Frequency-hopping sequences (FHSs) are widely use in recent applications such as Bluetooth, Wi-Fi, and so on. A one-coincidence frequency-hopping sequence (OC-FHS) set consists of FHSs with maximum Hamming autocorrelation 0 and maximum Hamming cross-correlation 1. In this paper, we present a new OC-FHS set of length p2- p over ℤp2, where p is an odd prime. The new OC-FHS set is constructed by using a primitive element of ℤp.

In optical communication systems supporting multiple quality-of-services, variable-weight optical orthogonal codes (VW-OOCs) are employed as spreading codes. In this paper, we present three new families of optimal VW-OOCs with length (q - 1)N and maximum correlation value 1, where q is a prime power and N is a positive integer with gcd(q - 1, N) = 1. These new optimal VW-OOCs can be obtained from optimal constant-weight optical orthogonal codesof length N. Compared with the previously known optimal VW-OOCs, these families can have codewords of higher weights.

The optical orthogonal codes (OOCs) have been widely used as spreading codes in communication systems employing the unipolar transmission. They are classified into constant-weight OOCs (CW-OOCs) and variable-weight OOCs (VW-OOCs) according to the number of distinct Hamming weights which their codewords have. In this paper, we present a new generic construction of VW-OOCs of length (q - 1)N from a CW-OOC of length N, where q is a prime power and gcd(q - 1, N) = 1. As a result, three new families of optimal VW-OOCs with a maximum correlation value 1 are obtained. In particular, these families can have the codewords of high weights, while most of the previously known optimal VW-OOCs have only codewords of weight less than 8.

Frequency-hopping sequences (FHSs) are employed to mitigate the interferences caused by the hits of frequencies in frequency-hopping spread spectrum systems. In this paper, we present two new constructions for FHS sets. We first give a new construction for FHS sets of length nN for two positive integers n and N with gcd(n, N) = 1. We then present another construction for FHS sets of length (q - 1)N, where q is a prime power satisfying gcd(q - 1, N) = 1. By these two constructions, we obtain infinitely many new optimal FHS sets with respect to the Peng-Fan bound as well as new optimal FHSs with respect to the Lempel-Greenberger bound, which have length nN or n(q -1)N. As a result, a great deal of flexibility may be provided in the choice of FHS sets for a given frequency-hopping spread spectrum system.

Short cycles in the Tanner graph of a low-density parity-check (LDPC) code may cause a severe performance degradation. In this paper, we investigate the cycle properties of quasi-cyclic LDPC (QC-LDPC) codes. We first analyze a necessary and sufficient condition for a cycle of a given length to exist, by using the sequence representation of a parity-check matrix for a QC-LDPC code. We then derive bounds which are necessary conditions for a QC-LDPC code to have a given girth in terms of its parameters. Our necessary conditions are applicable to any regular or irregular QC-LDPC codes as well as they improve the existing bounds for many classes of regular QC-LDPC codes.

In this paper, we investigate the partial-period correlation of Zadoff-Chu sequences. For a pair of Zadoff-Chu sequences, we define the linear phase-shifting sequences of one of them and analyze their full-period correlation properties with the other one. By linking them to the partial-period correlation of the given pair, we derive an upper bound on the magnitude of the partial-period correlation of Zadoff-Chu sequences.

In quasi-synchronous frequency-hopping multiple-access systems where relative delays between different users are restricted within a zone around the origin, low-hit-zone frequency-hopping sequences (LHZ-FHSs) are employed as spreading sequences. In this paper, we study LHZ-FHS sets obtained from the Cartesian product of some FHS sets. We first derive an upper bound on the Hamming correlation of FHSs constructed by the Cartesian product. We also give a general method to construct LHZ-FHS sets by the Cartesian product. We then present four new classes of optimal LHZ-FHS sets. The first three classes of sets are obtained from the product of Solomon's FHS sets and Kumar's FHS sets. The last one is constructed by the product of FHS sets based on interleaving techniques. These LHZ-FHS sets have new parameters not covered in the literature.

