Abstract
Asymptotically good sequences of ramp secret sharing schemes were given in [5] by using one-point algebraic geometric codes defined from asymptotically good towers of function fields. Their security is given by the relative generalized Hamming weights of the corresponding codes. In this paper we demonstrate how to obtain refined information on the RGHWs when the codimension of the codes is small. For general codimension, we give an improved estimate for the highest RGHW.
Highlights
Asymptotically good sequences of ramp secret sharing schemes were given in [5] by using one-point algebraic geometric codes defined from asymptotically good towers of function fields
Relative generalized Hamming weights (RGHWs) of two linear codes are fundamental for evaluating the security of ramp secret sharing schemes and wire-tap channels of type II [6, 7, 9, 12]
Until few years ago only the RGHWs of MDS codes and a few other examples of codes were known [8], but recently new results were discovered for one-point algebraic geometric codes [6], q-ary Reed-Muller codes [4] and cyclic codes [13]
Summary
Relative generalized Hamming weights (RGHWs) of two linear codes are fundamental for evaluating the security of ramp secret sharing schemes and wire-tap channels of type II [6, 7, 9, 12]. In [5] it was discussed how to obtain asymptotically good sequences of ramp secret sharing schemes by using one-point algebraic geometric codes defined from asymptotically good towers of function fields. From Garcia-Stichtenoth’s second tower [3] one obtains codes over any field Fq where q is an even power of a prime. For every function field Fν the following complete description of the Weierstrass semigroups corresponding to a sequence of rational places was given in [10]. Applying Proposition 1.3 to code pairs coming from Garcia-Stichtenoth’s second tower [3], an asymptotic result was given in [5, Theorem 23], which combined with Proposition 1.4 allows us to obtain the following result. Let (Fi)∞ i=1 be Garcia-Stichtenoth’s second tower of function fields over Fq, where q is an even power of a prime. Note that the bound (2) is sharper than (1) for √q1−1 ≤ R ≤ 1 − √q1−1
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