Fractional decoding of algebraic geometry codes over extension fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911835019542528 |
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| author | Camps-Moreno, Eduardo Matthews, Gretchen L. Santos, Welington |
| author_facet | Camps-Moreno, Eduardo Matthews, Gretchen L. Santos, Welington |
| contents | In this paper, we study algebraic geometry codes from curves over $\mathbb{F}_{q^\ell}$ through their virtual projections which are algebraic geometric codes over $\mathbb{F}_q$. We use the virtual projections to provide fractional decoding algorithms for the codes over $\mathbb{F}_{q^\ell}$. Fractional decoding seeks to perform error correction using a smaller fraction of $\mathbb{F}_q$-symbols than a typical decoding algorithm. In one instance, the bound on the number of correctable errors differs from the usual lower bound by the degree of a pole divisor of an annihilator function. In another, we view the virtual projections as interleaved codes to, with high probability, correct more errors than anticipated. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_07201 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractional decoding of algebraic geometry codes over extension fields Camps-Moreno, Eduardo Matthews, Gretchen L. Santos, Welington Information Theory Algebraic Geometry In this paper, we study algebraic geometry codes from curves over $\mathbb{F}_{q^\ell}$ through their virtual projections which are algebraic geometric codes over $\mathbb{F}_q$. We use the virtual projections to provide fractional decoding algorithms for the codes over $\mathbb{F}_{q^\ell}$. Fractional decoding seeks to perform error correction using a smaller fraction of $\mathbb{F}_q$-symbols than a typical decoding algorithm. In one instance, the bound on the number of correctable errors differs from the usual lower bound by the degree of a pole divisor of an annihilator function. In another, we view the virtual projections as interleaved codes to, with high probability, correct more errors than anticipated. |
| title | Fractional decoding of algebraic geometry codes over extension fields |
| topic | Information Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2404.07201 |