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Multi-point codes over Kummer extensions

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Abstract

This paper is concerned with the construction of algebraic geometric codes defined from Kummer extensions. It plays a significant role in the study of such codes to describe bases for the Riemann–Roch spaces associated with totally ramified places. Along this line, we present an explicit characterization of Weierstrass semigroups and pure gaps. Additionally, we determine the floor of a certain type of divisor introduced by Maharaj, Matthews and Pirsic. Finally, we apply these results to find multi-point codes with excellent parameters. As one of the examples, a presented code with parameters \([254,228,\geqslant 16]\) over \( {\mathbb {F}}_{64} \) yields a new record.

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Acknowledgements

This work was partially supported by the NSFC (Grant No. 11271381), the NSFC (Grant No.11431015), the NSFC (Grant No. 61472457) and China 973 Program (Grant No. 2011CB808000). This work was also partially supported by Guangdong Natural Science Foundation (Grant No. 2014A030313161) and the Natural Science Foundation of Shandong Province of China (ZR2016AM04).

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Correspondence to Shudi Yang.

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Communicated by G. Korchmaros.

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Hu, C., Yang, S. Multi-point codes over Kummer extensions. Des. Codes Cryptogr. 86, 211–230 (2018). https://doi.org/10.1007/s10623-017-0335-7

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