A note on bounds for q-ary covering codes

Bhandari, M. C.; Durairajan, C.

doi:10.1109/18.532916

Cited by 13 publications

(5 citation statements)

References 9 publications

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“…When R > 3, the following instances are not settled by Theorem 3 or 4: K(1, 10, 4), K (2,9,4), K(3, 7, 4), K (3,9,5), K(4, 6, 4), K (4,8,5), K(4, 10, 6), K (5,4,4), K(5, 6, 5), K (5,8,6), and K(5, 10, 7). The well-known inequality…”

Section: Theorem 3 (Kéri and öStergårdmentioning

confidence: 96%

Further results on the covering radius of small codes

Kéri

Östergrd²

2007

Discrete Mathematics

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The minimum number of codewords in a code with t ternary and b binary coordinates and covering radius R is denoted by K(t, b, R). In the paper, necessary and sufficient conditions for K(t, b, R) = M are given for M = 6 and 7 by proving that there exist exactly three families of optimal codes with six codewords and two families of optimal codes with seven codewords. The cases M 5 were settled in an earlier study by the same authors. For binary codes, it is proved that K(0, 2b + 4, b) 9 for b 1. For ternary codes, it is shown that K(3t + 2, 0, 2t) = 9 for t 2. New upper bounds obtained include K(3t + 4, 0, 2t) 36 for t 2. Thus, we have K(13, 0, 6) 36 (instead of 45, the previous best known upper bound).

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Section: Theorem 3 (Kéri and öStergårdmentioning

confidence: 96%

Further results on the covering radius of small codes

Kéri

Östergrd²

2007

Discrete Mathematics

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“…The upper bound follows from (2). For the lower bound we will prove that every (5,8,5)-partition matrix P has a 5-transversal by considering several cases.…”

Section: Exact Valuesmentioning

confidence: 98%

“…Entries in bold are exact. We use the inequality K q (n 1 + n 2 , R 1 + R 2 + 1) min K q (n 1 , R 1 ), K q (n 2 , R 2 ) (6) due to Bhandari and Durairajan [2].…”

Section: Referencementioning

confidence: 99%

Lower bounds on covering codes via partition matrices

Haas

Schlage‐Puchta

Quistorff

2009

Journal of Combinatorial Theory, Series A

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Let K q (n, R) denote the minimal cardinality of a q-ary code of length n and covering radius R. Let σ q (n, s; r) denote the minimal cardinality of a q-ary code of length n, which is s-surjective with radius r. In order to lower-bound K q (n, n − 2) and σ q (n, s; s − 2) we introduce partition matrices and their transversals. Our approach leads to a short new proof of a classical bound of Rodemich on K q (n, n − 2) and to the new bound K q (n, n − 2) 3q − 2n + 2, improving the first iff 5 n < q 2n − 4. We determine K q (q, q − 2) = q − 2 + σ 2 (q, 2; 0) if q 10. Moreover, we obtain the new powerful recursive bound K q+1 (n + 1, R + 1) min{2(q + 1), K q (n, R) + 1}.

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“…The following two inequalities that proved to be useful for setting lower bounds on the size of covering codes were published in [2] and [26], respectively.…”

Section: Tables: Lower and Upper Bounds And Classificationsmentioning

confidence: 99%

Covering and radius-covering arrays: Constructions and classification

Colbourn

Kéri

Soriano³

et al. 2010

Discrete Applied Mathematics

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The minimum number of rows in covering arrays (equivalently, surjective codes) and radius-covering arrays (equivalently, surjective codes with a radius) has been determined precisely only in special cases. In this paper, explicit constructions for numerous best known covering arrays (upper bounds) are found by a combination of combinatorial and computational methods. For radius-covering arrays, explicit constructions from covering codes are developed. Lower bounds are improved upon using connections to orthogonal arrays, partition matrices, and covering codes, and in specific cases by computation. Consequently for some parameter sets the minimum size of a covering array is determined precisely. For certain of these, a complete classification of all inequivalent covering arrays is determined, again using computational techniques. Existence tables for up to 10 columns, up to 8 symbols, and all possible strengths are presented to report the best current lower and upper bounds, and classifications of inequivalent arrays.

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A note on bounds for q-ary covering codes

Cited by 13 publications

References 9 publications

Further results on the covering radius of small codes

Further results on the covering radius of small codes

Lower bounds on covering codes via partition matrices

Covering and radius-covering arrays: Constructions and classification

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