Maria Bras-Amorós

Acute semigroups, the order bound on the minimum distance, and the Feng-Rao improvements

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/</spl lambda//sub 1/<...}, denote /spl nu//sub i/=#{j|/spl lambda//sub i/-/spl lambda//sub j//spl isin//spl Lambda/}. Given an acute numerical semigroup /spl Lambda/ we find the smallest nonnegative integer m for which the order bound on the minimum distance of one-point Goppa codes with associated semigroup /spl Lambda/ satisfies d/sub ORD/(C/sub i/)(:=min{/spl nu//sub j/|j>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/).

The Correction Capability of the Berlekamp–Massey–Sakata Algorithm with Majority Voting

Sakata’s generalization of the Berlekamp–Massey algorithm applies to a broad class of codes defined by an evaluation map on an order domain. In order to decode up to the minimum distance bound, Sakata’s algorithm must be combined with the majority voting algorithm of Feng, Rao and Duursma. This combined algorithm can often decode far more than (dmin −1)/2 errors, provided the errors are in general position. We give a precise characterization of the error correction capability of the combined algorithm. We also extend the concept behind Feng and Rao’s improved codes to decoding of errors in general position. The analysis leads to a new characterization of Arf numerical semigroups.

New Lower Bounds on the Generalized Hamming Weights of AG Codes

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.

Unique Decoding of Plane AG Codes via Interpolation

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.

On query self-submission in peer-to-peer user-private information retrieval

User-private information retrieval (UPIR) is the art of retrieving information without telling the information holder who you are. UPIR is sometimes called anonymous keyword search. This article discusses a UPIR protocol in which the users form a peer-to-peer network over which they collaborate in protecting the privacy of each other. The protocol is known as P2P UPIR. It will be explained why the P2P UPIR protocol may have a flaw in the protection of the privacy of the client in front of the server. Two alternative variations of the protocols are discussed. One of these will prove to resolve the privacy flaw discovered in the original protocol. Hence the aim of this article is to propose a modification of the P2P UPIR protocol. It is justified why the projective planes are still the optimal configurations for P2P UPIR for the modified protocol.

Unique Decoding of General AG Codes

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.

On Semigroups Generated by Two Consecutive Integers and Improved Hermitian Codes

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.

A Note on Numerical Semigroups

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

On the Geil---Matsumoto bound and the length of AG codes

The Geil–Matsumoto bound conditions the number of rational places of a function field in terms of the Weierstrass semigroup of any of the places. Lewittes’ bound preceded the Geil–Matsumoto bound and it only considers the smallest generator of the numerical semigroup. It can be derived from the Geil–Matsumoto bound and so it is weaker. However, for general semigroups the Geil–Matsumoto bound does not have a closed formula and it may be hard to compute, while Lewittes’ bound is very simple. We give a closed formula for the Geil–Matsumoto bound for the case when the Weierstrass semigroup has two generators. We first find a solution to the membership problem for semigroups generated by two integers and then apply it to find the above formula. We also study the semigroups for which Lewittes’s bound and the Geil–Matsumoto bound coincide. We finally investigate on some simplifications for the computation of the Geil–Matsumoto bound.

The Order Bound on the Minimum Distance of the One-Point Codes Associated to the Garcia–Stichtenoth Tower

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.