A mapping from ZNto ZMcan be directly applied for the design of a sequence of period N with alphabet size M, where ZNdenotes the ring of integers modulo N. The nonlinearity of such a mapping is closely related to the autocorrelation of the corresponding sequence. When M is a divisor of N, the sequence corresponding to a perfect nonlinear mapping has perfect autocorrelation, but it is not balanced. In this paper, we study balanced near-perfect nonlinear (NPN) mappings applicable for the design of sequence sets with low correlation. We first construct a new class of balanced NPN mappings from Z(p2-p) to Zpfor an odd prime p. We then present a general method to construct a frequency-hopping sequence (FHS) set from a nonlinear mapping. By applying it to the new class, we obtain a new optimal FHS set of period p2-p with respect to the Peng-Fan bound, whose FHSs are balanced and optimal with respect to the Lempel-Greenberger bound. Moreover, we construct a low-correlation sequence set with size p, period p2-p, and maximum correlation magnitude p from the new class of balanced NPN mappings, which is asymptotically optimal with respect to the Welch bound.

Optical orthogonal codes (OOCs) are widely used as spreading codes in optical fiber networks. An (N, w, λa, λc)-OOC with size L is a family of L {0,1}-sequences with length N, weight w, maximum autocorrelation λa, and maximum cross correlation λc. In this paper, we present two new constructions for OOCs with λa=λc=1 which are asymptotically optimal with respect to the Johnson bound. We first construct an asymptotically optimal (Mpn, M, 1,1)-OOC with size (pn-1)/M by using the structure of Zpn, the ring of integers modulo pn, where p is an odd prime with M|p-1, and n is a positive integer. We then present another asymptotically optimal (Mp1...pk, M, 1,1)-OOC with size (p1...pk-1)/M from a product of k finite fields, where piis an odd prime and M is a positive integer such that M| pi-1 for 1 ≤ i ≤ k. In particular, it is optimal in the case that k=1 and (M-1)2> p1-1.

In this paper, we investigate the cycle properties of quasi-cyclic low-density parity-check (QC-LDPC) codes. Using the sequence representation of a parity-check matrix for a QC-LDPC code, we analyze a necessary and sufficient condition for a cycle of a given length to exist. We then derive bounds which are necessary conditions for a QC-LDPC code to have a given girth in terms of its parameters. We also give a bound which is a sufficient condition for a QC-LDPC code of a given girth to be constructed by a greedy algorithm. The bounds derived here are applicable to any regular or irregular QC-LDPC codes as well as they improve the existing bounds in many classes of regular LDPC codes.

In this paper we investigate the cycle properties of regular quasi-cyclic low-density parity-check (QC-LDPC) codes. Using the sequence representation, we analyze the conditions for short cycles to exist in regular QC-LDPC codes and then give necessary conditions for avoiding short cycles in terms of their parameters. Numerical results show that our bounds on the number of rows in the parity-check matrices for regular QC-LDPC codes without cycles of length 4 or 6 are tighter than any other known bounds.

Optical orthogonal codes (OOCs) are widely used as spreading codes in optical fiber networks. For reliable and efficient data transmission in these systems, it is required to design OOCs with low correlation and large size. In this paper, we present two new constructions for asymptotically optimal OOCs with respect to the Johnson bound, whose maximum nontrivial correlation value is 1. We first construct an asymptotically optimal OOC of length (p - 1)p2for an odd prime p by using the structure of the ring of integers modulo p2. We then present an asymptotically optimal OOC of length Mp1p2from a product of two finite fields, where p1and p2are odd primes, and M is an integer such that M|pi- 1 for i = 1, 2.

In frequency-hopping multiple-access systems, the average Hamming correlation (AHC) among frequency-hopping sequences (FHSs) as well as the maximum Hamming correlation (MHC) is an important performance measure. Moreover, each FHS is required to be balanced for its robustness against jamming or fading environments. In this study, the authors investigate FHS sets with optimal AHC and (near-)optimal MHC, whose FHSs are balanced. The authors first show that any uniformly distributed FHS set has optimal AHC with respect to the Peng–Niu–Tang bound. The authors also present two classes of FHS sets with optimal AHC and (near-)optimal MHC, whose FHSs are (perfectly) balanced. The authors then analyse the AHC of FHS sets constructed by interleaving techniques, and present some new FHS sets with optimal AHC and MHC, whose FHSs are perfectly balanced.

