that if the cardinality of $\mathcal{P}$ is large enough then $A (\mathcal{P}; s)$ is close to the expected number $|\mathcal{P}|^3/q$.

scholarly journals Construction and Determination of Irreducible Polynomials in Galois elds, GF(2m)

2019 ◽  
pp. 1-6
Author(s):  
Abraham Aidoo ◽  
Kwasi Baah Gyam ◽  
Fengfan Yang
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Monic Polynomial ◽  
Galois Field ◽  
Irreducible Polynomials ◽  
Primitive Polynomials

This work is about Construction of Irreducible Polynomials in Finite fields. We defined some terms in the Galois field that led us to the construction of the polynomials in the GF(2m). We discussed the following in the text; irreducible polynomials, monic polynomial, primitive polynomials, eld, Galois eld or nite elds, and the order of a finite field. We found all the polynomials in $$F_2[x]$$ that is, $$P(x) =\sum_{i=1}^m a_ix^i : a_i \in F_2$$ with $$a_m \neq 0$$ for some degree $m$ whichled us to determine the number of irreducible polynomials generally at any degree in $$F_2[x]$$.

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