scholarly journals Period sets of linear recurrences over finite fields and related commutative rings

2021 ◽  
Vol 14 (3) ◽  
pp. 361-376
Author(s):  
Michael R. Bush ◽  
Danjoseph Quijada
Keyword(s):  
Finite Fields ◽  
Commutative Rings ◽  
Linear Recurrences

scholarly journals Uniform distribution of sequences in rings of integral matrices

1979 ◽  
Vol 20 (2) ◽  
pp. 169-178
Author(s):  
Harald Niederreiter ◽  
Jau-Shyong Shiue
Keyword(s):  
Uniform Distribution ◽  
Finite Fields ◽  
Commutative Rings ◽  
Algebraic Integers ◽  
Matrix Rings ◽  
Integral Matrices ◽  
Rings Of Algebraic Integers ◽  
Uniform Distribution Of Sequences ◽  
Satisfactory Theory ◽  
Noncommutative Setting

For various discrete commutative rings a concept of uniform distribution has already been introduced and studied, for example, for the ring of rational integers by Niven [9] (see also Kuipers and Niederreiter [2, Ch. 5]), for the rings of Gaussian and Eisenstein integers by Kuipers, Niederreiter, and Shiue [3], for rings of algebraic integers by Lo and Niederreiter [4], [7], and for finite fields by Gotusso [1] and Niederreiter and Shiue [8]. In the present paper, we shall show that a satisfactory theory of uniform distribution can also be developed in a noncommutative setting, namely for matrix rings over the rational integers.

scholarly journals FINITE QUASI-FROBENIUS MODULES AND LINEAR CODES

2004 ◽  
Vol 03 (03) ◽  
pp. 247-272 ◽  
Author(s):  
MARCUS GREFERATH ◽  
ALEXANDR NECHAEV ◽  
ROBERT WISBAUER
Keyword(s):  
Finite Fields ◽  
Linear Codes ◽  
General Setting ◽  
Commutative Rings ◽  
Finite Ring ◽  
Weight Enumerators ◽  
Frobenius Rings ◽  
Finite Frobenius Rings

The theory of linear codes over finite fields has been extended by A. Nechaev to codes over quasi-Frobenius modules over commutative rings, and by J. Wood to codes over (not necessarily commutative) finite Frobenius rings. In the present paper, we subsume these results by studying linear codes over quasi-Frobenius and Frobenius modules over any finite ring. Using the character module of the ring as alphabet, we show that fundamental results like MacWilliams' theorems on weight enumerators and code isometry can be obtained in this general setting.

scholarly journals Commutative rings with homomorphic power functions

1992 ◽  
Vol 15 (1) ◽  
pp. 91-102
Author(s):  
David E. Dobbs ◽  
John O. Kiltinen ◽  
Bobby J. Orndorff
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Structure Theorem ◽  
Commutative Rings ◽  
Prime Characteristic ◽  
Power Functions ◽  
Finite Case

A (commutative) ringR(with identity) is calledm-linear (for an integerm≥2) if(a+b)m=am+bmfor allaandbinR. Them-linear reduced rings are characterized, with special attention to the finite case. A structure theorem reduces the study ofm-linearity to the case of prime characteristic, for which the following result establishes an analogy with finite fields. For each primepand integerm≥2which is not a power ofp, there exists an integers≥msuch that, for each ringRof characteristicp,Rism-linear if and only ifrm=rpsfor eachrinR. Additional results and examples are given.

