Constructions for Difference Triangle Sets

An (n; k)-difference triangle set, or (n;k)-D1S, is a set X = fX i j1 i ng, where X i = fa ij j0 j kg; for 1 i n; are sets of integers called blocks, such that the differences aij 0 a ij for 1 i n and 0 j 6 = j 0 k, are all distinct and nonzero. An (n;k)-D1S is normalized if for 1 i n, we have 0 = a i0 < a i1 < 11 1 < a ik : All difference triangle sets considered in this correspondence are normalized. The scope of an (n;k)-D1S; X = fX i j1 i ng; is defined as m(X ) = max a a 2 n i=1 Xi :

The smallest scope possible for an (n;k)-D1S is m(n; k) = minfm(X)jX is an (n;k)-D1Sg: An (n;k)-D1S X is optimal if m(X ) = m(n; k): By counting differences, we easily obtain the trivial lower bound m(n; k) n k + 1 2 :

Better lower bounds can be found in the papers of Kløve [1], [2]. In particular, we have the following result [1, Theorem 2]. Theorem 1 (Kløve): For all n and k m(n; k) n k 2 0 2k p k + k + p k 4 : Given an (n; k)-D1S X; we can obtain an (n 0 1;k)-D1S by omitting from X the block containing the largest element. This operation is called a reduction. The operation of omitting the largest element from each block of X is called shortening, and this gives an (n;k 0 1)-D1S instead.

There is a restricted variant of difference triangle sets, called regular perfect systems of difference sets (see [3] and [4]), that is widely studied in combinatorial design theory. Let c be a positive integer. An (n;k)-D1S X = fX i j1 i ng; where X i = faijj0 j kg; is a regular (n;k + 1;c)-PSDS if n i=1 faij 0 a ij j0 j 0 < j kg = c; c + 1;111;c 0 1 + n k + 1 2 :

The existence of a regular (n; k + 1;c)-PSDS gives an (n;k)-D1S of scope c 0 1 + n k+1 2 :

Difference triangle sets have a number of interesting applications in data communications (see [2] and [5]). For all of these applications, difference triangle sets with small scopes are desirable. Hence, the determination of m(n; k) is of importance. Unfortunately, this is a rather difficult problem. The special case of determining m(1; k) is the well-known problem of finding Golomb rulers, which has continued to resist many attacks. Only recently was it shown that m(1; 18) = 246 [6]. In general, except for the following result (see, for example, [5]) on m(n; 1); m(n; 2); and m(n; 3); only finitely many values of m(n; k) are known.

Theorem 2: For n 1; we have m(n; 1) = n and m(n; 2) = 3n;

if n 0 or 1(mod4) 3n + 1; if n 2 or 3(mod4).

There are infinitely many values of n for which m(n; 3) = 6n:

The value of m(n; 1) is trivial. The value of m(n; 2) is folklore and is derived from the existence of Skolem and Langford sequences [7], [8]. The result on m(n; 3) follows from the results of Kotzig and Turgeon [9], and Rogers [10] on the existence of regular (n; 4;1)-PSDS. The following conjecture was made by Bermond [11].

Conjecture 1 (Bermond): For every n 4, we have m(n; 3) = 6n:

Bermond’s conjecture has been verified for 4 n 22 [12].

For each k 2 f1;2;3g; the above results indicate that there exists an n such that m(n; k) meets the trivial lower bound. However, this phenomenon cannot persist, as it was shown in [13], [14] that m(n; k) = n k+1 2 only if k 3; or k = 4 and n is an even integer at least six. The establishment of good bounds for m(n; k) is, therefore, of interest.

Our concern in this correspondence is on the constructive aspects of difference triangle sets. The aim is to provide combinatorial as well as algorithmic constructions for difference triangle sets of small scope, thereby improving some of the existing upper bounds on m(n; k): More information on difference triangle sets can be found in [1], [2], [5], [15], and [16].

In this section, if f and g are two nonnegative functions, we use the notation f g to mean that there is an absolute constant C such that f C g: A (v; k; n)-difference packing, or n-DP (v;k), is a set X = fXij1 i ng, where Xi = faijj1 j kg, for 1 i n, are sets of residues modulo v such that for 1 i; i 0 n; 1 j 6 = j 0 k, and 1 6 = 0 k; we have a ij 0 a ij a i 0 a i (mod v) only if i = i 0 ; j = j 0

; and = 0 : Difference packings and difference triangle sets are intimately related in many ways. In particular, the following observation is made by Chen, Fan, and Jin [16].

0018-9448/97$10.00 © 1997 IEEE Lemma 1 (Chen, Fan, and Jin): An n-DP (v; k) is an (n; k 0 1)-D1S, whose scope is at most v 0 1:

Furthermore, using Singer’s construction [17] of 1-DP(q 2 + q + 1;q + 1) for prime powers q and a technique of Colbourn and Colbourn [18], they constructed two infinite families of difference packings, one of which is given below.

Theorem 3 (Chen, Fan, and Jin): For any prime power q and prime n > q, there exists an n-DP(n(q 2 + q + 1);q + 1):

exists and equals one. Here, we show that the same conclusion holds even if one allows k to grow with n, provided that it does not grow too fast. The following result of Heath-Brown and Iwaniec [19] on differences between consecutive primes is useful. Theorem 4 (Heath-Brown and Iwaniec): Let p n denote

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