On nested code pairs from the Hermitian curve |
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| Affiliation: | 1. Institute of Mathematical Sciences, Ewha Womans University, Seoul, Republic of Korea;2. Department of Mathematics, Sungkyunkwan University, Suwon, Republic of Korea;3. Department of Mathematics, Ewha Womans University, Seoul, Republic of Korea;1. Department of Mathematics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic;2. Institut de Mathématiques de Toulon, campus La Garde, 83041 Toulon, France;1. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, 9000 Gent, Flanders, Belgium;2. Department of Mathematics and Data Science, University of Brussels (VUB), Pleinlaan 2, Building G, 1050 Elsene, Brussels, Belgium;3. Department of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia;1. Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan 430062, China;2. Wuhan Maritime Communication Research Institute, Wuhan 430079, China |
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| Abstract: | Nested code pairs play a crucial role in the construction of ramp secret sharing schemes [15] and in the CSS construction of quantum codes [14]. The important parameters are (1) the codimension, (2) the relative minimum distance of the codes, and (3) the relative minimum distance of the dual set of codes. Given values for two of them, one aims at finding a set of nested codes having parameters with these values and with the remaining parameter being as large as possible. In this work we study nested codes from the Hermitian curve. For not too small codimension, we present improved constructions and provide closed formula estimates on their performance. For small codimension we show how to choose pairs of one-point algebraic geometric codes in such a way that one of the relative minimum distances is larger than the corresponding non-relative minimum distance. |
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| Keywords: | Algebraic geometric code Asymmetric quantum code Hermitian curve Ramp secret sharing Relative minimum distance |
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