arXiv subjects

Maria Bras-Amorós

Publications and source records attributed to Maria Bras-Amorós.

40 records · Page 3Linked to original sources

arXiv2006

Duality for Several Families of Evaluation Codes

We consider generalizations of Reed-Muller codes, toric codes, and codes from certain plane curves, such as those defined by norm and trace functions on finite fields. In each case we are interested in codes defined by evaluating arbitrary subsets of monomials, and in identifying when the dual codes are also obtained by evaluating monomials. We then move to the context of order domain theory, in which the subsets of monomials can be chosen to optimize decoding performance using the Berlekamp-Massey-Sakata algorithm with majority voting. We show that for the codes under consideration these subsets are well-behaved and the dual codes are also defined by monomials.

cs.IT↗

arXiv2006

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

Garcia and Stichtenoth discovered two towers of function fields that meet the Drinfeld-Vlăduţ bound on the ratio of the number of points to the genus. For one of these towers, Garcia, Pellikaan and Torres derived a recursive description of the Weierstrass semigroups associated to a tower of points on the associated curves. In this article, a non-recursive 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.

cs.IT↗

arXiv2006

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. The formula for the redundancy of optimal Hermitian codes correcting a given number of errors answers an open question stated by Pellikaan and Torres in 1999.

cs.IT↗

arXiv2006

Redundancies of Correction-Capability-Optimized Reed-Muller Codes

This article is focused on some variations of Reed-Muller codes that yield improvements to the rate for a prescribed decoding performance under the Berlekamp-Massey-Sakata algorithm with majority voting. Explicit formulas for the redundancies of the new codes are given.

cs.IT↗