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 Bin Wen ISRN Combinatorics , 2013, DOI: 10.1155/2013/479408 Abstract: Frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are needed in FH-CDMA systems. In this paper, a construction of optimal sets of frequency hopping sequences is presented. The construction is based on the set-theoretic characterization of an optimal set of FH sequences. 1. Introduction Frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies. Frequency hopping sequences are an integral part of spread-spectrum communication systems such as FH-CDMA systems (for a description of such systems, see [1]). In modern radar and communication systems, frequency-hopping (FH) spread-spectrum techniques have become popular (see [2], for example). Assume that is a set of available frequencies, called an alphabet. Let be the set of all sequences of length over . Any element of is called a frequency hopping sequence (FHS) of length over . Given two FH sequences, and , we define their Hamming correlation to be where if and if , and where and all operations among position indices are performed modulo . If , is the Hamming autocorrelation. If , is the Hamming cross correlation. 2. Lower Bounds on the Correlations of FHSs FH sequences for FH-CDMA systems are required to have good Hamming correlations and large linear span [3]; the linear span is defined to be the length of the shortest linear feedback shift register that can produce the sequence. FH sequences’ design normally involves six parameters: the size of the frequency library , the sequence length , the family size of the subset , the maximum out-of-phase Hamming autocorrelations , the maximum Hamming cross correlations , and the linear span. It is generally desired that the family of FH sequences has the following properties: (1)the maximum out-of-phase Hamming autocorrelations should be as small as possible,(2)the maximum Hamming cross correlations should be as small as possible,(3)the family size for given , , , and should be as large as possible,(4)the linear span should be as large as possible. In order to evaluate the theoretical performance of the FH sequences, it is important to find some theoretical bounds for these parameters. Given , , and of , Lempel and Greenberger [4] and Peng and Fan [5] derived lower bounds on and of FH sequences in . We restate their results in this section, which will be used later as the criteria to determine whether the new FH sequences constructed in this paper are optimal or not. For any single FH sequence , let be the maximum
 Mathematics , 2011, Abstract: In frequency-hopping multiple-access (FHMA) systems, the average Hamming correlation (AHC) among frequency-hopping sequences (FHSs) as well as the maximum Hamming correlation (MHC) is an important performance measure. Therefore, it is a challenging problem to design FHS sets with good AHC and MHC properties for application. In this paper, we analyze the AHC properties of an FHS set, and present new constructions for FHS sets with optimal AHC. We first calculate the AHC of some known FHS sets with optimal MHC, and check their optimalities. We then prove that any uniformly distributed FHS set has optimal AHC. We also present two constructions of FHS sets with optimal AHC based on cyclotomy. Finally, we show that if an FHS set is obtained from another FHS set with optimal AHC by an interleaving, it has optimal AHC.
 Mathematics , 2010, Abstract: Optimal partitioned cyclic difference packings (PCDPs) are shown to give rise to optimal frequency-hopping sequences and optimal comma-free codes. New constructions for PCDPs, based on almost difference sets and cyclic difference matrices, are given. These produce new infinite families of optimal PCDPs (and hence optimal frequency-hopping sequences and optimal comma-free codes). The existence problem for optimal PCDPs in ${\mathbb Z}_{3m}$, with $m$ base blocks of size three, is also solved for all $m\not\equiv 8,16\pmod{24}$.
 Mathematics , 2010, Abstract: In this paper, using the theory of polynomial residue class rings, a new construction is proposed for frequency hopping patterns having optimal Hamming autocorrelation with respect to the well-known $Lempel$-$Greenberger$ bound. Based on the proposed construction, many new $Peng$-$Fan$ optimal families of frequency hopping sequences are obtained. The parameters of these sets of frequency hopping sequences are new and flexible.
 Computer Science , 2015, Abstract: Frequency hopping sequences (FHSs) with favorable partial Hamming correlation properties have important applications in many synchronization and multiple-access systems. In this paper, we investigate constructions of FHSs and FHS sets with optimal partial Hamming correlation. We first establish a correspondence between FHS sets with optimal partial Hamming correlation and multiple partition-type balanced nested cyclic difference packings with a special property. By virtue of this correspondence, some FHSs and FHS sets with optimal partial Hamming correlation are constructed from various combinatorial structures such as cyclic difference packings, and cyclic relative difference families. We also describe a direct construction and two recursive constructions for FHS sets with optimal partial Hamming correlation. As a consequence, our constructions yield new FHSs and FHS sets with optimal partial Hamming correlation.
