Covering Radius 1985-1994

Cohen, Gérard; Litsyn, Simon; Lobstein, Antoine; Mattson, H. F.

doi:10.1007/s002000050061

Cited by 36 publications

(20 citation statements)

References 184 publications

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“…The problem of determining the covering radius has been studied by many researchers [2], [5], [6], [7], [8], [9], [10], [11], [12], [14], [15], [17], [18], [19], [20], [21] among which [2], [5], [6], [11], [17], and [19] focused on its asymptotic behavior with respect to block length n under an exponentially increasing size M = e nR and a fixed rate R. Specifically, [5], [17], and [19] investigate this problem based on combinatorial techniques, while the studies in [2] and [11] introduce probabilistic approaches. All the mentioned works concentrated on codes transmitted over binary symmetric channel, where Hamming distance is the only distance measure.…”

Section: ρ(M S N ) = Minmentioning

confidence: 99%

Asymptotic Minimum Covering Radius of Block Codes

Chen¹,

Han²

2001

SIAM J. Discrete Math.

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Abstract. In this paper, we restudy the covering radius of block codes from an information theoretic point of view by ignoring the combinatorial formulation of the problem. In the new setting, the formula of the statistically defined minimum covering radius, for which the probability mass of uncovered space by M spheres can be made arbitrarily small, is reduced to a minimization of a statistically defined spectrum formula among codeword-selecting distributions. The advantage of the new view is that no assumptions need to be made on the code alphabet (such as finite, countable, etc.) and the distance measure (such as additive, symmetric, bounded, etc.) in the problem transformation, and hence the spectrum formula can be applied in most general situations. We next address a sufficient condition under which uniform codeword-selecting distribution minimizes the spectrum formula. With the condition, the asymptotic minimum covering radius for block codes under J-ary quantized channels and constant weight codes under Hamming distance measure are determined to display the usage of the spectrum formula.

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Section: ρ(M S N ) = Minmentioning

confidence: 99%

Asymptotic Minimum Covering Radius of Block Codes

Chen¹,

Han²

2001

SIAM J. Discrete Math.

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“…Thus, (7,16,1) Hamming code also generalizes to (7 + i, 2 i+4 , 1)-covering codes for all i ≥ 0. However, for many n ≥ 9, better (n, K, 1)-covering codes than the naive extension from (7,16,1) are known [5], [4]. In particular, (9, 62, 1) is such a code included in [4].…”

Section: The (N + I 2 I K R)-covering Codesmentioning

confidence: 99%

“…However, many covering codes have small R and large K relative to m [12], [5], [4]. Applying these covering codes directly to the entire array typically yields an unreasonable space overhead, even though the time is much improved.…”

Section: Reducing Space Overheadmentioning

confidence: 99%

“…We have chosen (n, K, R)-covering codes with combinations of minimum radius R and minimum number of codewords K, given the length of codewords n. Specifically, we consider four classes of codes: two classes for two different generalizations of Hamming code (7,16,1), one class for the generalization of (5, 7, 1) code, and one class for the generalization of (6, 12, 1) code. These are the only codes that yielded useful (s, t)-pairs among all the codes included in [12], [5], and [4].…”

Section: Applying Known Covering Codesmentioning

confidence: 99%

“…Second, we apply four known covering codes from [12], [5], and [4] to the partialsum problem to obtain algorithms with various space-time trade-offs. Third, we modify the requirements on covering codes to better reflect the partial-sum problem and devise new covering codes with respect to the new requirements.…”

Section: Contributionsmentioning

confidence: 99%

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Partial-sum queries in OLAP data cubes using covering codes

Bruck

1998

IEEE Trans. Comput.

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Abstract-A partial-sum query obtains the summation over a set of specified cells of a data cube. We establish a connection between the covering problem in the theory of error-correcting codes and the partial-sum problem and use this connection to devise algorithms for the partial-sum problem with efficient space-time trade-offs. For example, using our algorithms, with 44 percent additional storage, the query response time can be improved by about 12 percent; by roughly doubling the storage requirement, the query response time can be improved by about 34 percent.

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Resolution of a Conjecture on the Covering Radius of Linear Codes

Orozco,

Janwa

2024

Springer Proceedings in Mathematics &Amp; Statistics

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Covering Radius 1985-1994

Cited by 36 publications

References 184 publications

Asymptotic Minimum Covering Radius of Block Codes

Asymptotic Minimum Covering Radius of Block Codes

Partial-sum queries in OLAP data cubes using covering codes

Resolution of a Conjecture on the Covering Radius of Linear Codes

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