One eculiarity with deletion-correcting codes is that erfect t-deletion-correctingcodes of the same length over the same alhabet can have different numbers of codewords, becausethe balls of radius t with resect to the Levenshte˘ın distance may be of different sizes. There isinterest, therefore, in determining all ossible sizes of a erfect t-deletion-correcting code, giventhe length n and the alhabet size q. In this aer, we determine comletely the sectrum ofossible sizes for erfect q-ary 1-deletion-correcting codes of length three for all q, and erfect q-ary2-deletion-correcting codes of length four for almost all q, leaving only a small finite number of casesin doubt.

Motivated by applications in combinatorial group testing for consecutive positives,we investigate a block sequence of a maximum packing MP(t, k, v) which contains the blocks exactlyonce such that the collection of all blocks together with all unions of two consecutive blocks of thissequence forms an error correcting code with minimum distance d. Such a sequence is usually called ablock sequence with consecutive unions having minimum distance d, and denoted by BSCU(t, k, v|d). In this paper, we show that the necessary conditions for the existence of BSCU(3, 4, v|4)s of Steinerquadruple systems, namely, v≡2, 4 (mod 6) and v≥4, are also sufficient, excepting v=8, 10.

Frequency hopping (FH) sequences play a key rolein frequency hopping spread spectrum communication systems.In order to evaluate the performance of FH sequences, Lempeland Greenberger (1974) and Peng and Fan (2004) derived lowerbounds on their Hamming auto- and cross-correlations. In thispaper, we construct families of FH sequences with Hammingcorrelations meeting those bounds by combinatorial and algebraictechniques. We first construct optimal families consisting of asingle FH sequence with maximum Hamming correlation equalto 2 from a combinatorial approach. Then we investigate familiesconsisting of multiple FH sequences. We provide a combinatorialcharacterization for such families, and present a recursive methodto construct them by means of this characterization. We alsodescribe two algebraic constructions for such families of FHsequences, generalizing those of Ding, Moisio, and Yuan (2007). As a consequence, many new optimal families of FH sequencesare obtained.

The concept of group divisible codes, a generalizationof group divisible designs with constant block size, is introduced inthis paper. This new class of codes is shown to be useful in recursiveconstructions for constant-weight and constant-composition codes.Large classes of group divisible codes are constructed which enabledthe determination of the sizes of optimal constant-compositioncodes of weight three (and specified distance), leaving only fourcases undetermined. Previously, the sizes of constant-compositioncodes of weight three were known only for those of sufficiently largelength.

The radar array problem arises from the need to design frequencyhopping sequences with small out-of-phase autocorrelations. It assumesthe reflected signals have negligible Doppler shifts, so the correlationsare calculated along the time axis only. In this correspondence, asystematic construction for radar arrays is provided by means of homogeneousuniform difference matrices. A systematic construction for properlycentered permutation matrices, a special kind of homogeneous uniform differencematrices, is also provided, which partially solves the open problemsposed by Zhang and Tu.

The necessary conditions for existence of a triplewhist tournament TWh(v) are v≡0, 1 (mod 4). Bythe efforts of many authors through a century, these conditions are shown to be sufficient except forv=5, 9, 12, 13 and possibly for v=17. A triplewhist tournament Wh(v) is said to have the three personproperty if any two games in the tournament do not have three common players.We briefly denote sucha design as a 3PTWh(v). In this paper, we extend the known existence result for TWh(v)s and show that thenecessary conditions for existence of a 3PTWh(v), namely, v_8 and v≡0, 1 (mod 4), are also sufficientexcept for v=9, 12, 13 and possibly for v=17.

Frames are useful in dealing with resolvable designs such as resolvable balanced incomplete blockdesigns and triplewhist tournaments. Z-cyclic triplewhist tournament frames are also useful in the constructionsof Z-cyclic triplewhist tournaments. In this paper, the concept of an (h1, h2,..., hn; u)-regularZ-cyclic triplewhist tournament frame is defined, and used to establish several quite general recursive constructionsfor Z-cyclic triplewhist tournaments. As corollaries, we are able to unify many known constructionsfor Z-cyclic triplewhist tournaments. As an application, some new Z-cyclic triplewhist tournamentframes and Z-cyclic triplewhist tournaments are obtained. The known existence results of such designs arethen extended.

Frequency-hopping multiple-access (FHMA) spread-spectrum communication systems employing multiplefrequency shift keying as data modulation technique were investigated by Fuji-Hara,Miao and Mishima[R. Fuji-Hara, Y. Miao, M. Mishima, Optimal frequency hopping sequences: A combinatorial approach,IEEE Trans. Inform. Theory 50 (2004) 2408-2420] from a combinatorial approach, where a correspondencebetween frequency-hopping (FH) sequences and partition-type cyclic difference packings was established,and several combinatorial constructions were provided for FHMA systems with a single optimal FH sequence.In this paper, by means of this correspondence, we describe more combinatorial constructions forsuch optimal FH sequences. As a consequence, more new infinite series of optimal FH sequences are obtained.

In this paper, we construct CCRRS, complete coupling round robin schedules, for n teams eachconsisting of two pairs. The motivation for these schedules is a problem in scheduling bridge tournaments.We construct CCRRS(n) for n a positive integer, n_3, with the possible exceptions ofn ∈ {54, 62}. For n odd, we show that a CCRRS(n) can be constructed using a house with a specialproperty. For n even, a CCRRS(n) can be constructed from a Howell design, H(2n-2, 2n), witha special property called Property P. We use a combination of direct and recursive constructions toconstruct H(2n-2, 2n) with Property P. In order to apply our main recursive construction, we needgroup divisible designs with odd group sizes and odd block sizes. One of our main results is theexistence of these group divisible designs.

SONET/WDM networks using wavelength add-drop multiplexing can be constructed using certaingraph decompositions used to form a grooming', consisting of unions of primitive rings. The cost of such a decompositionis the sum, over all graphs in the decomposition, of the number of vertices of nonzero degree in thegraph. The existence of such decompositions with minimum cost, when every pair of sites employs no more than16 of the wavelength capacity, is determined with a finite number of possible exceptions. Indeed when the numberN of sites satisfies N=1 (mod 3), the determination is complete, and when N=2 (mod 3) the only valueleft undetermined is N=17. When N=0 (mod 3), a finite number of values of Nremain, the largest beingN=2580. The techniques developed rely heavily on tools from combinatorial design theory