Relative generalized Hamming weights of one-point algebraic geometric codes

Abstract

Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes [30,28]. In this paper we elaborate on the implication of these parameters and we devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes our bound is often tight. Furthermore, for these codes the relative generalized Hamming weights are often much larger than the corresponding generalized Hamming weights. * The result in this paper is in part submitted for possible presentation in IEEE Information Theory Workshop (ITW 2014) [16]. † olav@math.aau.dk ‡ stefano@math.aau.dk § ryutaroh@rmatsumoto.org ¶ diego@math.aau.dk yuanluo@sjtu.edu.cn 1 Keywords: linear code, Feng-Rao bound, Hermitian code, one-point algebraic geometric code, relative dimension/length profile, relative generalized Hamming weight, secret sharing, wiretap channel of type II.and RGHW/RDLP was only recently reported. In particular, few classes of linear codes have been examined for their RGHW/RDLP. In this paper we study RGHW of general linear codes by the Feng-Rao approach [17], and explore its consequences for one-point algebraic geometry (AG) codes [43,22] and in particular the Hermitian codes [42,40,47].The present paper starts with a discussion of known results regarding linear ramp secret sharing schemes and it continues with demonstrating that the RGHWs can also be used to express the best case information leakage. The main result of the paper is a method to estimate RGHW of one-point algebraic geometric codes. This is done by carefully applying the Feng-Rao bounds [17] for primary [1] as well as dual [11,12,37,22,32,21] codes. From this we derive a relatively simple bound which uses information on the corresponding Weierstrass semigroup [24,8]. As shall be demonstrated for Hermitian codes the new bound is often sharp. Moreover, for the same codes the RGHW are often much larger than the corresponding generalized Hamming weights (GHW) [44] which means that studies of RGHW cannot be substituted by those of GHW.The paper is organized as follows. Section 2 describes the use of RGHW in connection with linear ramp secret sharing schemes, and in connection with communication over the wiretap channel of type II. In Section 3 we apply the theory to the special case of MDS codes. In Section 4 we show -at the level of general linear codes -how to employ the Feng-Rao bounds to estimate RGHW. This method is then applied to one-point algebraic geometric codes in Section 5. We investigate Hermitian codes in Section 6, and treat the corresponding ramp secret sharing schemes in Section 7.2 Ramp secret sharing schemes and wiretap channels of type II Ramp secret sharing schemes were introduced in [4,46]. Let F q be the finite field with q elements. A ramp secret sharing scheme with t-privacy and rreconstruction is an algorithm that, given an input s ∈ F ℓ q , outputs a vector x ∈ F n q , the vector of shares that we want to share among n players, such tha...

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