Diego Ruano

On the order bounds for one-point AG codes

The order bound for the minimum distance of algebraic geometry codes was originally defined for the duals of one-point codes and later generalized for arbitrary algebraic geometry codes. Another bound of order type for the minimum distance of general linear codes, and for codes from order domains in particular, was given in [H. Andersen and O. Geil, Evaluation codes from order domain theory, Finite Fields and their Applications 14 (2008), pp. 92-123]. Here we investigate in detail the application of that bound to one-point algebraic geometry codes, obtaining a bound

Relative generalized Hamming weights of one-point algebraic geometric codes

d^*

On the structure of generalized toric codes

for the minimum distance of these codes. We establish a connection between

Feng-Rao decoding of primary codes

d^*

Construction and decoding of matrix-product codes from nested codes

and the order bound and its generalizations. We also study the improved code constructions based on

The Order Bound for Toric Codes

d^*

Decoding of matrix-product codes

. Finally we extend

List decoding algorithm based on voting in Gröbner bases for general one-point AG codes

d^*

Stabilizer quantum codes from J-affine variety codes and a new Steane-like enlargement

to all generalized Hamming weights.

List Decoding of Matrix-Product Codes from nested codes: an application to Quasi-Cyclic codes

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.