Difference Triangle Sets Research Articles

Construction of LDPC convolutional codes via difference triangle sets

In this paper, a construction of (n,k,delta ) LDPC convolutional codes over arbitrary finite fields, which generalizes the work of Robinson and Bernstein and the later work of Tong is provided. The sets of integers forming a (k, w)-(weak) difference triangle set are used as supports of some columns of the sliding parity-check matrix of an (n,k,delta ) convolutional code, where nin {mathbb {N}}, n>k. The parameters of the convolutional code are related to the parameters of the underlying difference triangle set. In particular, a relation between the free distance of the code and w is established as well as a relation between the degree of the code and the scope of the difference triangle set. Moreover, we show that some conditions on the weak difference triangle set ensure that the Tanner graph associated to the sliding parity-check matrix of the convolutional code is free from 2ell -cycles not satisfying the full rank condition over any finite field. Finally, we relax these conditions and provide a lower bound on the field size, depending on the parity of ell , that is sufficient to still avoid 2ell -cycles. This is important for improving the performance of a code and avoiding the presence of low-weight codewords and absorbing sets.

Monotonic Directed Designs

The notion of a monotonic directed design was introduced to construct difference triangle sets by Chu, Colbourn, and Golomb [SIAM J. Discrete Math., 18 (2005), pp. 741–748]. In this paper, we describe various constructions for monotonic directed designs and establish the necessary and sufficient conditions for the existence of a monotonic directed design with block size 3, and with block size 4 leaving two definite exceptions and six possible exceptions.

Linear Size Optimal $q$-ary Constant-Weight Codes and Constant-Composition Codes

An optimal constant-composition or constant-weight code of weight $w$ has linear size if and only if its distance $d$ is at least $2w-1$. When $d\geq 2w$, the determination of the exact size of such a constant-composition or constant-weight code is trivial, but the case of $d=2w-1$ has been solved previously only for binary and ternary constant-composition and constant-weight codes, and for some sporadic instances. This paper provides a construction for quasicyclic optimal constant-composition and constant-weight codes of weight $w$ and distance $2w-1$ based on a new generalization of difference triangle sets. As a result, the sizes of optimal constant-composition codes and optimal constant-weight codes of weight $w$ and distance $2w-1$ are determined for all such codes of sufficiently large lengths. This solves an open problem of Etzion. The sizes of optimal constant-composition codes of weight $w$ and distance $2w-1$ are also determined for all $w\leq 6$, except in two cases.

Difference Families in Z2d+1⊕Z2d+1 and Infinite Translation Designs in Z⊕Z

We analyse 3-subset difference families of Z2d+1⊕Z2d+1 arising as reductions (mod 2d+1) of particular families of 3-subsets of Z⊕Z. The latter structures, namely perfect d-families, can be viewed as 2-dimensional analogues of difference triangle sets having the least scope. Indeed, every perfect d-family is a set of base blocks which, under the natural action of the translation group Z⊕Z, cover all edges {(x,y),(x′,y′)} such that |x−x′|, |y−y′|≤d. In particular, such a family realises a translation invariant (G,K3)-design, where V(G)=Z⊕Z and the edges satisfy the above constraint. For that reason, we regard perfect families as part of the hereby defined translation designs, which comprise and slightly generalise many structures already existing in the literature. The geometric context allows some suggestive additional definitions. The main result of the paper is the construction of two infinite classes of d-families. Furthermore, we provide two sporadic examples and show that a d-family may exist only if d≡0,3,8,11 (mod 12).

A Recursive Construction For Regular Difference Triangle Sets

A difference triangle set (D$\Delta$S) is a collection of sets of integers having the property that every integer can be written in at most one way as the difference of two elements within a set of the collection. The standard objective is to minimize the largest difference represented, given a specified size of the collection and sizes of the sets that it contains. In order to construct D$\Delta$Ss, we present a new type of combinatorial design, monotonic directed $(v,k,\lambda)$-designs (MDDs). Using MDDs, we give a general recursive construction for difference triangle sets (D$\Delta$Ss). Several instances of this main construction are derived. One of these, the perfect construction, leads to an infinite family of regular (optimal) D$\Delta$Ss if the existence of a single regular D$\Delta$S is known.

