Further progress on difference families with block size 4 or 5

Variable-weight Optical orthogonal code (OOC) was introduced by G-C Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirement. In this paper, a construction for optimal variable-weight OOCs via cyclic difference families is given. Several new constructions for cyclic difference families are also given. By using these constructions, new optimal (n,W, 1,Q)-OOCs for 2 ≤ |W| ≤ 4 are constructed.

Several direct constructions via skew starters and Weil's theorem on character sum estimates are given in this paper for optimal (gv, 5, 1) optical orthogonal codes (OOCs) where 60 ≤ g ≤ 180 satisfying g ≡ 0 (mod 20) and v is a product of primes greater than 5. These improve the known existence results on optimal OOCs. Especially, we provide an optimal (v, 5, 1)‐OOC for any integer v ≡ 60, 420, 660, 780, 1020, 1140, 1380, 1740 (mod 1800). © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 54–69, 2005.

Optical orthogonal codes (OOCs) have application in OCDMA system because of their good correlation properties. In this paper, two correlation-detection methods, difference method and distance method, were studied. Calculate amount of these correlation-detection methods was analyzed. Constraint condition of optimal optical orthogonal codes was discussed. Besides, a construction method of optimal (n,w,1) optical orthogonal code was presented based on difference set. Optimal (n,w,1) optical orthogonal code which has 2 code words each set was constructed by this construction method. Codewords relativity simulation proved the validity of the method. This construction method has practicability because of codewords' good correlation properties and large capacity.

We prove the existence of a cyclic (4p, 4, 1)-BIBD—and hence, equivalently, that of a cyclic (4, 1)-GDD of type 4p—for any prime p \equiv 1 \pmod 6 such that (p−1)/6 has a prime factor q not greater than 19. This was known only for qe2, i.e., for p \equiv 1 \pmod 12. In this case an explicit construction was given for p \equiv 13 \pmod 24. Here, such an explicit construction is also realized for p \equiv 1 \pmod 24. We also give a strong indication about the existence of a cyclic (4p 4, 1)-BIBD for any prime p \equiv 1 \pmod 6, p>7. The existence is guaranteed for p>(2q3−3q2+1)2+3q2 where q is the least prime factor of (p−1)/6. Finally, we prove, giving explicit constructions, the existence of a cyclic (4, 1)-GDD of type 6p for any prime p>5 and the existence of a cyclic (4, 1)-GDD of type 8p for any prime p \equiv 1 \pmod 6. The result on GDD's with group size 6 was already known but our proof is new and very easy. All the above results may be translated in terms of optimal optical orthogonal codes of weight four with λe1.

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.

We consider the existence problem for a semi-cyclic holey group divisible design of type with block size 3, which is denoted by a 3-SCHGDD of type . When t is odd and or t is doubly even and , the existence problem is completely solved; when t is singly even, many infinite families are obtained. Applications of our results to two-dimensional balanced sampling plans and optimal two-dimensional optical orthogonal codes are also discussed.

The class of geometric orthogonal codes (GOCs) were introduced by Doty and Winslow (2016) for more robust macro-bonding in DNA origami. They observed that GOCs are closely related to optical orthogonal codes (OOCs). It is possible for GOCs to have size greater than OOCs of corresponding parameters due to slightly more relaxed constraints on correlations. However, the existence of GOCs exceeding the size of optimal OOCs of corresponding parameters have never been demonstrated. This paper gives the first infinite family of GOCs of size greater than optimal OOCs.

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.

Difference families from rings

Variable-weight optical orthogonal code (OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. In this paper, four general constructions for optimal cyclic packings and optimal variable-weight OOCs are presented. Many new infinite classes of optimal (v, W,1,Q)-OOCs are obtained. New infinite classes of optimal (v, W,1,Q)-OOCs for |W| ≥ 4 are easily obtained by the constructions.

We prove new existence and nonexistence results for modular Golomb rulers in this paper. We completely determine which modular Golomb rulers of order $k$ exist, for all $k\leq 11$, and we present a general existence result that holds for all $k \geq 3$. We also derive new nonexistence results for infinite classes of modular Golomb rulers and related structures such as difference packings, optical orthogonal codes, cyclic Steiner systems and relative difference families.

Seven infinite classes of relative difference families with variable block sizes are presented explicitly. In particular, a balanced (gv,g,K,1)-DF with g=Σk∈K[(k2-k)/2] is explicitly given for: (i) K={3,4,5} and every v coprime to 6; (ii) K={3,4,6}, {3,5,6} or {3,4,5,6} and every v coprime to 30. As far as the authors are aware, these difference families can be viewed as the first explicit constructions of infinite classes of optimal variable-weight optical orthogonal codes with more than two weights. It is observed, however, that there are infinitely many values of v for which an optimal (v,W,1,Q) -OOC exists, whatever the set of weights W and the weight distribution sequence Q are.

Four direct constructions, three of which are by way of skew starters, are given in this correspondence for optimal (/spl upsi/, 4, 1) optical orthogonal codes (OOC's). These improve the known existence results concerning optimal (/spl upsi/, 4, 1)-OOCs. In particular, it is shown that an optimal (/spl upsi/, 4, 1)-OOC exists for all positive integers /spl upsi//spl equiv/6 (mod 12) or /spl upsi//spl equiv/24 (mod 48). It is also shown that an optimal (12/spl upsi/, 4, 1)-OOC exists for any positive integer /spl upsi/ whose prime factors are all congruent to 1 modulo 4.

Cyclic (W,1,Q;v) Packings and Optimal Variable-weight Optical Orthogonal Codes

Algebraic constructions of asymptotically optimal two-dimensional optical orthogonal codes