D. M. Gordon, G. Kuperberg and O. Patashnik. New constructions for covering designs, J. Combin. Designs 3 (1995), 269--284.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....every t subset of X (t m) is contained in at least one block has been of interest to design theorists. So much so, that there is an entire website [7] devoted to such constructions, called (n;k; t) covering designs. Descriptions of some methods for constructing covering designs can be found in [6]. By taking the complement of each block in an (n;n k;r) covering design, we obtain an (n;k;r) uncovering. Some examples useful to us are given below (they were either found directly, or by taking complements of designs listed in [7] and then relabelling) For the affine group AGL(1;11) an ....

D.M. Gordon, G. Kuperberg & O. Patashnik (1995), New Constructions for Covering Designs, J. Comb. Des. 3, No. 4, 269-284.

....we are not proposing that our approximation algorithm be used in a practical implementation, it is of some interest to consider whether the constant 12 p 2 can be improved. One possibility is to use a more sophisticated method for constructing (n; M; 2) covering designs (see for example [5]) For a non trivial example of what a good design can accomplish, suppose that n = 15 and c = d = 1. The add drop lower bound is 105. Our approximation algorithm tells us to take M = 2, but suppose we take M = 3 instead and look for a (15; 3; 2) covering design. One example of this is the ....

D. M. Gordon, O. Patashnik, and G. Kuperberg, New constructions for covering designs, J. Combin. Designs, 3 (1995), pp. 269--284; http://sdcc12.ucsd.edu/~xm3dg/errata.html (errata).

....(k 1 ) v) X (k 2 ) v) X (kn ) v) with elements called blocks (of size k 1 ; k 2 ; k n ) so that every t set of X(v) is contained in exactly blocks. There is an extensive literature on the covering numbers C(v; k; t) For excellent surveys on the known results we refer to [3] and [5] In this work we continue the search for classes of covering numbers started in [2] 2 Main results Theorem 2.1 If n is a power of an odd prime, then C(n 2 Gamma n; n Gamma 1; 2) n 2 2n: 2 Proof. We start with an affine plane of order n; that is, a resolvable Steiner system ....

D.M. Gordon, G. Kuperberg and O. Patashnik, New Constructions for Covering Designs, Journal of Combinatorial Design 3(1995), 269-284.

....shows that C(n; n Gamma 4; t) 3t 3 Gamma b n 2 c whenever 3t 4 n 4t 4. Observe that Theorems 2, 3 and 4 leave a finite interval of covering numbers undetermined for any fixed values of k and t. For small values of k and t, the missing numbers can be found in the tables presented in [1]. As an example, consider the case t = k = 3. We have that C(n; n Gamma 3; 3) 4 for n 12, by Theorem 2. C(11; 8; 3) 5 by Theorem 3. C(10; 7; 3) 6 by Theorem 4. The remaining values of C(n; n Gamma 3; 3) are found in [1] C(9; 6; 3) 7, C(8; 5; 3) 8, C(7; 4; 3) 12 and C(6; 3; 3) ....

....k and t, the missing numbers can be found in the tables presented in [1] As an example, consider the case t = k = 3. We have that C(n; n Gamma 3; 3) 4 for n 12, by Theorem 2. C(11; 8; 3) 5 by Theorem 3. C(10; 7; 3) 6 by Theorem 4. The remaining values of C(n; n Gamma 3; 3) are found in [1]: C(9; 6; 3) 7, C(8; 5; 3) 8, C(7; 4; 3) 12 and C(6; 3; 3) 20. For the case k = 4 and t = 3, we have C(n; n Gamma 4; 3) 4 for n 16 from Theorem 2; C(n; n Gamma 4; 3) 5 for n = 14; 15 from Theorem 3; and C(n; n Gamma 3; 3) 6 for n = 12; 13 from Theorem 4. The remaining covering ....

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D. M. Gordon, G. Kuperberg and O. Patashnik. New constructions for covering designs, J. Combin. Designs 3 (1995), 269--284.

.... point of view D 7 (32; 8; 4) 620 C 1 (32; 8; 4) D 192 (32; 12; 4) 13888 C 168 (32; 12; 4) D 1883 (32; 16; 4) 36518 C 1847 (32; 16; 4) D 1872 (32; 20; 4) 13888 C 1848 (32; 20; 4) D 189 (32; 24; 4) 620 C 183 (32; 24; 4) Note that the first covering design is the record owner in [15]. Here we know that b 4 is indeed 2. A basis of the invariant space is given by J 3241 ; J 3242 : The denotations of the codes and the information on a t are from [9] CP means an extremal Type II code of length 32. C t a t (C) b t (C) u n Gammat v t ]f(u; v) CP 1 1 1 3 CP 2 1 1 4 CP 3 1 ....

