scholarly journals Self-Dual Normal Basis of a Galois Ring

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Irwansyah ◽  
Intan Muchtadi-Alamsyah ◽  
Aleams Barra ◽  
Ahmad Muchlis
Keyword(s):  
Free Module ◽  
Galois Ring ◽  
Normal Basis ◽  
Galois Rings ◽  
Orthogonal Matrices

LetR′=GR(ps,psml)andR=GR(ps,psm)be two Galois rings. In this paper, we show how to construct normal basis in the extension of Galois rings, and we also define weakly self-dual normal basis and self-dual normal basis forR′overR, whereR′is considered as a free module overR. Moreover, we explain a way to construct self-dual normal basis using particular system of polynomials. Finally, we show the connection between self-dual normal basis forR′overRand the set of all invertible, circulant, and orthogonal matrices overR.

scholarly journals Normal Bases on Galois Ring Extensions

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 702
Author(s):  
Aixian Zhang ◽  
Keqin Feng
Keyword(s):  
Signal Processing ◽  
Finite Field ◽  
Block Cipher ◽  
Field Extension ◽  
Galois Ring ◽  
Normal Basis ◽  
Galois Rings ◽  
Ring Extension ◽  
Galois Fields ◽  
Normal Bases

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.

Symbol-Pair Distances of Repeated-Root Negacyclic Codes of Length 2s over Galois Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 581-600
Author(s):  
Hai Q. Dinh ◽  
Hualu Liu ◽  
Roengchai Tansuchat ◽  
Thang M. Vo
Keyword(s):  
Finite Field ◽  
Galois Ring ◽  
Maximum Distance ◽  
Galois Rings ◽  
Constacyclic Codes ◽  
Chain Ring ◽  
Negacyclic Codes ◽  
Singleton Bound ◽  
Maximum Distance Separable ◽  
Special Case

Negacyclic codes of length [Formula: see text] over the Galois ring [Formula: see text] are linearly ordered under set-theoretic inclusion, i.e., they are the ideals [Formula: see text], [Formula: see text], of the chain ring [Formula: see text]. This structure is used to obtain the symbol-pair distances of all such negacyclic codes. Among others, for the special case when the alphabet is the finite field [Formula: see text] (i.e., [Formula: see text]), the symbol-pair distance distribution of constacyclic codes over [Formula: see text] verifies the Singleton bound for such symbol-pair codes, and provides all maximum distance separable symbol-pair constacyclic codes of length [Formula: see text] over [Formula: see text].

scholarly journals Cogredient Standard Forms of Symmetric Matrices over Galois Rings of Odd Characteristic

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Yonglin Cao
Keyword(s):  
Prime Number ◽  
Explicit Formula ◽  
Galois Ring ◽  
Symmetric Matrices ◽  
Galois Rings ◽  
Positive Integers

Let R=GR(ps,psm) be a Galois ring of characteristic ps and cardinality psm, where s and m are positive integers and p is an odd prime number. Two kinds of cogredient standard forms of symmetric matrices over R are given, and an explicit formula to count the number of all distinct cogredient classes of symmetric matrices over R is obtained.

EISENSTEIN LATTICES, GALOIS RINGS AND QUATERNARY CODES

2006 ◽  
Vol 02 (02) ◽  
pp. 289-303 ◽  
Author(s):  
PHILIPPE GABORIT ◽  
ANN MARIE NATIVIDAD ◽  
PATRICK SOLÉ
Keyword(s):  
Modular Lattice ◽  
Orthogonal Basis ◽  
Galois Ring ◽  
Galois Rings ◽  
Type Ii ◽  
Dual Codes ◽  
Type Ii Codes ◽  
Quaternary Codes ◽  
Double Circulant Codes ◽  
Circulant Codes

Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) ℤ4-codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual ℤ4-codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual 𝔽4-codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.

scholarly journals An Authentication Code over Galois Rings with Optimal Impersonation and Substitution Probabilities

2018 ◽  
Author(s):  
Juan Carlos Ku-Cauich ◽  
Guillermo Morales-Luna ◽  
Horacio Tapia-Recillas
Keyword(s):  
Special Class ◽  
Galois Ring ◽  
Gray Map ◽  
Galois Rings ◽  
Authentication Codes ◽  
Authentication Code