 Mathematics , 2006, Abstract: The sum capacity on a symbol-synchronous CDMA system having processing gain $N$ and supporting $K$ power constrained users is achieved by employing at most $2N-1$ sequences. Analogously, the minimum received power (energy-per-chip) on the symbol-synchronous CDMA system supporting $K$ users that demand specified data rates is attained by employing at most $2N-1$ sequences. If there are $L$ oversized users in the system, at most $2N-L-1$ sequences are needed. $2N-1$ is the minimum number of sequences needed to guarantee optimal allocation for single dimensional signaling. $N$ orthogonal sequences are sufficient if a few users (at most $N-1$) are allowed to signal in multiple dimensions. If there are no oversized users, these split users need to signal only in two dimensions each. The above results are shown by proving a converse to a well-known result of Weyl on the interlacing eigenvalues of the sum of two Hermitian matrices, one of which is of rank 1. The converse is analogous to Mirsky's converse to the interlacing eigenvalues theorem for bordering matrices.
 Mathematics , 2013, Abstract: The main results of this paper are generalizations some classical theorems about transversals for families of finite sets to some cases of families of infinite sets.
 Mathematics , 2005, Abstract: A method for constructing sets of sequences with zero-correlation zone (ZCZ sequences) and sequence sets with low cross correlation is proposed. The method is to use families of short sequences and complete orthogonal sequence sets to derive families of long sequences with desired correlation properties. It is a unification of works of Matsufuji and Torii \emph{et al.}, and there are more choices of parameters of sets for our method. In particular, ZCZ sequence sets generated by the method can achieve a related ZCZ bound. Furthermore, the proposed method can be utilized to derive new ZCZ sets with both longer ZCZ and larger set size from known ZCZ sets. These sequence sets are applicable in broadband satellite IP networks.
 Mathematics , 2010, Abstract: Frequency hopping (FH) sequences play a key role in frequency hopping spread spectrum communication systems. It is important to find FH sequences which have simultaneously good Hamming correlation, large family size and large period. In this paper, a new set of FH sequences with large period is proposed, and the Hamming correlation distribution of the new set is investigated. The construction of new FH sequences is based upon Whiteman's generalized cyclotomy. It is shown that the proposed FH sequence set is optimal with respect to the average Hamming correlation bound.
 Computer Science , 2014, Abstract: A subfamily ${\cal F}'$ of a set family ${\cal F}$ is said to $q$-{\em represent} ${\cal F}$ if for every $A \in {\cal F}$ and $B$ of size $q$ such that $A \cap B = \emptyset$ there exists a set $A' \in {\cal F}'$ such that $A' \cap B = \emptyset$. In this paper, we consider the efficient computation of $q$-representative sets for {\em product} families ${\cal F}$. A family ${\cal F}$ is a product family if there exist families ${\cal A}$ and ${\cal B}$ such that ${\cal F} = \{A \cup B~:~A \in {\cal A}, B \in {\cal B}, A \cap B = \emptyset\}$. Our main technical contribution is an algorithm which given ${\cal A}$, ${\cal B}$ and $q$ computes a $q$-representative family ${\cal F}'$ of ${\cal F}$. The running time of our algorithm is sublinear in $|{\cal F}|$ for many choices of ${\cal A}$, ${\cal B}$ and $q$ which occur naturally in several dynamic programming algorithms. We also give an algorithm for the computation of $q$-representative sets for product families ${\cal F}$ in the more general setting where $q$-representation also involves independence in a matroid in addition to disjointness. This algorithm considerably outperforms the naive approach where one first computes ${\cal F}$ from ${\cal A}$ and ${\cal B}$, and then computes the $q$-representative family ${\cal F}'$ from ${\cal F}$. We give two applications of our new algorithms for computing $q$-representative sets for product families. The first is a $3.8408^{k}n^{O(1)}$ deterministic algorithm for the Multilinear Monomial Detection ($k$-MlD) problem. The second is a significant improvement of deterministic dynamic programming algorithms for "connectivity problems" on graphs of bounded treewidth.
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