A Fast Erasure Deletion Generalized Minimum Distance Decoding for One-Point Algebraic-Geometry Codes

Shojiro SAKATA

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E84-A    No.10    pp.2376-2382
Publication Date: 2001/10/01
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)
Category: Coding Theory
one-point algebraic-geometry code,  generalized minimum distance decoding,  BMS algorithm,  erasure addition,  erasure deletion,  

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Before we gave a fast generalized minimum distance (GMD) decoding algorithm for one-point algebraic-geometry (AG) codes. In this paper, we propose another fast GMD decoding algorithm for these codes, where the present method includes an erasure deletion procedure while the past one uses an erasure addition procedure. Both methods find a minimal polynomial set of a given syndrome array, which is a candidate for an erasure-and-error locator polynomial set constrained with an erasure locator set of each size. Although both erasure addition and deletion GMD decoding algorithms have been established for one-dimensional algebraic codes such as RS codes, nothing but the erasure addition GMD decoding algorithm for multidimensional algebraic codes such as one-point AG codes have been given. The present erasure deletion GMD decoding algorithm is based on the Berlekamp-Massey-Sakata (BMS) algorithm from the standpoint of constrained multidimensional shift register synthesis. It is expected that both our past and present methods play a joint role in decoding for one-point AG codes up to the error correction bound.