On the Covering Radius of Small Codes Versus Dual Distance
نویسندگان
چکیده
منابع مشابه
On the covering radius of small codes versus dual distance
Tietäväinen’s upper and lower bounds assert that for block-length-n linear codes with dual distance d, the covering radius R is at most n2 − ( 2 − o(1)) √ dn and typically at least n2 − Θ( √ dn log nd ). The gap between those bounds on R − n2 is an Θ( √ log nd ) factor related to the gap between the worst covering radius given d and the sphere-covering bound. Our focus in this paper is on the c...
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Codes with minimum distance at least d and covering radius at most d− 1 are considered. The minimal cardinality of such codes is investigated. Herewith, their connection to covering problems is applied and a new construction theorem is given. Additionally, a new lower bound for the covering problem is proved. A necessary condition on an existence problem is presented by using a multiple coverin...
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Designing a good error-correcting code is a packing problem. The corresponding covering problem has received much less attention: now the codewords must be placed so that no vector of the space is very far from the nearest codeword. The two problems are quite different, and with a few exceptions good packings, i.e. codes with a large minimal distance, are usually not especially good coverings. ...
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In this two-part paper we introduce the notion of a stable code and give a new upper bound on the normalized covering radius ofa code. The main results are that, for fixed k and large n, the minimal covering radius t[n, k] is realized by a normal code in which all but one of the columns have multiplicity l; hence tin + 2, k] t[n, k] + for sufficiently large n. We also show that codes with n _-<...
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Cohen, G.D., S.N. Litsyn, On the covering radius of Reed-Muller codes, Discrete Mathematics 106/107 (1992) 147-155. 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-Mul...
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ژورنال
عنوان ژورنال: IEEE Transactions on Information Theory
سال: 2019
ISSN: 0018-9448,1557-9654
DOI: 10.1109/tit.2018.2857495