Using cyclic $(f,σ)$-codes over finite chain rings to construct $\mathbb{Z}_p$- and $\mathbb{F}_q[\![t]\!]$-lattices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917896869904384 |
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| author | Pumpluen, Susanne |
| author_facet | Pumpluen, Susanne |
| contents | We construct $\mathbb{Z}_p$-lattices and $\mathbb{F}_q[\![t]\!]$-lattices from cyclic $(f,σ)$-codes over finite chain rings, employing quotients of natural nonassociative orders and principal left ideals in carefully chosen nonassociative algebras. This approach generalizes the classical Construction A that obtains $\mathbb{Z}$-lattices from linear codes over finite fields or commutative rings to the nonassociative setting. We mostly use proper nonassociative cyclic algebras that are defined over field extensions of $p$-adic fields. This means we focus on $σ$-constacyclic codes to obtain $\mathbb{Z}_p$-lattices, hence $\mathbb{Z}_p$-lattice codes. We construct linear maximum rank distance (MRD) codes that are $\mathbb{Z}_p$-lattice codes employing the left multiplication of a nonassociative algebra over a finite chain ring.
Possible applications of our constructions include post-quantum cryptography involving $p$-adic lattices, e.g. learning with errors, building rank-metric codes like MRD-codes, or $p$-adic coset coding, in particular wire-tap coding. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10838 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Using cyclic $(f,σ)$-codes over finite chain rings to construct $\mathbb{Z}_p$- and $\mathbb{F}_q[\![t]\!]$-lattices Pumpluen, Susanne Rings and Algebras We construct $\mathbb{Z}_p$-lattices and $\mathbb{F}_q[\![t]\!]$-lattices from cyclic $(f,σ)$-codes over finite chain rings, employing quotients of natural nonassociative orders and principal left ideals in carefully chosen nonassociative algebras. This approach generalizes the classical Construction A that obtains $\mathbb{Z}$-lattices from linear codes over finite fields or commutative rings to the nonassociative setting. We mostly use proper nonassociative cyclic algebras that are defined over field extensions of $p$-adic fields. This means we focus on $σ$-constacyclic codes to obtain $\mathbb{Z}_p$-lattices, hence $\mathbb{Z}_p$-lattice codes. We construct linear maximum rank distance (MRD) codes that are $\mathbb{Z}_p$-lattice codes employing the left multiplication of a nonassociative algebra over a finite chain ring. Possible applications of our constructions include post-quantum cryptography involving $p$-adic lattices, e.g. learning with errors, building rank-metric codes like MRD-codes, or $p$-adic coset coding, in particular wire-tap coding. |
| title | Using cyclic $(f,σ)$-codes over finite chain rings to construct $\mathbb{Z}_p$- and $\mathbb{F}_q[\![t]\!]$-lattices |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2501.10838 |