D.K. Ray-Chaudhuri and R.M. Wilson, Solution of Kirkman's schoolgirl problem, Proc. Sympos. Pure Math. 19, American Math. Society, Providence RI (1971), pp. 187-203.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....R i T j is either zero or one for all 1 i; j r. When a system has two orthogonal resolutions, it is doubly resolvable. Kirkman [8] first asked about the existence of Kirkman triple systems in 1850, and solved the case when v = 15 (the Kirkman 15 schoolgirl problem) Ray Chaudhuri and Wilson [13] published the first solution to the existence question for KTSs for all v 3 (mod 6) There is a unique STS(9) up to isomorphism, and it is resolvable. Indeed, it underlies a unique KTS(9) Of the eighty nonisomorphic STS(15)s, four are resolvable; together they underlie seven nonisomorphic ....

D.K. Ray-Chaudhuri and R.M. Wilson, Solution of Kirkman's schoolgirl problem, Proc. Sympos. Pure Math. 19, American Math. Society, Providence RI (1971), pp. 187-203.

....R i T j is either zero or one for all 1 i; j r. When a system has two orthogonal resolutions, it is doubly resolvable. Kirkman [6] rst asked about the existence of Kirkman triple systems in 1850, and solved the case when v = 15 (the Kirkman 15 schoolgirl problem) Ray Chaudhuri and Wilson [10] published the rst solution to the existence question for KTSs for all v 3 (mod 6) There is a unique STS(9) up to isomorphism, and it is resolvable. Indeed, it underlies a unique KTS(9) Of the eighty nonisomorphic STS(15)s, four are resolvable; together they underlie seven nonisomorphic ....

D.K. Ray-Chaudhuri and R.M. Wilson, Solution of Kirkman's schoolgirl problem, Proc. Sympos. Pure Math. 19, American Math. Society, Providence RI (1971), pp. 187-203.

....; R v Gamma1 2 such that each R i partitions the set V . In other words, 1. R 1 ; R 2 ; R v Gamma1 2 B. 2. R i T R j = for i 6= j. 3. If x 2 V then x appears in exactly one triple of each R i . A resolvable Steiner triple system is a Kirkman triple system. Theorem 2.1. [11] There is a Kirkman triple system of order v if and only if v j 3 (mod 6) We use this theorem to prove: Theorem 2.2. Let (V; B) be a TS(v; 1) with v j 3 (mod 6) Then (V; B) can be enclosed in a TS(v s; 2) whenever 0 s (v Gamma 1) 2 and s j 0; 1 (mod 3) Proof. Let (V; B 0 ) be a KTS(v) ....

Ray-Chaudhuri D. K. and Wilson R. M., Solution of Kirkman's school-girl problem, Proc. Symp. Pure Math. 19, Amer. Math. Soc. Providence, pp. 187--203.

....the two properties coincide for dimensions corresponding to STSs. A signal set can be developed from an STS by partitioning the triples into some partial parallel classes of the required size. Table III is an example of STS with 13 elements. The sets of triples f[0, 1, 4] 5, 7, 12] [8, 10, 2], 11, 9, 3]g, f[2, 3, 6] 7, 9, 1] 10, 12, 4] 11, 0, 5]g and f[7, 8, 11] 12, 1, 6] 2, 4, 9] 3, 5, 10]g are some partial parallel classes forming a signal set (which is not optimal) Experimental work shows that such a design approach can be very effective in constructing high ....

....A signal set can be developed from an STS by partitioning the triples into some partial parallel classes of the required size. Table III is an example of STS with 13 elements. The sets of triples f[0, 1, 4] 5, 7, 12] 8, 10, 2] 11, 9, 3]g, f[2, 3, 6] 7, 9, 1] 10, 12, 4] 11, 0, 5]g and f[7, 8, 11], 12, 1, 6] 2, 4, 9] 3, 5, 10]g are some partial parallel classes forming a signal set (which is not optimal) Experimental work shows that such a design approach can be very effective in constructing high dimensional MTMFSK multiplexing signal sets, mostly producing optimal designs [12] ....

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D. K. Ray-Chaudhuri, Richard M. Wilson "Solution of Kirkman's Schoolgirl Problem", Proceedings of the Symposium on Mathematics XIX, 1971, pp. 187-203, American Mathematical Society.

....case, a TS(6t 3; 1) exists having (2t 1) 3t 1) blocks. A partial parallel class of m = 2t 1 triples is a parallel class, and hence we require a resolvable TS(6t 3; 1) such designs are called Kirkman triple systems. Their existence for all t 0 is established by Ray Chaudhuri and Wilson [12]. The existence of Kirkman triple systems yields MKSS(6t 3)s for all t 0. These signal sets are balanced; indeed every element occurs in the same number of triples. When d = 2, a maximum PTS(v; 1) has (2t) 3t 1) blocks, and so s 3t 1 when m = 2t. Then starting with a Kirkman triple system ....

D.K. Ray-Chaudhuri and R.M. Wilson, Solution of Kirkman's school-girl problem, Proc. Symp. Pure Math., 19, Amer. Math. Soc., Providence, R.I., 1971, 187-203.

....13 For 2 k 4, an S(2; k; v) can be partitioned into S(1; k; v) if and only if v j k mod k(k Gamma 1) The case k = 2 is trivial. In fact, a resolvable S(2; 2; v) is a one factorization of K v , the complete graph on v vertices. The proof of Proposition 13 for the case k = 3 can be found in [30] (this is the well known Kirkman s schoolgirl problem) whereas for k = 4 see [20] 23 Proposition 14 ( 26, 27, 40] An S(3; 3; v) can be partitioned into S(2; 3; v) if and only if v 6= 7 and either v j 1 mod 6 or v j 3 mod 6. Proposition 15 ( 1, 43] For any n 2 an S(3; 4; 4 n ) can be ....

D. K. Ray-Chaudhuri and R. M. Wilson, Solution of Kirkman's Schoolgirl Problem, Amer. Math. Soc. Proc. Symp. Pure Math., Vol. 19 (1971), pp. 187--204.

....of the infinite points, then we have a 4 GDD of type t u (t(u Gamma 1) 2) 1 . Conversely, from a 4 GDD of this type, one can obtain a resolvable 3 GDD. The problem of resolvable 3 GDDs was first solved in the cases t = 1; 3 (as Kirkman triple systems) by Ray Chaudhuri and Wilson [14]; and next in the case t = 2 (as nearly Kirkman triple systems) by Kotzig and Rosa [12] Baker and Wilson [2] Brouwer [5] and Rees and Stinson [16] The problem for general t was studied by Rees and Stinson [16] Assaf and Hartman [1] and then completed by Rees [15] The necessary conditions for ....

D. K. Ray-Chaudhuri and R. M. Wilson. Solution of Kirkman's school-girl problem. AMS Proc. Symp. Pure Math. 19 (1971), 187--203.

....shows that the problem of constructing [n; c; 3; 4] ERC, c j 3 (mod 6) with optimal check disk overhead, having a 1 balanced ordering is equivalent to the following problem. Problem 8. 7 Determine those v for which there exists an anti Pasch KTS(v) The existence of KTS(v) has long been settled [27]; the condition v j 3 (mod 6) is both necessary and sufficient. Work on the existence problem for anti Pasch STS(v) is also well under way. However, Problem 8.7 appears not to have been studied, perhaps due to the lack in motivation. This is not the case now. We settle here the existence of ....

D. K. Ray-Chaudhuri and R. M. Wilson. Solution of Kirkman's schoolgirl problem. Proc. Symp. Pure Math. Amer. Math. Soc., 19:187--204, 1971.

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