Asymptotically Optimal Covering Designs

Gordon, Daniel M.; Patashnik, Oren; Kuperberg, Greg; Spencer, Joel

doi:10.1006/jcta.1996.0077

Cited by 25 publications

(18 citation statements)

References 5 publications

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“…No lower bounds on the number of uncovered pairs are known, but substantial computer simulations of the random greedy algorithm have been carried out by Balińska and Wieczorek [2] for triangle packing and by Gordon, Kuperberg, Patashnik, and Spencer [6] for more general packing problems. These simulations indicate that the correct order of magnitude for the number of uncovered pairs is n 3/2+o (1) and, indeed, Joel Spencer has offered $200 for a proof.…”

Section: Introductionmentioning

confidence: 99%

On Random Greedy Triangle Packing

Grable¹

1997

Electron. J. Combin.

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The behaviour of the random greedy algorithm for constructing a maximal packing of edge-disjoint triangles on $n$ points (a maximal partial triple system) is analysed with particular emphasis on the final number of unused edges. It is shown that this number is at most $n^{7/4+o(1)}$, "halfway" from the previous best-known upper bound $o(n^2)$ to the conjectured value $n^{3/2+o(1)}$. The more general problem of random greedy packing in hypergraphs is also considered.

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Section: Introductionmentioning

confidence: 99%

On Random Greedy Triangle Packing

Grable¹

1997

Electron. J. Combin.

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“…An alternative proof of Rödl's theorem was subsequently obtained by Spencer [39], while constructions for covering designs which (asymptotically) meet this bound were obtained by Gordon et al [17,18].…”

Section: Covering Designsmentioning

confidence: 99%

Generalized covering designs and clique coverings

et al. 2011

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“…A covering design is collection of subsets of size of , called blocks, such that every subset of of size is contained in at least one block (changing "at least one" to "exactly one" thus makes this a Steiner system). The smallest number of blocks in a covering design is usually denoted by and called the covering number (see [11], [18], and references therein). Thus, if is an MDS code, then…”

Section: But Thenmentioning

confidence: 99%

On the stopping distance and the stopping redundancy of codes

Schwartz

Vardy

2006

IEEE Trans. Inform. Theory

131

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Abstract-It is now well known that the performance of a linear code under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for . Several recent papers refer to this parameter as the stopping distance of . This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for depends on the corresponding choice of a parity-check matrix. It is easy to see that , where is the minimum Hamming distance of , and we show that it is always possible to choose a parity-check matrix for (with sufficiently many dependent rows) such that = . We thus introduce a new parameter, the stopping redundancy of , defined as the minimum number of rows in a parity-check matrix for such that the corresponding stopping distance ( ) attains its largest possible value, namely, ( ) = . We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 34 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes.

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Asymptotically Optimal Covering Designs

Cited by 25 publications

References 5 publications

On Random Greedy Triangle Packing

On Random Greedy Triangle Packing

Generalized covering designs and clique coverings

On the stopping distance and the stopping redundancy of codes

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