On quadratic residue codes and hyperelliptic curves
For an odd prime p and each non-empty subset S⊂GF(p), consider the hyperelliptic curve X S defined by y 2 =f S (x), where f S (x) = ∏ a∈S (x-a). Using a connection between binary quadratic residue codes and hyperelliptic curves over GF(p), this paper investigates how coding theory bound...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2008-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Online Access: | http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/606 |
| Summary: | For an odd prime p and each non-empty subset S⊂GF(p), consider the hyperelliptic curve X S defined by y 2 =f S (x), where f S (x) = ∏ a∈S (x-a). Using a connection between binary quadratic residue codes and hyperelliptic curves over GF(p), this paper investigates how coding theory bounds give rise to bounds such as the following example: for all sufficiently large primes p there exists a subset S⊂GF(p) for which the bound |X S (GF(p))| > 1.39p holds. We also use the quasi-quadratic residue codes defined below to construct an example of a formally self-dual optimal code whose zeta function does not satisfy the ``Riemann hypothesis.'' |
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| ISSN: | 1462-7264 1365-8050 |