For an integer k ≥ 1, let , 1≤ qi ≤ k,be prime powers such that qi - Mif + 1 for some integers Miand f. In this paper, the k-fold cyclotomy of Fqk× ⋯ × Fqkas a nontrivial generalization of the conventional cyclotomy (k = 1 case) and its application to frequency-hopping sequences (FHSs) are presented, where Fqis the finite field with q elements. First, the definitions of k-fold cyclotomic classes and k-fold cyclotomic numbers are given. And then, their basic properties including k-fold diagonal sums are derived. Based on them, new optimal FHS sets of length N and frequency set size M or M + 1 with respect to the Peng-Fan bound are constructed for a product N of distinct odd primes and a di visor M of N - 1. Furthermore, new optimal FHSs of length N and frequency set size M with respect to the Lempel-Greenberger bound are constructed when N has at least one prime factor which is 3 modulo 4 and (N - 1)/M is an even integer. Our constructions give several new optimal parameters not covered in the literature, which are summarized in Table I.

For an integer k ≥ 1, k-fold cyclotomic numbers of Fq1× ... × Fqkare introduced, where Fqis the finite field with q elements and qi's are powers of distinct primes. They are a generalization of the conventional cyclotomic numbers (k = 1 case). Some of their basic properties including k-fold diagonal sums are derived. As an application of the k-fold cyclotomy, frequency-hopping sequences (FHSs) of length p1... pkare constructed for distinct odd primes p1, ..., pk, which are optimal with respect to the Lempel-Greenberger bound and the Peng-Fan bound.

A frequency-hopping sequence (FHS) of lengthvand frequency set sizeMis called a(v,M,¿)-FHS if its maximum out-of-phase Hamming autocorrelation is¿. Three new classes of optimal FHSs with respect to the Lempel-Greenberger bound are presented in this paper. First, new optimal (p,M,f)-FHSs are constructed when p = Mf +1 is an odd prime such thatfis even and p ¿ 3 mod 4 . And then, a construction for optimal (kp,p,k)-FHSs is given for any odd prime p and a positive integer K1,p1(p1+ 2 ),2m-1,or p1m-1, where p1and p1+2 are odd primes. Finally, several new optimal FHSs with maximum out-of-phase Hamming autocorrelation 1 or 2 are also presented. In particular, the existence of optimal (v,N,1)-FHSs is proven for any integer N ¿ 3 and any integer v with N +1 ¿ v ¿ 2 N-1, as well as the existence of optimal (2N +1,N,2)-FHSs is shown for any integer N ¿ 3. These classes of optimal FHSs have new parameters which are not covered in the literature.

In this paper we present a method to construct low correlation zone (LCZ) sequence sets of period kN for some integers k and N which are relatively prime. We construct kN-periodic LCZ sequences by combining an N-periodic sequence having good autocorrelation with a k times k Hadamard matrix. We also give some examples of our construction, which are optimal or nearly optimal with respect to the Tang-Fan-Matsufuji bound. Our construction gives flexible parameters in the sense of period and LCZ size.

In this paper we construct new classes of optimal frequency-hopping sequences (FHSs) with respect to the Lempel-Greenberger bound and the Peng-Fan bound by interleaving techniques which are used to construct a sequence of length kN from k sequences of length N. We first give two generic constructions for optimal FHS sets from some known optimal FHS sets by interleaving techniques and present some examples of new optimal FHS sets. We then design new optimal FHSs whose parameters include those of the known optimal constructions with length kN and frequency set size n for some positive integers k and N. We also construct optimal FHS sets of length kp from power-residue sequences for any odd prime p and 2 les k < p. In particular, our constructions give several new parameters not covered in the literature, which are summarized in I and II.

In this correspondence, we present new construction methods for quaternary low-correlation zone (LCZ) sequence sets from a binary sequence with good autocorrelation. We show that the sets obtained by our methods are optimal or nearly optimal with respect to the Tang–Fan–Matsufuji bound and that our construction methods are more flexible than any other previous constructions in the sense of period, family size, and zone size. We also give a construction method for a quaternary LCZ sequence set from a binary LCZ sequence set. Finally, we give a new construction method for quaternary Hadamard matrices by the inverse of the Gray map.

In this correspondence, we study the statistical stability properties of pm-periodic binary sequences in terms of their linear complexity and k-error linear complexity, where p is n prime number and 2 is a primitive root modulo p2. We show that their linear complexity and k-error linear complexity take a value only from some specific ranges. We then present the minimum value k for which the k-error linear complexity is strictly less than the linear complexity in a new viewpoint different from the approach by Meidl. We also derive the distribution of pm-periodic binary sequences with specific k-error linear complexity. Finally, we get an explicit formula for the expectation value of the k-error linear complexity and give its lower and upper bounds, when k les [p/2].