Ryoh Fuji-Hara

Optical orthogonal codes: their bounds and new optimal constructions

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.

Combinatorial constructions of optimal optical orthogonal codes with weight 4

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/

Optimal frequency hopping sequences: a combinatorial approach

: 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 Weils 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).

Sets of Frequency Hopping Sequences: Bounds and Optimal Constructions

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.

Further combinatorial constructions for optimal frequency-hopping sequences

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.

Optimal (9 v , 4, 1) Optical Orthogonal Codes

Frequency-hopping multiple-access (FHMA) spread-spectrum communication systems employing multiple frequency 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 correspondence between 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 for such optimal FH sequences. As a consequence, more new infinite series of optimal FH sequences are obtained.

Computing Logarithms in Finite Fields of Characteristic Two

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

Hyperplane partitions and difference systems of sets

p \equiv 13 \ \mbox{mod} \ 24

The existence of orthogonal resolutions of lines in AG ( n , q )

, and Weils theorem on character sums is used to settle the cases

Complete sets of disjoint difference families and their applications

p \equiv 1,5,17 \ \mbox{mod} \ 24