scholarly journals Primitive Idempotents of Irreducible Cyclic Codes of Length n

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Yuqian Lin ◽  
Qin Yue ◽  
Yansheng Wu
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Matrix Method ◽  
Cyclic Codes ◽  
Prime Divisors ◽  
Primitive Idempotents ◽  
Irreducible Cyclic Codes

Let Fq be a finite field with q elements and n a positive integer. In this paper, we use matrix method to give all primitive idempotents of irreducible cyclic codes of length n, whose prime divisors divide q-1.

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 $a\in \mathbb{N}$.

A note on weight distributions of irreducible cyclic codes

2014 ◽  
Vol 06 (03) ◽  
pp. 1450041
Author(s):  
Anuradha Sharma ◽  
Amit K. Sharma
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Cyclic Codes ◽  
Error Performance ◽  
Space Communications ◽  
Irreducible Cyclic Codes ◽  
Weight Distributions ◽  
Long Time ◽  
Interesting Object ◽  
Object Of Study

Irreducible cyclic codes form an important family of cyclic codes and have applications in space communications. Their weight distributions measure their error performance relative to several channels, and hence have been an interesting object of study for a long time. In this note, we provide a method to determine the weight distributions of q-ary irreducible cyclic codes of length n, where q is a prime power and n is a positive integer coprime to q. This method is more effective for irreducible cyclic codes of some special lengths. We also list some optimal irreducible cyclic codes, which attain the distance bounds given in Grassl's Table [Code Tables: Bounds on the parameters of various types of codes, http://www.codetables.de ].

The weight distributions of irreducible cyclic codes of length 2npm

2018 ◽  
Vol 11 (06) ◽  
pp. 1850085
Author(s):  
Monika Sangwan ◽  
Pankaj Kumar
Keyword(s):  
Finite Field ◽  
Explicit Formula ◽  
Weight Distribution ◽  
Cyclic Codes ◽  
Primitive Root ◽  
Irreducible Cyclic Codes ◽  
Weight Distributions ◽  
Primitive Root Modulo ◽  
Explicit Expressions

Let [Formula: see text] be a primitive root modulo [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct odd primes. Let [Formula: see text] be a finite field. For such pair of [Formula: see text] and [Formula: see text], the explicit expressions of minimal and generating polynomials over [Formula: see text] are obtained for all irreducible cyclic codes of length [Formula: see text]. In Sec. 4, it is observed that the weight distributions of all irreducible cyclic codes of length [Formula: see text] over [Formula: see text] can be computed easily with the help of the results obtained in [P. Kumar, M. Sangwan and S. K. Arora, The weight distribution of some irreducible cyclic codes of length [Formula: see text] and [Formula: see text], Adv. Math. Commun. 9 (2015) 277–289]. An explicit formula is also given to compute the weight distributions of irreducible cyclic codes of length [Formula: see text] over [Formula: see text].

scholarly journals A new characterization of L2(p2)

2020 ◽  
Vol 18 (1) ◽  
pp. 907-915
Author(s):  
Zhongbi Wang ◽  
Chao Qin ◽  
Heng Lv ◽  
Yanxiong Yan ◽  
Guiyun Chen
Keyword(s):  
Positive Integer ◽  
Irreducible Character ◽  
Character Degrees ◽  
Prime Divisors

Abstract For a positive integer n and a prime p, let {n}_{p} denote the p-part of n. Let G be a group, \text{cd}(G) the set of all irreducible character degrees of G , \rho (G) the set of all prime divisors of integers in \text{cd}(G) , V(G)=\left\{{p}^{{e}_{p}(G)}|p\in \rho (G)\right\} , where {p}^{{e}_{p}(G)}=\hspace{.25em}\max \hspace{.25em}\{\chi {(1)}_{p}|\chi \in \text{Irr}(G)\}. In this article, it is proved that G\cong {L}_{2}({p}^{2}) if and only if |G|=|{L}_{2}({p}^{2})| and V(G)=V({L}_{2}({p}^{2})) .

