Abstract In this work we revisit the fundamental findings by Chen et al. (in: Advances in Cryptology—EUROCRYPT 2007. Lecture Notes in Computer Science, Springer, Berlin, 2007) on general information transfer in linear ramp secret sharing schemes to conclude that their method not only gives a way to establish worst case leakage (Chen et al. 2007; Kurihara et al. in IEICE Trans Fundam E95-A(11):2067–2075, 2012) and best case recovery (Chen et al. 2007; Geil et al. in IEEE Trans Inf Theory 60(10):5938–5949, 2014) , but can also lead to additional insight on non-qualifying sets for any prescribed amount of information. We then apply this insight to schemes defined from monomial-Cartesian codes and by doing so we demonstrate that the good schemes from (Galindo et al. in IEEE Trans Inf Theory 64(4, part 1):2444–2459, 2018, Sec. IV) have a second layer of security. Elaborating further, when given designed partial recovery numbers, in a new construction the focus is entirely on ensuring that the access structure possesses desirable second layer security, rather on what is the worst case information leakage in terms of number of participants. The particular structure of largest possible sets being not able to determine any amount of information suggests that we coin the concept of considerate ramp secret sharing schemes of which the proposed new construction is a well-structured example.

LCD codes are linear codes with important cryptographic applications. Recently, a method has been presented to transform any linear code into an LCD code with the same parameters when it is supported on a finite field with cardinality larger than 3. Hence, the study of LCD codes is mainly open for binary and ternary fields. Subfield subcodes of J-affine variety codes are a generalization of BCH codes which have been successfully used for constructing good quantum codes. We describe binary and ternary LCD codes constructed as subfield subcodes of J-affine variety codes and provide some new and good LCD codes coming from this construction.

Two new constructions of linear code pairs C2⊂ C1are given for which the codimension and the relative minimum distances M1(C1, C2) and M1(C2⊥, C1⊥) are good. By this, we mean that for any two out of the three parameters the third parameter of the constructed code pair is large. Such pairs of nested codes are indispensable for the determination of good linear ramp secret sharing schemes. They can also be used to ensure reliable communication over asymmetric quantum channels. The new constructions result from carefully applying the Feng-Rao bounds to a family of codes defined from multivariate polynomials and Cartesian product point sets.

Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes [23], [22]. 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.

Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes. In this paper, we elaborate on the implication of these parameters and 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.

We generalize the list decoding algorithm for Hermitian codes proposed by Lee and O'Sullivan [15] based on Gröbner bases to general one-point AG codes, under an assumption weaker than one used by Beelen and Brander [4]. By using the same principle, we also generalize the unique decoding algorithm for one-point AG codes over the Miura-Kamiya Cabcurves proposed by Lee, Bras-Amorós and O'Sullivan [14] to general one-point AG codes, without any assumption. Finally we extend the latter unique decoding algorithm to list decoding, modify it so that it can be used with the Feng-Rao improved code construction, prove equality between its error correcting capability and half the minimum distance lower bound by Andersen and Geil [3] that has not been done in the original proposal, and remove the unnecessary computational steps so that it can run faster.

Olav Geil; Tracey Ho and Desmond S. LunNetwork Coding: An Introduction. Cambridge University Press (2008). ISBN 9780521873109. 184 pp. £30/$60. Hardcover., The

This paper presents the missing codes evaluation map in the order domain theory is a generalization of the code construction E(s) to order domains of arbitrary transcendence degree and the introduction of a class of improved codes E/spl tilde/(s). The code construction to the order domain setting, the minimum distance estimation and the generalized Hamming weights of the new codes are also discussed.

In two previous papers, the first by Feng, Rao, Berg, and Zhu (see ibid., vol.43, p.1799-810, 1997) and the second by Feng, Zhu, Shi, and Rao (see Proc. 35th. Afferton Conf. Communication, Control and Computing, p.205-14, 1997), the authors use a generalization of Bezout's theorem to estimate the minimum distance and generalized Hamming weights for a class of error correcting codes obtained by evaluation of polynomials in points of an algebraic curve. The main aim of this article is to show that instead of using this rather complex method the same results and some improvements can be obtained by using the so-called footprint from Grobner basis theory. We also develop the theory further such that the minimum distance and the generalized Hamming weights not only can be estimated but also can actually be determined.