Frequency hopping spread spectrum and direct sequence spread spectrum are two main spread coding technologies in communication systems. Frequency hopping sequences are needed in frequency hopping code-division multiple-access (FH-CDMA) systems. In this paper, four algebraic and a combinatorial constructions of optimal sets of frequency hopping sequences with new parameters are presented, and a number of bounds on sets of frequency hopping sequences are described.

Frequency hopping multiple access (FHMA) spread-spectrum communication systems employing multiple frequency-shift keying (MFSK) as data modulation technique are investigated from a combinatorial approach. A correspondence between optimal frequency hopping (FH) sequences and partition-type difference packings is first established. By virtue of this correspondence, FHMA systems with a single optimal FH sequence each are constructed from various combinatorial structures such as affine geometries, cyclic designs, and difference families. Combinatorial recursive constructions are also presented. Many new infinite series of optimal FH sequences are thus obtained. These new FH sequences are also useful in ultra wideband (UWB) communication systems.

A (v,k,/spl lambda/) optical orthogonal code C is a family of (0,1) sequences of length v and weight k satisfying the following correlation properties: 1) /spl Sigma//sub 0/spl les/t/spl les/v-1/x/sub t/x/sub t+i//spl les//spl lambda/ for any x=(x/sub 0/, x/sub 1/, ..., x/sub v-1/)/spl isin/C and any integer i/spl ne/0(mod v); 2) /spl Sigma//sub 0/spl les/t/spl les/v-1/x/sub t/y/sub t+i//spl les//spl lambda/ for any x=(x/sub 0/, x/sub 1/, ..., x/sub v-1/)/spl isin/C, y=(y/sub 0/, y/sub 1/, ..., y/sub v-1/)/spl isin/C with x/spl ne/y, and any integer i, where the subscripts are taken modulo v. A (v,k,/spl lambda/) optical orthogonal code (OOC) with /spl lfloor/(1/k)/spl lfloor/(v-1/k-2)/spl lfloor/(v-2/k-2)/spl lfloor//spl middot//spl middot//spl middot//spl lfloor/(v-/spl lambda//k-/spl lambda/)/spl rfloor/$: M/spl rfloor//spl rfloor//spl rfloor/ codewords is said to be optimal. OOCs are essential for success of fiber-optic code-division multiple-access (CDMA) communication systems. The use of an optimal OOC enables the largest possible number of asynchronous users to transmit information efficiently and reliably. In this paper, various combinatorial constructions for optimal (v,4,1) OOCs, such as those via skew starters and Weil's theorem on character sums, are given for v/spl equiv/0 (mod 12). These improve the known existence results on optimal OOCs. In particular, it is shown that an optimal (v,4,1) OOC exists for any positive integer v/spl equiv/0 (mod 24).

Optimal (9p,4,1) optical orthogonal codes (OOCs) are constructed for all primes p congruent to 1 modulo 4. Direct constructions with explicit codewords are presented for the case $p \equiv 13 \ \mbox{mod} \ 24$, and Weil's theorem on character sums is used to settle the cases $p \equiv 1,5,17 \ \mbox{mod} \ 24$. By applying a known recursive construction, optimal (9v,4,1)-OOCs are obtained for all v, a product of primes congruent to 1 modulo 4.

A (v, k, /spl lambda//sub a/, /spl lambda//sub c/) optical orthogonal code (OOC) C is a family of (0, 1)-sequences of length v and weight k satisfying the following two correlation properties: (1) /spl Sigma//sub 0/spl les/t/spl les/v-1/x/sub t/x/sub t+i//spl les//spl lambda//sub a/ for any x=(x/sub 0/,x/sub 1/,/spl middot//spl middot//spl middot/,x/sub v-1/)/spl isin/C and any integer i not equivalent 0 mod v; and (2) /spl Sigma//sub 0/spl les/t/spl les/v-1/x/sub t/y/sub t+i//spl les//spl lambda//sub b/ for any x=(x/sub 0/,x/sub 1/,/spl middot//spl middot//spl middot/, x/sub v-1/) /spl isin/ C, y=(y/sub 0/,y/sub 1/,/spl middot//spl middot//spl middot/,y/sub v-1/) /spl isin/C with x/spl ne/y, and any integer i, where the subscripts are taken modulo v. The study of optical orthogonal codes is motivated by an application in optical code-division multiple-access communication systems. In this paper, upper bounds on the size of an optical orthogonal code are discussed. Several new constructions for optimal optical orthogonal codes with weight k/spl ges/4 and correlation constraints /spl lambda//sub a/=/spl lambda//sub c/=1 are described by means of optimal cyclic packings. Many new infinite series of such optimal optical orthogonal codes are thus produced.

In 1982, G.J. Simmons introduced a mathematical model of authentication theory. Recently, E.F. Brickell (1984) and D.R. Stinson (1988) have shown some combinatorial constructions for optimal authentication systems. In the present work, game-theoretical models are proposed for authentication systems that have multilevel costs on sources, and lower bounds of the game values are obtained. Some constructions of the optimal authentication systems that attain the bounds are shown.>