scholarly journals Monomial codes seen as invariant subspaces

2017 ◽  
Vol 15 (1) ◽  
pp. 1099-1107 ◽  
Author(s):  
María Isabel García-Planas ◽  
Maria Dolors Magret ◽  
Laurence Emilie Um
Keyword(s):  
Finite Field ◽  
Linear Algebra ◽  
Linear Transformation ◽  
Cyclic Codes ◽  
Invariant Subspaces ◽  
Hyperinvariant Subspaces ◽  
Computationally Expensive ◽  
The Relationship

Abstract It is well known that cyclic codes are very useful because of their applications, since they are not computationally expensive and encoding can be easily implemented. The relationship between cyclic codes and invariant subspaces is also well known. In this paper a generalization of this relationship is presented between monomial codes over a finite field 𝔽 and hyperinvariant subspaces of 𝔽n under an appropriate linear transformation. Using techniques of Linear Algebra it is possible to deduce certain properties for this particular type of codes, generalizing known results on cyclic codes.

Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields

2012 ◽  
Vol 55 (2) ◽  
pp. 418-423 ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Bilinear Form ◽  
Symmetric Bilinear Form

AbstractGiven a positive integern, a finite fieldofqelements (qodd), and a non-degenerate symmetric bilinear formBon, we determine the largest possible cardinality of pairwiseB-orthogonal subsets, that is, for any two vectorsx,y∈ Ε, one hasB(x,y) = 0.

Several classes of permutation trinomials from Niho exponents over finite fields of characteristic 3

2019 ◽  
Vol 18 (04) ◽  
pp. 1950069
Author(s):  
Qian Liu ◽  
Yujuan Sun
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Coding Theory ◽  
Permutation Polynomials ◽  
Combinatorial Designs ◽  
Dickson Polynomial ◽  
Niho Exponents ◽  
Characteristic 3 ◽  
Permutation Trinomials ◽  
Algebraic Forms

Permutation polynomials have important applications in cryptography, coding theory, combinatorial designs, and other areas of mathematics and engineering. Finding new classes of permutation polynomials is therefore an interesting subject of study. Permutation trinomials attract people’s interest due to their simple algebraic forms and additional extraordinary properties. In this paper, based on a seventh-degree and a fifth-degree Dickson polynomial over the finite field [Formula: see text], two conjectures on permutation trinomials over [Formula: see text] presented recently by Li–Qu–Li–Fu are partially settled, where [Formula: see text] is a positive integer.

Representations by Hermitian Forms in a Finite Field of Characteristic Two

1977 ◽  
Vol 29 (1) ◽  
pp. 169-179 ◽  
Author(s):  
John D. Fulton
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Multiplicative Group ◽  
Group Homomorphism ◽  
Hermitian Forms ◽  
Characteristic Two ◽  
Field Automorphism

Throughout this paper, we let q = 2W,﹜ w a positive integer, and for u = 1 or 2, we let GF(qu) denote the finite field of cardinality qu. Let - denote the involutory field automorphism of GF(q2) with GF(q) as fixed subfield, where ā = aQ for all a in GF﹛q2). Moreover, let | | denote the norm (multiplicative group homomorphism) mapping of GF(q2) onto GF(q), where |a| — a • ā = aQ+1.

scholarly journals The distribution of irreducible polynomials in several indeterminates over a finite field

1968 ◽  
Vol 16 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Stephen D. Cohen
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Equivalence Class ◽  
Ordered Set ◽  
Irreducible Polynomials ◽  
Totally Positive

We consider non-zero polynomials f(x1, …, xk) in k variables x1, …, xk with coefficients in the finite field GF[q] (q = pn for some prime p and positive integer n). We assume that the polynomials have been normalised by selecting one polynomial from each equivalence class with respect to multiplication by non-zero elements of GF[q]. By the degree of a polynomial f(x1, …, xk) will be understood the ordered set (m1, …, mk), where mi is the degree of f(x1 ,…, xk) in x1(i = 1, 2, …, K). The degree (m,…, mk) of a polynomial will be called totally positive if mi>0, i = 1, 2, …, k.

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