New Results on Codes with Covering Radius 1 and Minimum Distance 2

Östergård, Patric R. J.; Quistorff, Jörn; Wassermann, Alfred

doi:10.1007/s10623-005-6404-3

Cited by 8 publications

(8 citation statements)

References 16 publications

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“…Assume to the contrary, that C ⊂ R × S × T is a code with covering radius 1, minimum distance 2 and |C| = K(r, s, t; 2) = m ≤ m. together in the row and the column of that entry is at least r. Therefore the propositions of Theorem 19 are satisfied and (5) holds, contradicting (6). Hence K(r, s, t; 2) > m. An easy modification yields the bound K (r − 1, s, t; 2) > m for r ≥ 2.…”

Section: From This We Deducementioning

confidence: 94%

On mixed codes with covering radius 1 and minimum distance 2

Haas

Quistorff

2007

Electronic Notes in Discrete Mathematics

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Let R, S and T be finite sets with |R| = r, |S| = s and |T | = t. A code C ⊂ R × S × T with covering radius 1 and minimum distance 2 is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality K(r, s, t; 2). These bounds turn out to be best possible in many instances. Focussing on the special case t = s we determine K(r, s, s; 2) when r divides s, when r = s − 1, when s is large, relative to r, when r is large, relative to s, as well as K(3r, 2r, 2r; 2). Some open problems are posed. Finally, a table with bounds on K(r, s, s; 2) is given.

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Section: From This We Deducementioning

confidence: 94%

On mixed codes with covering radius 1 and minimum distance 2

Haas

Quistorff

2007

Electronic Notes in Discrete Mathematics

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“…If necessary, replace some 0's in B by 1's until the total number of 1's in the new matrix B is m. Since A is non-extendable, every element of R appears in the row and the column of an 0-entry in A. Thus for every 0-entry in B the number of 1's together in the row and the column of that entry is at least r. Therefore the propositions of Theorem 19 are satisfied and (5) holds, contradicting(6). Hence K(r, s, t; 2) > m. An easy modification yields the bound K (r − 1, s, t; 2) > m for r ≥ 2.…”

mentioning

confidence: 92%

“…q-ary codes with covering radius (at most) 1 and minimum distance (at least) 2 as well as the corresponding non-extendable partial multiquasigroups have been studied in [9,7,8,1,6]. Equivalent objects are pairwise non-attacking rooks which cover all cells of a generalized chessboard and non-extendable partial Latin hypercubes.…”

Section: Introductionmentioning

confidence: 99%

“…Denote by K q (n, 1, 2) the minimal cardinality of a code C ⊂ Q n with covering radius 1 and minimum distance 2. Well-known results are K q (2, 1, 2) = q and K q (3, 1, 2) = q 2 /2 as well as K 2 (4, 1, 2) = 4, K 2 (5, 1, 2) = 8, K 2 (6, 1, 2) = 12, K 3 (4, 1, 2) = 9, K 4 (4, 1, 2) = 28 and K q (n + 1, 1, 2) ≤ q • K q (n, 1, 2), see [4,3,8,6].…”

Section: Introductionmentioning

confidence: 99%

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On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

Haas¹,

Quistorff²

2007

Electron. J. Combin.

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Let $R$, $S$ and $T$ be finite sets with $|R|=r$, $|S|=s$ and $|T|=t$. A code $C\subset R\times S\times T$ with covering radius $1$ and minimum distance $2$ is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality $K(r,s,t;2)$. These bounds turn out to be best possible in many instances. Focussing on the special case $t=s$ we determine $K(r,s,s;2)$ when $r$ divides $s$, when $r=s-1$, when $s$ is large, relative to $r$, when $r$ is large, relative to $s$, as well as $K(3r,2r,2r;2)$. Some open problems are posed. Finally, a table with bounds on $K(r,s,s;2)$ is given.

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“…The covering problem for finite fields and finite rings has long been studied by researchers in connection with coding theoretical and combinatorial applications (see [1][2][3]). …”

Section: Introductionmentioning

confidence: 99%

The covering problem for finite rings with respect to the RT-metric

Yıldız

Şiap

Bilgin

et al. 2010

Applied Mathematics Letters

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a b s t r a c tIn this work, the cardinality of the minimal R-covers of finite rings with respect to the RT-metric is established. By generalizing the result in Nakaoka and dos Santos (2010) [1], the minimal cardinalities of 0-short coverings of finite chain rings are calculated. The connection between R-short coverings of rings with respect to the RT-metric and the 0-short coverings of rings is demonstrated, and with the help of this connection, the problem of finding the minimal cardinalities of R-short coverings of finite chain rings is solved.

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New Results on Codes with Covering Radius 1 and Minimum Distance 2

Cited by 8 publications

References 16 publications

On mixed codes with covering radius 1 and minimum distance 2

On mixed codes with covering radius 1 and minimum distance 2

On Mixed Codes with Covering Radius $1$ and Minimum Distance $2$

The covering problem for finite rings with respect to the RT-metric

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