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On the norm and covering radius of the first-order Reed-Muller codes

✍ Scribed by Xiang-dong Hou

Book ID
114540640
Publisher
IEEE
Year
1997
Tongue
English
Weight
142 KB
Volume
43
Category
Article
ISSN
0018-9448

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We present lower and upper bounds on the covering radius of Reed-Muller codes, yielding asymptotical improvements on known results. The lower bound is simply the sphere covering one (not very new). The upper bound is derived from a thorough use of a lemma, the 'essence of Reed-Mullerity'. The idea

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Let R(r, m) be the rth order Reed-Muller code of length 2 '~, and let p(r, m) be its covering radius. We prove that if 2 \_< k -< m -r -1, then p(r + k, m + k) > #(r, m) + 2(k -1). We also prove that if m -r > 4, 2 < k < m -r -1, and R(r, m) has a coset with minimal weight pfr, m) which does not co