More on Codes Over Finite Quotients of Polynomial Rings
Let <inline-formula> <tex-math notation="LaTeX">$q=p^{r}$ </tex-math></inline-formula> be a prime power, <inline-formula> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> be the finite field of order q and...
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IEEE
2025-01-01
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| Online Access: | https://ieeexplore.ieee.org/document/10845791/ |
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| author | Emad Kadhim Al-Lami Reza Sobhani Alireza Abdollahi Javad Bagherian |
| author_facet | Emad Kadhim Al-Lami Reza Sobhani Alireza Abdollahi Javad Bagherian |
| author_sort | Emad Kadhim Al-Lami |
| collection | DOAJ |
| description | Let <inline-formula> <tex-math notation="LaTeX">$q=p^{r}$ </tex-math></inline-formula> be a prime power, <inline-formula> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> be the finite field of order q and <inline-formula> <tex-math notation="LaTeX">$f(x)$ </tex-math></inline-formula> be a monic polynomial in <inline-formula> <tex-math notation="LaTeX">${\mathbb {F}}_{q}[x]$ </tex-math></inline-formula>. Set <inline-formula> <tex-math notation="LaTeX">${\mathbb {A}}:={\mathbb {F}}_{q}[x]/{\left \lt {{ f(x) }}\right \gt }$ </tex-math></inline-formula>. In this paper we continue the study (started by T. P. Berger and N. El Amrani) of <inline-formula> <tex-math notation="LaTeX">$\mathbb {A}$ </tex-math></inline-formula>-codes of length l over <inline-formula> <tex-math notation="LaTeX">$\mathbb {A}$ </tex-math></inline-formula>, i.e. <inline-formula> <tex-math notation="LaTeX">$\mathbb {A}$ </tex-math></inline-formula>-submodules of <inline-formula> <tex-math notation="LaTeX">${\mathbb {A}}^{l}$ </tex-math></inline-formula>. We introduce two types of unique generating sets, called type I and type II basis of divisors, for an <inline-formula> <tex-math notation="LaTeX">$\mathbb {A}$ </tex-math></inline-formula>-code. Using this, we present a building-up construction so that one can obtain all distinct <inline-formula> <tex-math notation="LaTeX">$\mathbb {A}$ </tex-math></inline-formula>-codes of length l, with their basis of divisors. We complete the classification for the special case <inline-formula> <tex-math notation="LaTeX">$l=2$ </tex-math></inline-formula> and enumerate all the <inline-formula> <tex-math notation="LaTeX">$\mathbb {A}$ </tex-math></inline-formula>-codes of length 2. As an example, we list all binary index-2 quasi-cyclic codes of lengths 16 and 32, and all ternary index-2 quasi-cyclic codes of lengths 6 and 18, which are best-known codes. |
| format | Article |
| id | doaj-art-fa9bc796a7484a8b8b07387df7474a16 |
| institution | Kabale University |
| issn | 2169-3536 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | IEEE |
| record_format | Article |
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| spelling | doaj-art-fa9bc796a7484a8b8b07387df7474a162025-01-29T00:00:54ZengIEEEIEEE Access2169-35362025-01-0113153391534510.1109/ACCESS.2025.353164410845791More on Codes Over Finite Quotients of Polynomial RingsEmad Kadhim Al-Lami0Reza Sobhani1https://orcid.org/0000-0001-6876-307XAlireza Abdollahi2https://orcid.org/0000-0001-7277-4855Javad Bagherian3Department of Applied Mathematics and Computer Science, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, IranDepartment of Applied Mathematics and Computer Science, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, IranDepartment of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, IranDepartment of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, IranLet <inline-formula> <tex-math notation="LaTeX">$q=p^{r}$ </tex-math></inline-formula> be a prime power, <inline-formula> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> be the finite field of order q and <inline-formula> <tex-math notation="LaTeX">$f(x)$ </tex-math></inline-formula> be a monic polynomial in <inline-formula> <tex-math notation="LaTeX">${\mathbb {F}}_{q}[x]$ </tex-math></inline-formula>. Set <inline-formula> <tex-math notation="LaTeX">${\mathbb {A}}:={\mathbb {F}}_{q}[x]/{\left \lt {{ f(x) }}\right \gt }$ </tex-math></inline-formula>. In this paper we continue the study (started by T. P. Berger and N. El Amrani) of <inline-formula> <tex-math notation="LaTeX">$\mathbb {A}$ </tex-math></inline-formula>-codes of length l over <inline-formula> <tex-math notation="LaTeX">$\mathbb {A}$ </tex-math></inline-formula>, i.e. <inline-formula> <tex-math notation="LaTeX">$\mathbb {A}$ </tex-math></inline-formula>-submodules of <inline-formula> <tex-math notation="LaTeX">${\mathbb {A}}^{l}$ </tex-math></inline-formula>. We introduce two types of unique generating sets, called type I and type II basis of divisors, for an <inline-formula> <tex-math notation="LaTeX">$\mathbb {A}$ </tex-math></inline-formula>-code. Using this, we present a building-up construction so that one can obtain all distinct <inline-formula> <tex-math notation="LaTeX">$\mathbb {A}$ </tex-math></inline-formula>-codes of length l, with their basis of divisors. We complete the classification for the special case <inline-formula> <tex-math notation="LaTeX">$l=2$ </tex-math></inline-formula> and enumerate all the <inline-formula> <tex-math notation="LaTeX">$\mathbb {A}$ </tex-math></inline-formula>-codes of length 2. As an example, we list all binary index-2 quasi-cyclic codes of lengths 16 and 32, and all ternary index-2 quasi-cyclic codes of lengths 6 and 18, which are best-known codes.https://ieeexplore.ieee.org/document/10845791/Polynomial ringpolynomial quotient ringgenerator matrixparity check matrixbasis of divisors |
| spellingShingle | Emad Kadhim Al-Lami Reza Sobhani Alireza Abdollahi Javad Bagherian More on Codes Over Finite Quotients of Polynomial Rings IEEE Access Polynomial ring polynomial quotient ring generator matrix parity check matrix basis of divisors |
| title | More on Codes Over Finite Quotients of Polynomial Rings |
| title_full | More on Codes Over Finite Quotients of Polynomial Rings |
| title_fullStr | More on Codes Over Finite Quotients of Polynomial Rings |
| title_full_unstemmed | More on Codes Over Finite Quotients of Polynomial Rings |
| title_short | More on Codes Over Finite Quotients of Polynomial Rings |
| title_sort | more on codes over finite quotients of polynomial rings |
| topic | Polynomial ring polynomial quotient ring generator matrix parity check matrix basis of divisors |
| url | https://ieeexplore.ieee.org/document/10845791/ |
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