OMEGA: a system for the effective construction, coding and decoding of block error-correcting codes

In this work, we show how to implement effective constructions, coding and decoding of algebraic codes by means of Omega, a system specifically designed and programmed for general mathematical computations. For alternant codes, the main class we consider (which includes BCH, RS and classical Goppa codes), we give an implementation of the Euclidean division BM decoding algorithm. For cyclic codes we implement the Meggitt decoder, and to illustrate how it works we provide an implementation of the Meggitt syndrome tables for the two Golay codes. Finally, we present several other groups of functions and the computations and problems (still almost in the area of error-correcting codes) they solve. Resum L’objecte d’aquest treball és explicar com es poden implementar d’una manera efectiva, mitjançant el programa de manipulació simbòlica OMEGA, algunes de les construccions i operacions més importants de la teoria de codis correctors algebraics. Per als codis alternants, la classe més important que considerem, i que inclou els codis BCH, RS i de Goppa clàssics, presentem una implementació de l’algorisme de descodificació de Berlekamp-Massey. Per als codis cíclics, implementem l’algorisme de descodificació de Meggitt, i il·lustrem el seu funcionament, mitjançant la construcció de les corresponents taules de síndromes de Meggitt, per als codis de Golay. Finalment, presentem diversos altres grups de funcions, així com els càlculs i problemes (encara circumscrits gairebé a l’àrea de codis correctors) que ens permeten resoldre. * Author for correspondence: Sebastià Xambó, Departament de Matemàtica Aplicada II (MA2) i Facultat de Matemàtiques i Estadística (FME), Universitat Politècnica de Catalunya (UPC). Pau Gargallo, 5. Campus Sud-Edif. U. 08028 Barcelona, Catalonia (Spain). Tel. 34 93 401 69 26. Fax: 34 93 401 72 84. Email: [email protected]. Currently the author is president of the Catalan Society for Mathematics. The OMEGA team 1998/1999, led by the author, consists of Daniel Marquès, Ramon Eixarch, Marc Castells, David Arso and Pere Garriga. tions as well as other functions related to the computational aspects of error-correcting codes; in section 7, finally, we write down some conclusions and make a few additional remarks on the expressive power of W, including its multilingual features. For more specialized applications in the coding area, see [26]; for other applications of OMEGA, see [27], the users manual, and [25], an OMEGA package for computations in intersection theory (two functions of this package are shown in section 6). 1. A first glimpse on OMEGA In this section we provide a quick tour to some of the main features of the programme OMEGA (and of the language W). The reader interested in effective methods in coding can skim over this section, and may return to it later if needed. OMEGA is written in C++. Here we run a version compiled with Visual C++ for Windows 98 called OMEGA/Athens/1999. This system should be useful for teaching several subjects (e.g. linear algebra, geometry, algebra, calculus, number theory, combinatorics, algebraic geometry) in the universities and for research on many fields of mathematics and physics. The programme OMEGA/Athens has a command line of the form