scholarly journals Revisiting the Factorization of x n + 1 over Finite Fields with Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Arunwan Boripan ◽  
Somphong Jitman
Keyword(s):  
Finite Fields ◽  
Recursive Methods ◽  
Negacyclic Codes ◽  
Rigorous Treatment ◽  
Enumeration Formula

The polynomial x n + 1 over finite fields has been of interest due to its applications in the study of negacyclic codes over finite fields. In this paper, a rigorous treatment of the factorization of x n + 1 over finite fields is given as well as its applications. Explicit and recursive methods for factorizing x n + 1 over finite fields are provided together with the enumeration formula. As applications, some families of negacyclic codes are revisited with more clear and simpler forms.

scholarly journals SRIM and SCRIM factors of xn + 1 over finite fields and their applications

2020 ◽  
pp. 2050098 ◽  
Author(s):  
Arunwan Boripan ◽  
Somphong Jitman
Keyword(s):  
Finite Fields ◽  
Algebraic Structures ◽  
Negacyclic Codes

Self-Reciprocal Irreducible Monic (SRIM) and Self-Conjugate-Reciprocal Irreducible Monic (SCRIM) factors of [Formula: see text] over finite fields have become of interest due to their rich algebraic structures and wide applications. In this paper, these notions are extended to factors of [Formula: see text] over finite fields. Characterization and enumeration of SRIM and SCRIM factors of [Formula: see text] over finite fields are established. Simplification and recessive formulas for the number of such factors are given. Finally, applications in the study of complementary negacyclic codes are discussed.

scholarly journals Hulls of cyclic and negacyclic codes over finite fields

2015 ◽  
Vol 33 ◽  
pp. 232-257 ◽  
Author(s):  
Ekkasit Sangwisut ◽  
Somphong Jitman ◽  
San Ling ◽  
Patanee Udomkavanich
Keyword(s):  
Finite Fields ◽  
Negacyclic Codes

scholarly journals Skew Constacyclic Codes over Finite Fields and Finite Chain Rings

2016 ◽  
Vol 2016 ◽  
pp. 1-17 ◽  
Author(s):  
Hai Q. Dinh ◽  
Bac T. Nguyen ◽  
Songsak Sriboonchitta
Keyword(s):  
Finite Fields ◽  
Bch Codes ◽  
Constacyclic Codes ◽  
Finite Chain ◽  
Dual Codes ◽  
Negacyclic Codes ◽  
Finite Chain Rings

This paper overviews the study of skewΘ-λ-constacyclic codes over finite fields and finite commutative chain rings. The structure of skewΘ-λ-constacyclic codes and their duals are provided. Among other results, we also consider the Euclidean and Hermitian dual codes of skewΘ-cyclic and skewΘ-negacyclic codes over finite chain rings in general and overFpm+uFpmin particular. Moreover, general decoding procedure for decoding skew BCH codes with designed distance and an algorithm for decoding skew BCH codes are discussed.

scholarly journals Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields

2015 ◽  
Vol 9 (4) ◽  
pp. 437-447 ◽  
Author(s):  
Amita Sahni ◽  
Poonam Trama Sehgal
Keyword(s):  
Finite Fields ◽  
Negacyclic Codes

scholarly journals Optimal quinary negacyclic codes with minimum distance four

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jinmei Fan ◽  
Yanhai Zhang
Keyword(s):  
Finite Fields ◽  
Minimum Distance ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Negacyclic Codes ◽  
Generator Polynomial ◽  
Polynomials Over Finite Fields ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>Based on solutions of certain equations over finite yields, a necessary and sufficient condition for the quinary negacyclic codes with parameters <inline-formula><tex-math id="M1">\begin{document}$ [\frac{5^m-1}{2},\frac{5^m-1}{2}-2m,4] $\end{document}</tex-math></inline-formula> to have generator polynomial <inline-formula><tex-math id="M2">\begin{document}$ m_{\alpha^3}(x)m_{\alpha^e}(x) $\end{document}</tex-math></inline-formula> is provided. Several classes of new optimal quinary negacyclic codes with the same parameters are constructed by analyzing irreducible factors of certain polynomials over finite fields. Moreover, several classes of new optimal quinary negacyclic codes with these parameters and generator polynomial <inline-formula><tex-math id="M3">\begin{document}$ m_{\alpha}(x)m_{\alpha^e}(x) $\end{document}</tex-math></inline-formula> are also presented.</p>

scholarly journals Existence conditions for self-orthogonal negacyclic codes over finite fields

2015 ◽  
Vol 9 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Liren Lin ◽  
◽  
Hongwei Liu ◽  
Bocong Chen ◽  
◽  
...  
Keyword(s):  
Finite Fields ◽  
Negacyclic Codes ◽  
Existence Conditions

scholarly journals On LCD Negacyclic Codes over Finite Fields

2017 ◽  
Vol 31 (4) ◽  
pp. 1065-1077 ◽  
Author(s):  
Binbin Pang ◽  
Shixin Zhu ◽  
Zhonghua Sun
Keyword(s):  
Finite Fields ◽  
Negacyclic Codes

scholarly journals The Average Hull Dimension of Negacyclic Codes over Finite Fields

2018 ◽  
Vol 23 (3) ◽  
pp. 41
Author(s):  
Somphong Jitman ◽  
Ekkasit Sangwisut
Keyword(s):  
Finite Fields ◽  
Lower Bounds ◽  
Coding Theory ◽  
Linear Codes ◽  
Upper And Lower Bounds ◽  
Negacyclic Codes ◽  
Average Dimension

Hulls of linear codes have been extensively studied due to their wide applications and links with the efficiency of some algorithms in coding theory. In this paper, the average dimension of the Euclidean hull of negacyclic codes of length n over finite fields F q , denoted by E ( n , − 1 , q ) , has been investigated. The formula for E ( n , − 1 , q ) has been determined. Some upper and lower bounds of E ( n , − 1 , q ) have been given as well. Asymptotically, it has been shown that either E ( n , − 1 , q ) is zero or it grows the same rate as n.

scholarly journals Negacyclic Codes of Length $2^mp^n$ Over Finite Fields

IEEE Access ◽  
2019 ◽  
Vol 7 ◽  
pp. 121874-121880 ◽  
Author(s):  
Jing Huang
Keyword(s):  
Finite Fields ◽  
Negacyclic Codes
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