D.M. Gordon, G. Kuperberg, O. Patashnik, New Constructions for Covering Designs, preprint (1995). http://sdcc12.ucsd.edu/ xm3dg/cover.html

....= 5 for 5 2 k n 9 4 k; and C(n; n Gamma k; 2) 6 for 9 4 k n 7 4 k. Observe that Theorems 2, 3 and 4 leave a finite interval of covering numbers undetermined for any fixed values of k and t. For small values of k and t, the missing numbers can be found in the tables presented in [1]. As an example, consider the case t = k = 3. We have that C(n; n Gamma 3; 3) 4 for n 12, by Theorem 2. C(11; 8; 3) 5 by Theorem 3. C(10; 7; 3) 6 by Theorem 4. The remaining values of C(n; n Gamma 3; 3) are found in [1] C(9; 6; 3) 7, C(8; 5; 3) 8, C(7; 4; 3) 12 and C(6; 3; 3) ....

....k and t, the missing numbers can be found in the tables presented in [1] As an example, consider the case t = k = 3. We have that C(n; n Gamma 3; 3) 4 for n 12, by Theorem 2. C(11; 8; 3) 5 by Theorem 3. C(10; 7; 3) 6 by Theorem 4. The remaining values of C(n; n Gamma 3; 3) are found in [1]: C(9; 6; 3) 7, C(8; 5; 3) 8, C(7; 4; 3) 12 and C(6; 3; 3) 20. For the case k = 4 and t = 3, we have C(n; n Gamma 4; 3) 4 for n 16 from Theorem 2; C(n; n Gamma 4; 3) 5 for n = 14; 15 from Theorem 3; and C(n; n Gamma 3; 3) 6 for n = 12; 13 from Theorem 4. The remaining covering ....

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D. M. Gordon, G. Kuperberg and O. Patashnik. New constructions for covering designs, J. Combin. Designs 3 (1995), 269--284.

....such a family B is denoted by C (v; k; t) If = 1, we omit and write C(v; k; t) A more general de nition of covering designs is discussed, for example, in [7, 19, 25] The methods in this paper can straightforwardly be generalized to that approach. For recent surveys of covering designs, see [12, 17, 26]. The problem of determining values of C (v; k; t) is highly nontrivial even for relatively small values of the parameters. Hence, exact values are known only in special cases. For other sets of parameters, there is a gap between the best known lower and upper bounds. Upper bounds are obtained by ....

....Ones In the sequel, some results are discussed where upper bounds (for example, the new bounds from Table III) are used to get other upper bounds. For example, it is not diOEcult to show that C(v; k; t) C(v Gamma 1; k; t) C(v Gamma 1; k Gamma 1; t Gamma 1) 5) In the next few bounds (cf. [12]) the distribution of points aoeects the result. For a covering design with M blocks, let a point occur in M 1 blocks and let a (possibly dioeerent) point be absent from M 0 blocks. For any design, by suitable point selection the following averaging formulas hold: M 0 b(v Gamma k)M=vc; M 1 ....

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D. M. Gordon, O. Patashnik, and G. Kuperberg, New constructions for covering designs, J. Combin. Des. 3 (1995), 269284.

....and Zakharova [103] Bounded distance decoding of Reed Solomon codes beyond d=2 was considered in Dumer [51] Sidelnikov [144] Sudan [153] List cascade decoding and Theorem 3. 27 are due to Zyablov and Pinsker [170] Constructing covering designs is discussed in Furedi [67] Gordon et al. [78], 77] Grable [79] Paper [78] also lists tables of good covering designs for n up to 32. Deterministic construction of coverings with linearly growing k and t is discussed in Dumer [54] Fedorenko [61] 3.4 Soft decision decoding As we have mentioned in the introduction, soft decision ....

....distance decoding of Reed Solomon codes beyond d=2 was considered in Dumer [51] Sidelnikov [144] Sudan [153] List cascade decoding and Theorem 3. 27 are due to Zyablov and Pinsker [170] Constructing covering designs is discussed in Furedi [67] Gordon et al. 78] 77] Grable [79] Paper [78] also lists tables of good covering designs for n up to 32. Deterministic construction of coverings with linearly growing k and t is discussed in Dumer [54] Fedorenko [61] 3.4 Soft decision decoding As we have mentioned in the introduction, soft decision decoders use information about the ....

D. M. Gordon, O. Patashnik, and G. Kuperberg, "New constructions for covering designs," J. Combin. Designs, 3 (4) (1995), 269--284.

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D. M. Gordon, G. Kuperberg and O. Patashnik. New constructions for covering designs, J. Combin. Designs 3 (1995), 269--284.

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D. M. Gordon, O. Patashnik, G. Kuperberg. New Constructions for Covering Designs, Journal of Combinatorial Designs, 3(4) (1995), 269-284.

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D. M. Gordon, O. Patashnik, G. Kuperberg. New Constructions for Covering Designs, Journal of Combinatorial Designs, 3(4) (1995), 269-284.

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