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On Correctable Errors of Binary Linear Codes (2010)
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BibTeX
@MISC{Yasunaga10oncorrectable,
author = {Kenji Yasunaga and Toru Fujiwara},
title = {On Correctable Errors of Binary Linear Codes},
year = {2010}
}
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Abstract
The error correction capability of binary linear codes with minimum distance decoding, in particular the number of correctable/uncorrectable errors, is investigated for general linear codes and the first-order Reed–Muller codes. For linear codes, a lower bound on the number of uncorrectable errors is derived. The bound for uncorrectable errors with a weight of half the minimum distance asymptotically coincides with the corresponding upper bound for Reed–Muller codes and random linear codes. For the first-order Reed–Muller codes, the number of correctable/uncorrectable errors with a weight of half the minimum distance plus one is determined. This result is equivalent to deriving the number of Boolean functions of variables with nonlinearity. The monotone error structure and its related notions larger half and trial set, which were introduced by Helleseth, Kløve, and Levenshtein, are mainly used to derive the results.
Keyphrases
binary linear code correctable error minimum distance correctable uncorrectable error first-order reed muller code uncorrectable error error correction capability related notion monotone error structure trial set reed muller code general linear code boolean function kl ve corresponding upper bound linear code minimum distance decoding random linear code
