Assmus Jr, E.F., On the Reed-Muller codes, Discrete Mathematics 106/107 (1992) 25-33. We give a brief but complete account of all the essential facts concerning the Reed-Muller and punctured Reed-Muller codes. The treatment is new and includes an easy, direct proof of the fact that the punctured R
On Berman's characterization of the Reed-Muller codes
β Scribed by E.F. Assmus Jr.
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 358 KB
- Volume
- 56
- Category
- Article
- ISSN
- 0378-3758
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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