Relative generalized Hamming weights of one-point algebraic geometric codes

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 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

AG codes and AG quantum codes from the GGS curve

In this paper, algebraic-geometric (AG) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all $\mathbb F_{q^2}$-rational points of the curve is determined; the Feng-Rao designed minimum distance is computed for infinite families of such codes, as well as the automorphism group. As a result, some linear codes with better relative parameters with respect to one-point Hermitian codes are discovered. Classes of quantum and convolutional codes are provided relying on the constructed AG codes

Codes and Curves

When information is transmitted, errors are likely to occur. Coding theory examines effi cient ways of packaging data so that these errors can be detected, or even corrected. The traditional tools of coding theory have come from combinatorics and group theory. Lately, however, coding theorists have added techniques from algebraic geometry to their toolboxes. In particular, by re-interpreting the Reed- Solomon codes, one can see how to defi ne new codes based on divisors on algebraic curves. For instance, using modular curves over fi nite fi elds, Tsfasman, Vladut, and Zink showed that one can defi ne a sequence of codes with asymptotically better parameters than any previously known codes. This monograph is based on a series of lectures the author gave as part of the IAS/PCMI program on arithmetic algebraic geometry. Here, the reader is introduced to the exciting fi eld of algebraic geometric coding theory. Presenting the material in the same conversational tone of the lectures, the author covers linear codes, including cyclic codes, and both bounds and asymptotic bounds on the parameters of codes. Algebraic geometry is introduced, with particular attention given to projective curves, rational functions and divisors. The construction of algebraic geometric codes is given, and the Tsfasman-Vladut-Zink result mentioned above is discussed

On asymptotically good ramp secret sharing schemes

Asymptotically good sequences of linear ramp secret sharing schemes have been intensively studied by Cramer et al. in terms of sequences of pairs of nested algebraic geometric codes. In those works the focus is on full privacy and full reconstruction. In this paper we analyze additional parameters describing the asymptotic behavior of partial information leakage and possibly also partial reconstruction giving a more complete picture of the access structure for sequences of linear ramp secret sharing schemes. Our study involves a detailed treatment of the (relative) generalized Hamming weights of the considered codes

Parameters of AG codes from vector bundles

AbstractWe investigate the parameters of the algebraic–geometric codes constructed from vector bundles on a projective variety defined over a finite field. In the case of curves we give a method of constructing weakly stable bundles using restriction of vector bundles on algebraic surfaces and illustrate the result by some examples

ON LINEAR CODES CONSTRUCTED BY AN HERMITE CURVE

Society has changed from analog era to the digital age. Telecommunications are carried out by using bit strings of 0 and 1, but the bit inversion (replacement of 0 and 1) may occur by noises, etc. We call it an error. Such an error often occurs in the digital technology, and how to correct the errors is a problem of much practical interest. The theory of code plays a big role here. Coding theory is an indispensable technology in modern telecommunication. As one of the theories of codes, we have the theory of algebraic geometric code, which is based on the theory of algebraic curve. The features of algebraic geometric codes are that the code parameters are estimated with inequalities and that any linear codes are realized by algebraic geometric codes. In this research, we use algebraic curves and finite fields to construct linear codes with specific parameters

Algebraic geometric codes on surfaces

For a given algebraic variety $V$ defined over a finite field and a very ample divisor $D$ on $V$, we give a construction of a linear code $C_{V,D}$. If $V$ is a curve, we recover the algebraic geometric Goppa codes. We are interested here in the case where $V$ is an algebraic surface, and we give in some cases the parameters of such corresponding codes. We compare these parameters to the Singleton bound and to those of Goppa codes. In order to compute these parameters, we use the Riemann-Roch theorem for surfaces