Ideal-Based k-Zero-Divisor Hypergraph of Commutative Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 655-672
Author(s):  
K. Selvakumar ◽  
M. Subajini
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Zero Divisor ◽  
Necessary Conditions ◽  
Commutative Rings ◽  
Nonzero Ideal ◽  
Fixed Integer ◽  
Zero Divisors ◽  
Vertex Set ◽  
The Ideal

Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] a fixed integer. The ideal-based [Formula: see text]-zero-divisor hypergraph [Formula: see text] of [Formula: see text] has vertex set [Formula: see text], the set of all ideal-based [Formula: see text]-zero-divisors of [Formula: see text], and for distinct elements [Formula: see text] in [Formula: see text], the set [Formula: see text] is an edge in [Formula: see text] if and only if [Formula: see text] and the product of the elements of any [Formula: see text]-subset of [Formula: see text] is not in [Formula: see text]. In this paper, we show that [Formula: see text] is connected with diameter at most 4 provided that [Formula: see text] for all ideal-based 3-zero-divisor hypergraphs. Moreover, we find the chromatic number of [Formula: see text] when [Formula: see text] is a product of finite fields. Finally, we find some necessary conditions for a finite ring [Formula: see text] and a nonzero ideal [Formula: see text] of [Formula: see text] to have [Formula: see text] planar.

scholarly journals Matrix-Product Codes over Commutative Rings and Constructions Arising from σ , δ -Codes

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Mhammed Boulagouaz ◽  
Abdulaziz Deajim
Keyword(s):  
Lower Bound ◽  
Commutative Ring ◽  
Finite Fields ◽  
Hamming Distance ◽  
Commutative Rings ◽  
Matrix Product ◽  
Sufficient Condition ◽  
Generator Matrix ◽  
Product Codes ◽  
Product Code

A well-known lower bound (over finite fields and some special finite commutative rings) on the Hamming distance of a matrix-product code (MPC) is shown to remain valid over any commutative ring R . A sufficient condition is given, as well, for such a bound to be sharp. It is also shown that an MPC is free when its input codes are all free, in which case a generator matrix is given. If R is finite, a sufficient condition is provided for the dual of an MPC to be an MPC, a generator matrix for such a dual is given, and characterizations of LCD, self-dual, and self-orthogonal MPCs are presented. Finally, the results of this paper are used along with previous results of the authors to construct novel MPCs arising from σ , δ -codes. Some properties of such constructions are also studied.

Ultrafunctor

1975 ◽  
Vol 27 (2) ◽  
pp. 372-375 ◽  
Author(s):  
Andy R. Magid
Keyword(s):  
Finite Fields ◽  
Abelian Groups ◽  
Commutative Rings ◽  
Picard Group ◽  
Brauer Group

Let G be a functor from commutative rings to abelian groups and let ﹛Rt : i ∈ S﹜ be a family of commutative rings indexed by the set S. Let be an ultrafilter on S, and let denote the ultraproduct of the Rt with respect to . This paper studies the problem of computing from the G(Rj) via the mapThe functors studied are Pic = Picard group, Br = Brauer group, U = units, and the functors K0, K1, SK1, K2 of Algebraic K-Theory. For G = Pic, U, K1 and SK1, (*) is always a monomorphism. An example is given to show that even if all the Rt are finite fields the map (*) has a kernel for G = K2.

scholarly journals TERM REWRITE RULES FOR FINITE FIELDS

1991 ◽  
Vol 01 (03) ◽  
pp. 353-369 ◽  
Author(s):  
STANLEY BURRIS ◽  
JOHN LAWRENCE
Keyword(s):  
Finite Fields ◽  
Equational Theory ◽  
Commutative Rings ◽  
Finite Sets ◽  
First Case ◽  
Finite Set ◽  
Complete Set ◽  
Rewrite Rules

Let F1, …, Fk be finite fields with distinct characteristics. We give a finite set of equations which axiomatize the equational theory of F1, …, Fk and then use these axioms to find a finite set of AC-term rewrite rules which is complete for this theory. This gives finite sets of complete AC-term rewrite rules for most instances of xm ≈ x rings by adding new rules to the usual AC-term rewrite rules for commutative rings. The first case for which we do not find a complete set of AC-term rewrite rules is x22 ≈ x, and we doubt that such rules can be found. If R is a set of AC-term rewrite rules from which one can derive x(y + z) → xy + xz, then we show R cannot be complete for x22 ≈ x rings.

Finite Fields

1996 ◽  
Author(s):  
Rudolf Lidl ◽  
Harald Niederreiter
Keyword(s):  
Finite Fields
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