Nielsen, Johan Sebastian Rosenkilde

Afhandlingen indholder tre paradigmer for afkodning i polynomiel tid af Reed–Solomon-koder ud over den halve minimumsafstand: Guruswami–Sudan, Power, og Wu afkodning. Hovedresultatet er at den beregningsmæssige kerne af alle tre metoder kan udformes som et problem der kan løses med modul-minimering. Ved at benytte de hurtigste kendte metoder til dette generelle problem, opnår vi realiseringer af hvert afkodningsparadigme, som er lige så hurtige eller hurtigere end alle tidligere kendte. Et vigtigt element i udformningen er en kraftigt generaliseret form af den klassiske nøgleligning, de såkaldte “2D nøgleligninger”. Vi viser hvordan man kan løse denne slags ligninger ved hjælp af modul-minimering eller ved hjælp af vores nye Demand–Driven-algoritme, som også er baseret på modul-minimering. I afhandlingen er afkodningsparadigmerne selvstændigt udledt og analyseret, ofte på nye måder eller mere dybdegående end tidligere. Vi præsenterer et antal nye resultater, blandt andet: en hurtig maximum-likelihood listeafkoder baseret på Guruswami–Sudan algoritmen; en ny variant af Power-afkodning, Power Gao, samt nye indsigter i Power afkodning; og en ny, modul-baseret metode for at udføre rational interpolation i Wu algoritmen. Vi viser også hvordan man kan afkode Hermitekoder ved hjælp af Guruswami–Sudan algoritmen eller Power-afkodning som er hurtigere end tidligere kendt, og hvordan man kan Wu listeafkode binære Goppa-koder.We investigate three paradigms for polynomial-time decoding of Reed–Solomon codes beyond half the minimum distance: the Guruswami–Sudan algorithm, Power decoding and the Wu algorithm. The main results concern shaping the computational core of all three methods to a problem solvable by module minimisation; by applying the fastest known algorithms for this general problem, we then obtain realisations of each paradigm which are as fast or faster than all previously known methods. An element of this is the “2D key equation”, a heavily generalised form of the classical key equation, and we show how to solve such using module minimisation, or using our new Demand–Driven algorithm which is also based on module minimisation.The decoding paradigms are all derived and analysed in a self-contained manner, often in new ways or examined in greater depth than previously. Among a number of new results, we give: a fast maximum-likelihood list decoder based on the Guruswami–Sudan algorithm; a new variant of Power decoding, Power Gao, along with some new insights into Power decoding; and a new, module based method for performing rational interpolation for theWu algorithm. We also show how to decode Hermitian codes using Guruswami–Sudan or Power decoding faster than previously known, and we show how to Wu list decode binary Goppa codes

Online Research Database In Technology

List Decoding of Algebraic Codes

http://orbit.dtu.dk/ws/files/90302039/jsrn_thesis.pdf

List Decoding of Algebraic Codes

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Online Research Database In Technology