On nested code pairs from the Hermitian curve

Research output: Working paperResearch

Abstract

Nested code pairs play a crucial role in the construction of ramp secret sharing schemes [16] and in the CSS construction of quantum codes [15]. 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.
Original languageEnglish
PublisherarXiv.org
Number of pages28
Publication statusPublished - 2018

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Hermitian Curve
Minimum Distance
Codimension
Algebraic-geometric Codes
Secret Sharing Scheme
Choose
Closed
Estimate

Cite this

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title = "On nested code pairs from the Hermitian curve",
abstract = "Nested code pairs play a crucial role in the construction of ramp secret sharing schemes [16] and in the CSS construction of quantum codes [15]. 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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On nested code pairs from the Hermitian curve. / Christensen, René Bødker; Geil, Hans Olav.

arXiv.org, 2018.

Research output: Working paperResearch

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