Keywords: trades, signed designs, trade-off, trading signed design. 1. .... blocks exist, then the algorithm is more successful in reaching the desirable solution. In.
Designs, Codes and Cryptography, 11, 279–288 (1997)
c 1997 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. °
Trading Signed Designs and Some New 4-(12, 5, 4) Designs G. B. KHOSROVSHAHI Institute for Studies in Theoretical Physics and Mathematics, and Department of Mathematics, University of Tehran, P.O. Box 19395-1795, Tehran, Iran A. NOWZARI-DALINI AND R. TORABI Institute for Studies in Theoretical Physics and Mathematics, University of Tehran, P.O. Box 19395-1795, Tehran, Iran Communicated by: D. Jungnickel Received April 5, 1995; Revised January 12, 1996; Accepted June 25, 1996 Abstract. Variations of the trade-off method exist in the literature of design theory and have been utilized by some authors to produce some t-designs with or without repeated blocks. In this paper we explore a new version of this algorithmic method (i) to produce 20 nonisomorphic and rigid 4-(12, 5, 4) designs, (ii) to study the spectrum of support sizes of 4-(12, 5, 4) designs. Along these, we also present a new design invariant for testing isomorphism among designs and a new way of representing t-designs. Keywords: trades, signed designs, trade-off, trading signed design
1.
Introduction and Notations
Let v, k and t be positive integers satisfying t ≤ k ≤ v. Let λ be a nonnegative integer. Let X be a v-set and denote by Pi (X ) the set of all i-subsets of X , 0 ≤ i ≤ v. We order given ordering. The matrix Ptkv = ( p AB ), for A ∈ Pt (X ) the elements of Pi (X )¡ with ¡some v¢ v¢ and B ∈ Pk (X ) is a t × k (0, 1) matrix whose rows and columns are indexed by the elements of Pt (X ) and Pk (X ), respectively and is defined as follows. ½ p AB =
1 A ⊆ B, 0 otherwise.
Next, we consider the following system of linear equations Ptkv F = λet ,
(∗)
¡¢ where et = (1, 1, · · · , 1)t of size vt . An integral solution of (∗) is called a t-(v, k, λ) signed design. A nonnegative integral solution of (∗) is called a t-(v, k, λ) design (or briefly a t-design). An integral solution of (∗) for λ = 0 is called a t-(v, k) trade. If the components of F belongs to {0, −1, 1}, then the (signed) design or trade is called simple, otherwize it is called (signed) design or trade with repeated blacks.
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It is well known [2,10] that the necessary and sufficient conditions for the existence of a (v−i t−i ) , for i = 0, . . . , t-1, to be integers. These conditions are t-(v, k, λ) signed design is λ k−i ( t−i ) not sufficient for the existence of t-designs. Let F = ( f 1 , f 2 , · · · , f (vk ) ) be a t-design and B be a collection of the elements of Pk (X ) ¡¢ (called blocks) such that every Bi ∈ Pk (X ) for 1 ≤ i ≤ vk appears f i times in B. Clearly, every element of Pt (X ) occurs in B, λ times. In this notation, every t-(v, k, λ) design is denoted by D = (X, B). This¡is the ¢ traditional way of defining a t-design. (X, Pk (X )) is a which is called the trivial design. t-(v, k, λ∗ ) design with λ∗ = v−t k−t Let D = (X, B) be a simple t-(v, k, λ) design. Then D = (X, Pk (X )\B) is a t-(v, k, λ∗ λ) design which is called the complement of D. For any t and k = t + 1, any simple t-(v, k) trade is called minimal if it has 2t+1 nonzero components. The aim of this paper is to discuss a method we call “trading signed designs”. In [5] an algorithm to produce a signed design for any parameters t-(v, k, λ) that satisfy the necessary conditions is given. Our method to produce a t-design from a given signed design consists of killing off the negative blocks of the signed design via augmenting it by minimal trades. By applying this method we were able to construct 20 new nonisomorphic simple 4-(12, 5, 4) designs. To check the nonisomorphism of these designs we employed a new design isomorphism invariant based on minimal trades. In Section 3 we give a report on this. Also, by using the minimal trades we present a new method for the representation of designs. If the number of blocks of a t- (v, k, λ) design is large, then the presentation of the design is a serious problem, unless the design has a rich internal structure. We offer a remedy for this problem too. Finally, we obtain some 4-(12, 5, 4) designs with repeated blocks by exploiting again the method of trade-off. Thus we initiate the study of the spectrum of the support sizes of this family of designs. A comprehensive discussion of trades and the trade-off method is given in [4,5].
2.
Trading Signed Designs: Description of a Method
Let F = ( f 1 , . . . , f (vk ) ) be a t-(v, k, λ) signed design and let T be an arbitrary t-(v, k) trade. Then Ptkv (F + T ) = Ptkv F + Ptkv T = λet . Therefore, F + T is also a t-(v, k, λ) signed design. In general, augmentation of a (signed) design by a trade to obtain a desirable (signed) design is called “trade-off”. The variations of trade-off which are employed here in this research, are in a sense, minimization problems in the context of combinatorial optimization, and the method which is utilized here, is usually phrased as “local search”.
TRADING SIGNED DESIGNS AND SOME NEW 4-(12, 5, 4) DESIGNS
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For a given set of parameters, let S be the set of the solutions of (∗). For F1 , F2 ∈ S, F1 is a neighbour of F2 if F1 − F2 is a minimal trade. The cost function ϕ for our problem can be defined in various forms. Here are some examples: (vk ) X | f i |, ϕ1 (F) = i=1
ϕ2 (F) = −
(vk ) X
fi ,
( fi=1 ) i
