Skew generalized quasi-cyclic codes over non-chain ring $F_q+vF_q$
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| Format: | Preprint |
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2025
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| author | Suxena, Kundan Debnath, Indibar Prakash, Om |
| author_facet | Suxena, Kundan Debnath, Indibar Prakash, Om |
| contents | For a prime $p$, let $F_q$ be the finite field of order $q= p^d$. This paper presents the study on skew generalized quasi-cyclic (SGQC) codes of length $n$ over the non-chain ring $F_q+vF_q$ where $v^2=v$ and $θ_t$ is the Galois automorphism. Here, first, we prove the dual of an SGQC code of length $n$ is also an SGQC code of the same length and derive a necessary and sufficient condition for the existence of a self-dual SGQC code. Then, we discuss the $1$-generator polynomial and the $ρ$-generator polynomial for skew generalized quasi-cyclic codes. Further, we determine the dimension and BCH type bound for the 1-generator skew generalized quasi-cyclic codes. As a by-product, with the help of MAGMA software, we provide a few examples of SGQC codes and obtain some $2$-generator SGQC codes of index $2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_19926 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Skew generalized quasi-cyclic codes over non-chain ring $F_q+vF_q$ Suxena, Kundan Debnath, Indibar Prakash, Om Information Theory 94B05, 94B15, 94B60 For a prime $p$, let $F_q$ be the finite field of order $q= p^d$. This paper presents the study on skew generalized quasi-cyclic (SGQC) codes of length $n$ over the non-chain ring $F_q+vF_q$ where $v^2=v$ and $θ_t$ is the Galois automorphism. Here, first, we prove the dual of an SGQC code of length $n$ is also an SGQC code of the same length and derive a necessary and sufficient condition for the existence of a self-dual SGQC code. Then, we discuss the $1$-generator polynomial and the $ρ$-generator polynomial for skew generalized quasi-cyclic codes. Further, we determine the dimension and BCH type bound for the 1-generator skew generalized quasi-cyclic codes. As a by-product, with the help of MAGMA software, we provide a few examples of SGQC codes and obtain some $2$-generator SGQC codes of index $2$. |
| title | Skew generalized quasi-cyclic codes over non-chain ring $F_q+vF_q$ |
| topic | Information Theory 94B05, 94B15, 94B60 |
| url | https://arxiv.org/abs/2504.19926 |