arXiv subjects

Emmanuela Orsini

Publications and source records attributed to Emmanuela Orsini.

3 recordsLinked to original sources

arXiv2015

On the shape of the general error locator polynomial for cyclic codes

A general result on the explicit form of the general error locator polynomial for all cyclic codes is given, along with several results for infinite classes of cyclic codes with $t=2$ and $t=3$. From these, a theoretically justification of the sparsity of the general error locator polynomial is obtained for all cyclic codes with $t\leq 3$ and $n<63$, except for three cases where the sparsity is proved by a computer check. Moreover, we discuss some consequences of our results to the understanding of the complexity of bounded-distance decoding of cyclic codes.

cs.IT↗

arXiv2011

Improved decoding of affine-variety codes

General error locator polynomials are polynomials able to decode any correctable syndrome for a given linear code. Such polynomials are known to exist for all cyclic codes and for a large class of linear codes. We provide some decoding techniques for affine-variety codes using some multidimensional extensions of general error locator polynomials. We prove the existence of such polynomials for any correctable affine-variety code and hence for any linear code. We propose two main different approaches, that depend on the underlying geometry. We compute some interesting cases, including Hermitian codes. To prove our coding theory results, we develop a theory for special classes of zero-dimensional ideals, that can be considered generalizations of stratified ideals. Our improvement with respect to stratified ideals is twofold: we generalize from one variable to many variables and we introduce points with multiplicities.

math.AC↗

arXiv2009

Computing the distance distribution of systematic non-linear codes

The most important families of non-linear codes are systematic. A brute-force check is the only known method to compute their weight distribution and distance distribution. On the other hand, it outputs also all closest word pairs in the code. In the black-box complexity model, the check is optimal among closest-pair algorithms. In this paper we provide a Groebner basis technique to compute the weight/distance distribution of any systematic non-linear code. Also our technique outputs all closest pairs. Unlike the check, our method can be extended to work on code families.

cs.DM↗