Two new systematic authentication codes based on the Gray map over a Galois ring are introduced. The first introduced code attains optimal impersonation and substitution probabilities. The second code improves space sizes but it does not attain optimal probabilities. Besides it is conditioned to the existence of a special class of bent maps on Galois rings.

scholarly journals An Authentication Code over Galois Rings with Optimal Impersonation and Substitution Probabilities

2018 ◽  
Vol 23 (3) ◽  
pp. 46
Author(s):  
Juan Ku-Cauich ◽  
Guillermo Morales-Luna ◽  
Horacio Tapia-Recillas
Keyword(s):  
Special Class ◽  
Galois Ring ◽  
Gray Map ◽  
Galois Rings ◽  
Authentication Codes ◽  
Authentication Code

Two new systematic authentication codes based on the Gray map over a Galois ring are introduced. The first introduced code attains optimal impersonation and substitution probabilities. The second code improves space sizes, but it does not attain optimal probabilities. Additionally, it is conditioned to the existence of a special class of bent maps on Galois rings.

COORDINATE SETS OF GENERALIZED GALOIS RINGS

2004 ◽  
Vol 03 (01) ◽  
pp. 31-48 ◽  
Author(s):  
S. GONZÁLEZ ◽  
C. MARTÍNEZ ◽  
I. F. RÚA ◽  
V. T. MARKOV ◽  
A. A. NECHAEV
Keyword(s):  
Prime Number ◽  
Closed Subset ◽  
Galois Ring ◽  
Galois Rings ◽  
Characteristic P ◽  
Nonassociative Ring

A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic pn, for some prime number p, such that its top-factor [Formula: see text] is a semifield. It is well-known that if S is an associative Galois Ring (GR), then it contains a multiplicatively closed subset isomorphic to [Formula: see text], the so-called Teichmüller Coordinate Set (TCS). In this paper we show that the existence of a TCS characterizes GR in the class of all GGR S such that the multiplicative loop [Formula: see text] is right (or left) primitive.

scholarly journals Mass Formula for Self-Dual Codes over Galois Rings GR(p3, r)

2019 ◽  
Vol 12 (4) ◽  
pp. 1701-1716
Author(s):  
Trilbe Lizann Espina Vasquez ◽  
Gaudencio Jr. Cempron Petalcorin
Keyword(s):  
Positive Integer ◽  
Mass Formula ◽  
Galois Ring ◽  
The Self ◽  
Galois Rings ◽  
Dual Codes ◽  
Characteristic P

Let $p$ be an odd prime and $r$ a positive integer. Let $\text{GR}(p^3,r)$ be the Galois ring of characteristic $p^3$ and cardinality $p^{3r}$. In this paper, we investigate the self-dual codes over $\text{GR}(p^3,r)$ and give a method to construct self-dual codes over this ring. We establish a mass formula for self-dual codes over $\text{GR}(p^3,r)$ and classify self-dual codes over $\text{GR}(p^3,2)$ of length 4 for $p=3,5$.

scholarly journals Polynomials with minimal value set over Galois rings

1991 ◽  
Vol 14 (3) ◽  
pp. 471-474
Author(s):  
Maria T. Acosta-De-Orozco ◽  
Javier Gomez-Calderon
Keyword(s):  
Galois Ring ◽  
Galois Rings ◽  
Value Set

LetGR(pn,m)denote the Galois ring of orderpn,m, wherepis a prime. In this paper we define and characterize minimal value set polynomials overGR(pn,m).

scholarly journals Generalized affine transformation monoids on Galois rings

2006 ◽  
Vol 2006 ◽  
pp. 1-6
Author(s):  
Yonglin Cao
Keyword(s):  
Algebraic Structure ◽  
Affine Transformation ◽  
Transformation Group ◽  
Galois Ring ◽  
Explicit Description ◽  
Green’S Relations ◽  
Galois Rings ◽  
Transformation Monoid ◽  
Green's Relations ◽  
Transformation Monoids

LetAbe a ring with identity. The generalized affine transformation monoidGaff(A)is defined as the set of all transformations onAof the formx↦xu+a(for allx∈A), whereu,a∈A. We study the algebraic structure of the monoidGaff(A)on a finite Galois ringA. The following results are obtained: an explicit description of Green's relations onGaff(A); and an explicit description of the Schützenberger group of every-class, which is shown to be isomorphic to the affine transformation group for a smaller Galois ring.

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