Talk abstract:

Realization and Interpolation via Gröbner Bases

Patrick Fitzpatrick
Department of Mathematics
National University if Ireland, Cork
fitzpat@ucc.ie
http://ogham.ucc.ie

The solution of a system of polynomial congruences (in one or several variables) can be interpreted as a search for canonical elements in a module and is amenable to analysis using Gröbner bases. Classical problems such as partial realization and (scalar) rational interpolation can be solved using this approach. We review the background and applications to decoding alternant codes, and consider generalizations to the matrix case and to the case in which the underlying scalars are from a Galois ring.

References

1. P. Fitzpatrick, On the key equation, IEEE Trans. on Information Theory, 41 (1995), 1290-1302.

2. S.M. Jennings, Gröbner basis view of the Welch-Berlekamp algorithm for Reed--Solomon codes, IEE Proc. Commun.142 (1995), 349-351.

3. P. Fitzpatrick, On the scalar rational interpolation problem, Math. of Control, Signals, and Systems, 9 (1996), 352-369.

4. P. Fitzpatrick, Solving a multivariable congruence by change of term order, J. Symb. Comp., 24 (1997), 505-510.

5. P. Fitzpatrick, Rational approximation using Gröbner bases: some numerical results, Math. in Signal Processing IV, J.G. McWhiter, I.K. Proudler, eds, Clarendon Press, Oxford, 1997, 1998, 35-46.

6. P. Fitzpatrick and S.M. Jennings, Comparison of two algorithms for decoding alternant codes, Applicable Algebra in Engineering, Communication and Computation 9 (1998) 3, 211-220.

7. P. Fitzpatrick, Errors and erasures decoding of BCH codes, IEE Proc.-E, to appear.

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