Using Directed Hill-Climbing for the Construction of Difference Triangle Sets

This correspondence describes the use of the directed hill-climbing algorithm for finding improved upper bounds of difference triangle sets.

A Tabu Search algorithm for difference triangle sets and Golomb rulers

We propose a Tabu Search heuristic to design optimal Golomb rulers and Difference Triangle Sets. These two highly combinatorial problems have a number of applications in information transmission in telecommunications, and in particular in the construction of convolutional self-orthogonal codes. Our tabu algorithm obtains remarkable results and finds many Difference Triangle Sets with smaller spans than those obtained so far.

A note on the equivalence between strict optical orthogonal codes and difference triangle sets

Zhang (see IEEE Trans. Commun., vol.47, p.967-973, July 1999) proposed a special family of optical address codes, called strict optical orthogonal codes (S-OOCs), was proposed for fiber-optic code-division multiple-access (FO-CDMA) networks. Such codes can strictly guarantee both cross-correlation and autocorrelation functions constrained to have the value one in fully asynchronous data communications and ultra fast switching. In Zhang's work the theory and designs of S-OOC, plus several examples, comparison tables, and performance analyses were presented. In this article, we set up the equivalence between S-OOC and so-called difference triangle sets (DTS), which have been extensively studied previously. Thus, all the known constructions, bounds, and analyses for DTS can be directly applied to S-OOC.

Difference triangle sets from affine planes

This article obtains new difference triangle sets using affine planes. Difference triangle sets have a number of interesting applications in data communications. In these applications, it is desirable to have difference triangle sets with small scopes.

Diffuse difference triangle sets

A particular kind of difference triangle sets (DTSs) called diffuse DTS (DDTS) are considered. Their combinatorial structure is underlying the construction of all known types of self-orthogonal diffuse codes. A number of constructions of DDTS are described, and lower and upper bounds on the maximal element of an optimal DDTS are given. The asymptotic behaviour of the maximal element is studied. Finally, tables of DDTS and of their minimal possible maximal elements are presented.

Improved LP Lower Bounds for Difference Triangle Sets

In 1991 Lorentzen and Nilsen showed how to use linear programming to prove lower bounds on the size of difference triangle sets. In this note we show how to improve these bounds by including additional valid linear inequalities in the LP formulation. We also give some new optimal difference triangle sets found by computer search.

Constructions for difference triangle sets

Difference triangle sets are useful in many practical problems of information transmission. This article studies combinatorial and computational constructions for difference triangle sets having small scopes. Our algorithms have been used to produce difference triangle sets whose scopes are the best currently known.

Further results on difference triangle sets

Further results on the upper bounds for difference triangle sets (DTS) are derived from disjoint difference sets and additive sequences of permutations, which greatly improve the known bounds.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonsymmetric configurations with natural index

The existence of nonsymmetric configurations ( v r , b k ) is discussed here for the first time as a general problem. By using results about resolvable and near-resolvable Steiner systems as well as difference triangle sets, the existence of all configurations with k = 3 is proved. For k⩾4 many infinite series of configurations with natural index are constructed, i.e. configurations where the number of blocks b is a multiple of the number of points v.

Disjoint difference sets, difference triangle sets, and related codes

Disjoint difference sets (DDS), difference triangle sets (DTS), and related codes are discussed and a recursive construction for DDS is given. With this construction and the relationship between DDS and DTS, many new upper bounds for DTS and some better orthogonal codes are obtained. >

Application of linear programming to the optimal difference triangle set problem

Difference triangle sets have a number of applications in telecommunications. The authors describe a linear programming model for finding improved lower bounds on the size of such sets. Two integer linear programming formulations for finding minimal difference triangle sets are presented. The linear programming relaxation of the second formulation is strengthened with a family of valid inequalities and solved to yield improved lower bounds. Numerical results for selected difference triangle sets are given. >

Bounds and construction for difference triangle sets

The definition of difference triangle sets (DTSs) and is short survey of some of their important applications is given. A lower bound on the maximal element in any (I, J)-(DTS) is then given, and a construction of DTSs is described. Finally, tables of the best known lower and upper bounds on optimal DTSs are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Bounds on the size of optimal difference triangle sets

Applications of difference triangle sets are briefly described. New lower and upper bounds on the size of optimal difference triangle sets are given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>