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Factorizations of Binomial Polynomials and Enumerations of LCD and Self-Dual Constacyclic Codes.
- Source :
- IEEE Transactions on Information Theory; Mar2019, Vol. 65 Issue 3, p1740-1751, 12p
- Publication Year :
- 2019
-
Abstract
- Constacyclic codes are well-known generalizations of cyclic and negacyclic codes. Due to their rich algebraic structure, constacyclic codes are used to construct quantum codes and symbol-pair codes. Let ${\mathbb {F}}_{q}$ be a finite field with order $q$ , where $q$ is a positive power of a prime $p$. Suppose that $n$ is a positive integer and the product of distinct prime factors of $n$ divides $q-1$ , i.e., $rad(n)\mid (q-1)$. In this paper, we explicitly factorize the polynomial $x^{n}-\lambda $ for each $\lambda \in {\mathbb {F}}_{q}^{*}$. As applications, first, we obtain all repeated-root $\lambda $ -constacyclic codes and their dual codes of length $np^{s}$ over ${\mathbb {F}}_{q}$ ; second, we determine all simple-root LCD cyclic codes and LCD negacyclic codes of length $n$ over ${\mathbb {F}}_{q}$ ; third, we list all self-dual repeated-root negacyclic codes of length $np^{s}$ over ${\mathbb {F}}_{q}$. In contrast to known results, the lengths of constacyclic codes in this paper have more flexible parameters. [ABSTRACT FROM AUTHOR]
- Subjects :
- FACTORIZATION
BINOMIAL equations
LINEAR codes
POLYNOMIALS
MATHEMATICS theorems
Subjects
Details
- Language :
- English
- ISSN :
- 00189448
- Volume :
- 65
- Issue :
- 3
- Database :
- Complementary Index
- Journal :
- IEEE Transactions on Information Theory
- Publication Type :
- Academic Journal
- Accession number :
- 134886966
- Full Text :
- https://doi.org/10.1109/TIT.2018.2864200