A flag of codes$C_{0} \subsetneq C_{1} \subsetneq \cdots \subsetneq C_{s} \subseteq \mathbb {F}_{q} ^{n}$is said to satisfy theisometry-dual propertyif there exists${\mathbf{x}}\in (\mathbb {F}_{q}^{*})^{n}$such that the code$C_{i}$isx-isometric to the dual code$C_{s-i}^\perp $for all$i=0,\ldots, s$. For$P$and$Q$rational places in a function field$\mathcal {F}$, we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic geometry codes$C_{\mathcal {L}}(D, a_{0}P+bQ)\subsetneq C_{\mathcal {L}}(D, a_{1}P+bQ)\subsetneq {\dots } \subsetneq C_{\mathcal {L}}(D, a_{s}P+bQ)$, where the divisor$D$is the sum of pairwise different rational places of$\mathcal {F}$and$P, Q$are not in$\mathop {\mathrm {supp}}\nolimits (D)$. We characterize those sequences in terms of$b$for general function fields. We then apply the result to the broad class of Kummer extensions$\mathcal {F}$defined by affine equations of the form$y^{m}=f(x)$, for$f(x)$a separable polynomial of degree$r$, where$\gcd (r, m)=1$. For$P$the rational place at infinity and$Q$the rational place associated to one of the roots of$f(x)$, and for$D$an$Aut(\mathcal {F}/ \mathbb {F}_{q})$-invariant sum of rational places of$\mathcal {F}$, such that$P, Q \notin \mathop {\mathrm {supp}}\nolimits D$, it is shown that the flag of two-point algebraic geometry codes has the isometry-dual property if and only if$m$divides$2b+1$. At the end we illustrate our results by applying them to two-point codes over several well know function fields.

Anonymization for privacy-preserving data publishing, also known as statistical disclosure control (SDC), can be viewed under the lens of the permutation model. According to this model, any SDC method for individual data records is functionally equivalent to a permutation step plus a noise addition step, where the noise added is marginal, in the sense that it does not alter ranks. Here, we propose metrics to quantify the data confidentiality and utility achieved by SDC methods based on the permutation model. We distinguish two privacy notions: in our work, anonymity refers to subjects and hence mainly to protection against record re-identification, whereas confidentiality refers to the protection afforded to attribute values against attribute disclosure. Thus, our confidentiality metrics are useful even if using a privacy model ensuring an anonymity level ex ante. The utility metric is a general-purpose metric that can be conveniently traded off against the confidentiality metrics, because all of them are bounded between 0 and 1. As an application, we compare the utility-confidentiality trade-offs achieved by several anonymization approaches, including privacy models (k-anonymity and ε-differential privacy) as well as SDC methods (additive noise, multiplicative noise and synthetic data) used without privacy models.

A sharp upper bound for the maximum integer not belonging to an ideal of a numerical semigroup is given and the ideals attaining this bound are characterized. Then, the result is used, through the so-called Feng-Rao numbers, to bound the generalized Hamming weights of algebraic-geometry codes. This is further developed for Hermitian codes and the codes on one of the Garcia-Stichtenoth towers, as well as for some more general families.

A unique decoding algorithm for general AG codes, namely multipoint evaluation codes on algebraic curves, is presented. It is a natural generalization of the previous decoding algorithm which was only for one-point AG codes. As such, it retains the same advantages of fast speed, regular structure, and direct message recovery. Upon this generalization, we add a technique from the Guruswami-Sudan list decoding that boosts the decoding speed significantly. Compared with other known decoding algorithms for general AG codes, it has a similar decoding performance and allows streamlined practical implementation by its simple and regular structure.

We present a unique decoding algorithm of algebraic geometry (AG) codes on plane curves, Hermitian codes in particular, from an interpolation point of view. The algorithm successfully corrects errors of weight up to half of the order bound on the minimum distance of the AG code. It is the first decoding algorithm to combine some features of the interpolation-based list decoding with the performance of the syndrome decoding with the majority voting scheme. The regular structure of the algorithm allows a straightforward parallel implementation.

The way an author or a group of authors are cited tells more about the real impact of their work than authorship and collaborations. Indeed, the connections within the scientific community can be more accurately elicited from the co-citation graph than from the collaboration graph. We suggest some indices that can be drawn from the co-citation graph in order to capture the relevance of individual authors and the relevance of groups of authors.

This correspondence is a short extension to the previous article Bras-Amoroacutes, 2004. In that work, some results were given on one-point codes related to numerical semigroups. One of the crucial concepts in the discussion was the so-called nu-sequence of a semigroup. This sequence has been used in the literature to derive bounds on the minimum distance as well as for defining improvements on the dimension of existing codes. It was proven in that work that the nu-sequence of a semigroup uniquely determines it. Here this result is extended to another object related to a semigroup, the oplus operation. This operation has also been important in the literature for defining other classes of improved codes. It is also proven here that, although the infinite set of values in the nu-sequence (resp. the oplus values) uniquely determines the associated semigroup, no finite part of it can determine it, because it is shared by infinitely many semigroups. In that reference the proof of the fact that the nu-sequence of a numerical semigroup uniquely determines it is constructive. The result here presented shows that, however, that construction can not be performed as an algorithm with finite input

Analysis of the Berlekamp-Massey-Sakata algorithm for decoding one-point codes leads to two methods for improving code rate. One method, due to Feng and Rao, removes parity checks that may be recovered by their majority voting algorithm. The second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic. In this work, formulae are given for the redundancies of Hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance. The results proceed from an analysis of numerical semigroups generated by two consecutive integers.

Garcia and Stichtenoth discovered a tower of function fields that meets the Drinfeld-Vladut bound on the ratio of the number of points to the genus. For this tower, Pellikaan, Stichtenoth, and Torres derived a recursive description of the Weierstrass semigroups associated to a tower of points on the associated curves. In this correspondence, a nonrecursive description of the semigroups is given and from this the enumeration of each of the semigroups is derived as well as its inverse. This enables us to find an explicit formula for the order (Feng-Rao) bound on the minimum distance of the associated one-point codes.

Summary form only given. We adapted shape-adaptive coding and the BISK algorithm to new sign and refinement encoders, with the novelty of encoding separately the refinement bits for each set of coefficients having the same prefix. This allows the algorithm to capitalize the refinement redundancy among each of these sets. The proposed sign and refinement encoders may be considered independently and may be integrated to other bit plane encoders. However, the search scheme suggests a new complete bit plane encoder defined by an exhaustive two-valued shape-adaptive search (ETSE). Even though ETSE does not include arithmetic coding, the coding performance of ETSE is competitive when compared to other wavelet-based encoders which include arithmetic coding. Furthermore, as BISK, ETSE may perform compression of images with non-regular boundary

We introduce a new class of numerical semigroups, which we call the class of acute semigroups and we prove that they generalize symmetric and pseudosymmetric numerical semigroups, Arf numerical semigroups, and the semigroups generated by an interval. For a numerical semigroup /spl Lambda/={/spl lambda//sub 0/i})=/spl nu//sub i+1/ for all i/spl ges/m. We prove that the only numerical semigroups for which the sequence (/spl nu//sub i/) is always nondecreasing are ordinary numerical semigroups. Furthermore, we show that a semigroup can be uniquely determined by its sequence (/spl nu//sub i/).