Determining sets

We give several constructions of determining sets in various settings. We also demonstrate the connections between determining sets, skew karcs and linear codes of minimum distance 5. 1. Motivation. Let P be a non-empty set of elements which we shall call points. Let B be a non-empty set of subsets of P which we shall call blocks. We consider the use of a point/block system S = (P, B) in a message sending scenario in the sense that elements of B will be thought of as messages (on the point set P) relayed from one person or station to another. The set P need not necessarily be finite, but finiteness will be assumed in some of what follows. For example let P = {I, 2,3,4,5,6,}, B = {{I, 2, 3, 5, }, {2, 4}, {l, 3, 6}}. If A decides to send {I, 2, 3, 5} to B as a message, A may instead send the subset {5} or the subset {2,3} for instance, and B can easily establish that {1, 2, 3, 5} was intended. (This is because {I, 2, 3, 5} is the unique block in the message set containing the sets {5} and {2,3}.) Notice that if a set of blocks contains two blocks one of which is a subset of the other, then no subset of the smaller block uniquely determines that block. However, if no block in B is a subset of any other, each block has at least one subset which uniquely determines it simply take the full block. The above motivates the following definitions. The triple (P, B, C) is called a critical system ifP is a non-empty set of elements called points, B a non-empty set of subsets of P called blocks and C a non-empty set of subsets of P called critical sets such that each block contains at least one critical set and each critical set is contained in a unique block. Returning to the message sending scenario, one could ask, why not just use the full block as a message. There are two principal reasons. First of all, in a very large (finite) system, a great deal of space can be saved by transmitting only a small portion of each block. Secondly, as was shown in [4], in a cryptologic setting it is useful to have several choices of critical sets for each block. * Research supported in part by NSERC grant OGP045831 Australasian Journal of Combinatorics 22(2000), pp.167-176 In section 2, we consider critical systems from the point of view of antichains. Section 3 gives recursive constructions for critical systems. The connection between critical systems and blocking sets is facilitated in section 4 by the introduction of determining sets. We show how determining sets can be constructed in Desarguesian projective planes and prove that a planar determining set can be extended to a determining set in higher dimensions. In the final section, we introduce a concept complementary to that of determining set: skew n-arc, and prove that skew n-arcs in PG(m, q) are equivalent to linear codes of minimum distance 5. An improvement to known parameters of such codes is also presented. 2. Antichains. An antichain in a poset (£, f'J) with relation rv is a subset of the elements of £, no two of which are related [1]. When £ is the set of all subsets of a fixed point set P, ordered by inclusion, for any antichain B the triple (P, B, B) is a critical system. Conversely, given a critical system (P, B, C), choose a subset of C' of C such that each block has a unique critical set in C'. Then C' is clearly an antichain. The following well-known result of E. Sperner gives an upper bound on the number of blocks forming an antichain on a fixed point set. Sperner (1928) [22) Let A be an antichain of subsets ofa v-set P. Then IAI :; ([;j)' Moreover, the case of equality occurs precisely when each block has size [¥]. (Here [¥] for v odd is without loss of generality either V;l or ~.) The construction of maximal antichains having vectors with different numbers of l's has been of interest to a number of people. In particular, it may be required that a certain given set of vectors (blocks) appears in the maximal antichain. Many of the algorithms used in the constructions of antichains are based on the following theorem due to Dilworth. Dilworth (1950) [12}. In any poset P, the maximum size of an antichain is equal to the minimum number of chains in a chain decomposition of P. Dantzig and Hoffman [11] then use a linear programming approach to find chain decompositions in finite posets. Anderson [1] gives further information on, and references to, antichains. Note that any t-design with A = 1 forms a critical system where the points and blocks are the same in each system and the critical sets are the t-subsets. Thus each block of size k contains (~) minimal critical sets. 3. Recursive Constructions We first of all give the obvious direct product construction of critical systems on finite point sets. We shall consider a block of a critical system as the corresponding row in a fixed incidence matrix for the system. For Si = (Pi, Bi , Ci ), 1 :; i :; n, a critical system on Vi points, define TI7=1 Bi