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|Title: ||On the key equation over a commutative ring|
|Authors: ||Norton, G.H.|
|Keywords: ||Alternant code|
|Issue Date: ||2000|
|Publisher: ||© Springer Verlag|
|Citation: ||NORTON and SALAGEAN, 2000. On the key equation over a commutative ring. Designs, codes and cryptology, 20, pp. 125-141|
|Abstract: ||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 (shortest linear recurrence) of the syndrome sequence
and that it can be obtained by Algorithm MR of Norton.
When R is a local ring, we show that the syndrome sequence may have more than one (monic) minimal polynomial, but all the minimal polynomials coincide modulo the maximal ideal of R. We characterise the 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.|
|Description: ||This article was published in the journal, Designs, codes and cryptology [© Springer Verlag] and is also available at: http://www.springerlink.com/openurl.asp?genre=journal&issn=0925-1022|
|Appears in Collections:||Published Articles (Computer Science)|
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