5 research outputs found
Improved Bounds on the Threshold Gap in Ramp Secret Sharing
Producción CientÃficaAbstract: In this paper we consider linear secret sharing schemes over a finite field Fq, where the secret is a vector in Fâ„“q and each of the n shares is a single element of Fq. We obtain lower bounds on the so-called threshold gap g of such schemes, defined as the quantity r−t where r is the smallest number such that any subset of r shares uniquely determines the secret and t is the largest number such that any subset of t shares provides no information about the secret. Our main result establishes a family of bounds which are tighter than previously known bounds for ℓ≥2. Furthermore, we also provide bounds, in terms of n and q, on the partial reconstruction and privacy thresholds, a more fine-grained notion that considers the amount of information about the secret that can be contained in a set of shares of a given size. Finally, we compare our lower bounds with known upper bounds in the asymptotic setting.Danish Council for Independent Research (grant DFF-4002- 00367)Ministerio de EconomÃa, Industria y Competitividad (grants MTM2015-65764-C3-2-P / MTM2015-69138- REDT)RYC-2016-20208 (AEI/FSE/UE)Junta de Castilla y León (grant VA166G18
On nested code pairs from the Hermitian curve
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.Comment: 28 page