arXiv subjects

Kwankyu Lee

Publications and source records attributed to Kwankyu Lee.

10 recordsLinked to original sources

arXiv2017

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.

cs.IT↗

arXiv2014

Decoding of Differential AG Codes

The interpolation-based decoding that was developed for general evaluation AG codes is shown to be equally applicable to general differential AG codes. A performance analysis of the decoding algorithm, which is parallel to that of its companion algorithm, is reported. In particular, the decoding capacities of evaluation AG codes and differential AG codes are seen to be nicely interrelated. As an interesting special case, a decoding algorithm for classical Goppa codes is presented.

cs.IT↗

arXiv2012

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 and regular structure with the previous algorithm. Compared with other known decoding algorithms for general AG codes, it is much simpler in its description and implementation.

cs.IT↗

arXiv2012

Unique Decoding of Plane AG Codes Revisited

We reformulate a recently introduced interpolation-based unique decoding algorithm of algebraic geometry codes using the theory of Gröbner bases of modules on the coordinate ring of the base curve. With the same decoding performance, the new algorithm has a more conceptual description that lets us better understand the majority voting procedure central in the interpolation-based unique decoding.

cs.IT↗

arXiv2011

Unique Decoding of Plane AG Codes via Interpolation

We present a unique decoding algorithm of algebraic geometry 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. The decoding algorithm is the first to combine some features of the interpolation based list decoding with the performance of the syndrome decoding with majority voting scheme. The regular structure of the algorithm allows a straightforward parallel implementation.

cs.IT↗

arXiv2008

Algebraic Soft-Decision Decoding of Hermitian Codes

An algebraic soft-decision decoder for Hermitian codes is presented. We apply Koetter and Vardy's soft-decision decoding framework, now well established for Reed-Solomon codes, to Hermitian codes. First we provide an algebraic foundation for soft-decision decoding. Then we present an interpolation algorithm finding the Q-polynomial that plays a key role in the decoding. With some simulation results, we compare performances of the algebraic soft-decision decoders for Hermitian codes and Reed-Solomon codes, favorable to the former.

cs.IT↗

arXiv2007

List Decoding of Reed-Solomon Codes from a Groebner Basis Perspective

The interpolation step of Guruswami and Sudan's list decoding of Reed-Solomon codes poses the problem of finding the minimal polynomial of an ideal with respect to a certain monomial order. An efficient algorithm that solves the problem is presented based on the theory of Groebner bases of modules. In a special case, this algorithm reduces to a simple Berlekamp-Massey-like decoding algorithm.

math.AC↗

arXiv2007

Interpolation-based Decoding of Alternant Codes

We formulate the classical decoding algorithm of alternant codes afresh based on interpolation as in Sudan's list decoding of Reed-Solomon codes, and thus get rid of the key equation and the linear recurring sequences in the theory. The result is a streamlined exposition of the decoding algorithm using a bit of the theory of Groebner bases of modules.

cs.IT↗

arXiv2006

List Decoding of Hermitian Codes using Groebner Bases

List decoding of Hermitian codes is reformulated to allow an efficient and simple algorithm for the interpolation step. The algorithm is developed using the theory of Groebner bases of modules. The computational complexity of the algorithm seems comparable to previously known algorithms achieving the same task, and the algorithm is better suited for hardware implementation.

cs.IT↗

arXiv2005

Distance-Increasing Maps of All Length by Simple Mapping Algorithms

Distance-increasing maps from binary vectors to permutations, namely DIMs, are useful for the construction of permutation arrays. While a simple mapping algorithm defining DIMs of even length is known, existing DIMs of odd length are either recursively constructed by merging shorter DIMs or defined by much complicated mapping algorithms. In this paper, DIMs of all length defined by simple mapping algorithms are presented.

cs.IT↗