TY - JOUR
T1 - On nested code pairs from the Hermitian curve
AU - Christensen, René Bødker
AU - Geil, Olav
PY - 2020
Y1 - 2020
N2 - Nested code pairs play a crucial role in the construction of ramp secret sharing schemes [Kurihara et al. 2012] and in the CSS construction of quantum codes [Ketkar et al. 2006]. 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.
AB - Nested code pairs play a crucial role in the construction of ramp secret sharing schemes [Kurihara et al. 2012] and in the CSS construction of quantum codes [Ketkar et al. 2006]. 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.
KW - algebraic geometric codes
KW - Asymmetric quantum code
KW - Hermitian curve
KW - ramp secret sharing
KW - relative minimum distance
UR - http://www.scopus.com/inward/record.url?scp=85090040903&partnerID=8YFLogxK
U2 - 10.1016/j.ffa.2020.101742
DO - 10.1016/j.ffa.2020.101742
M3 - Journal article
SN - 1071-5797
VL - 68
JO - Finite Fields and Their Applications
JF - Finite Fields and Their Applications
M1 - 101742
ER -