Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces

Norihiro NAKASHIMA; Hajime MATSUI

doi:10.1587/transfun.E99.A.733

Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces

Norihiro NAKASHIMA, Hajime MATSUI

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A projective Reed-Muller (PRM) code, obtained by modifying a Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and the dual code of a PRM code are known, and some decoding examples have been presented for low-dimensional projective spaces. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of correctable errors of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of the minimum distance decoding.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E99-A No.3 pp.733-741

Publication Date: 2016/03/01

Publicized

Online ISSN: 1745-1337

DOI: 10.1587/transfun.E99.A.733

Type of Manuscript: PAPER

Category: Coding Theory

Authors

Norihiro NAKASHIMA
Toyota Technological Institute
Hajime MATSUI
Toyota Technological Institute

Keyword

error-correcting codes, affine variety codes, Gröbner basis, Berlekamp-Massey-Sakata algorithm, discrete Fourier transform

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Norihiro NAKASHIMA, Hajime MATSUI, "Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces" in IEICE TRANSACTIONS on Fundamentals, vol. E99-A, no. 3, pp. 733-741, March 2016, doi: 10.1587/transfun.E99.A.733.
Abstract: A projective Reed-Muller (PRM) code, obtained by modifying a Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and the dual code of a PRM code are known, and some decoding examples have been presented for low-dimensional projective spaces. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of correctable errors of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of the minimum distance decoding.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1587/transfun.E99.A.733/_p

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@ARTICLE{e99-a_3_733,
author={Norihiro NAKASHIMA, Hajime MATSUI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces},
year={2016},
volume={E99-A},
number={3},
pages={733-741},
abstract={A projective Reed-Muller (PRM) code, obtained by modifying a Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and the dual code of a PRM code are known, and some decoding examples have been presented for low-dimensional projective spaces. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of correctable errors of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of the minimum distance decoding.},
keywords={},
doi={10.1587/transfun.E99.A.733},
ISSN={1745-1337},
month={March},}

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TY - JOUR
TI - Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 733
EP - 741
AU - Norihiro NAKASHIMA
AU - Hajime MATSUI
PY - 2016
DO - 10.1587/transfun.E99.A.733
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E99-A
IS - 3
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - March 2016
AB - A projective Reed-Muller (PRM) code, obtained by modifying a Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and the dual code of a PRM code are known, and some decoding examples have been presented for low-dimensional projective spaces. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of correctable errors of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of the minimum distance decoding.
ER -