caa a22 4500 475807421 CHVBK 20180406123750.0 cr unu---uuuuu 170329e20000601xx s 000 0 eng 10.1023/A:1008385407717 doi (NATIONALLICENCE)springer-10.1023/A:1008385407717 On the Key Equation Over a Commutative Ring [Elektronische Daten] [Graham Norton, Ana Salagean-Mandache] We define alternant codes over a commutative ring R and a corresponding key equation. We show that when the ring is a domain, e.g. the p-adic integers, the error-locator polynomial is the unique monic minimal polynomial (equivalently, the unique shortest linear recurrence) of the finite sequence of syndromes and that it can be obtained by Algorithm MR of Norton. WhenR is a local ring, we show that the syndrome sequence may have more than one (monic) minimal polynomial, but that all the minimal polynomials coincide modulo the maximal ideal ofR . We characterise the set of minimal polynomials when R is a Hensel ring. We also apply these results to decoding alternant codes over a local ring R: it is enough to find any monic minimal polynomial over R and to find its roots in the residue field. This gives a decoding algorithm for alternant codes over a finite chain ring, which generalizes and improves a method of Interlando et. al. for BCH and Reed-Solomon codes over a Galois ring. Kluwer Academic Publishers, 2000 alternant code nationallicence domain nationallicence local ring nationallicence minimal polynomial nationallicence decoding algorithm nationallicence Norton Graham Algebraic Coding Research Group, Centre for Communications Research, University of Bristol, U.K. aut Salagean-Mandache Ana Department of Mathematics, Nottingham Trent University, U.K. aut Designs, Codes and Cryptography Kluwer Academic Publishers 20/2(2000-06-01), 125-141 0925-1022 20:2<125 2000 20 10623 https://doi.org/10.1023/A:1008385407717 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1008385407717 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Norton Graham Algebraic Coding Research Group, Centre for Communications Research, University of Bristol, U.K aut NATIONALLICENCE 700 1- Salagean-Mandache Ana Department of Mathematics, Nottingham Trent University, U.K aut NATIONALLICENCE 773 0- Designs, Codes and Cryptography Kluwer Academic Publishers 20/2(2000-06-01), 125-141 0925-1022 20:2<125 2000 20 10623 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer