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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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
IEEE
2025-01-01
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| Series: | IEEE Access |
| Subjects: | |
| Online Access: | https://ieeexplore.ieee.org/document/10845791/ |
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| Summary: | 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. |
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| ISSN: | 2169-3536 |