Yanxun Chang

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/

A new class of optimal optical orthogonal codes with weight five

: 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).

Further results on optimal optical orthogonal codes with weight 4

A (v,k,1) optical orthogonal code (OOC), or briefly a (v, k, 1)-OOC, C, is a family of (0,1) sequences of length v and weight k satisfying the following two properties: 1) /spl Sigma//sub 0/spl les/t/spl les/v-1/x/sub t/x/sub t+i//spl les/1 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/1 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 reduced modulo v. A (v, k,1)-OOC is optimal if it contains /spl lfloor/(v-1)/k(k-1)/spl rfloor/ codewords. In this note, we establish that there exists an optimal (3/sup s/5v, 5,1)-OOC for any nonnegative integer s whenever visa product of primes congruent to 1 modulo 4. This improves the known existence results concerning optimal OOCs.

Constructions for strictly cyclic 3-designs and applications to optimal OOCs with λ=2

Abstract By a ( v , k ,1)-OOC we mean an optical orthogonal code of length v , weight k , and correlation constraints 1. In this paper, we take advantage of the equivalence between such codes and cyclic packings of pairs to make further investigation regarding the existence of a ( v ,4,1)-OOC. It is proved that an optimal ( v ,4,1)-OOC exists whenever v =3 n u with u a product of primes congruent to 1 modulo 4, or v =2 n u with u a product of primes congruent to 1 modulo 6, where n is an arbitrary positive integer and n ≠2 in the case v =2 n u . A strong indication about the existence of an optimal (2 2 u ,4,1)-OOC with u a product of primes congruent to 1 modulo 6 has been given in (M. Buratti, Des. Codes Cryptogr. 26 (2002) 111–125). The results in this paper are obtained mainly by means of a great deal of direct constructions, including using Weils theorem with more than one independent variations.

Constructions of External Difference Families and Disjoint Difference Families

In this paper we give some recursive constructions for strictly cyclic 3-designs. Using these constructions we have some infinite families of strictly cyclic Steiner quadruple systems and optimal optical orthogonal codes with weight 4 and index 2. As corollaries, many known constructions for strictly cyclic Steiner quadruple systems and optimal optical orthogonal codes are unified. We also notice that there does not exist an optimal (n,4,2)-OOC for any n=0 (mod24). Thus we introduce the concept of strictly cyclic maximal packing quadruple systems to deal with the cases of n=0 (mod24) for (n,4,2)-OOCs. By our recursive constructions, some infinite families are also given on strictly cyclic maximal packing quadruple systems.

TWO-DIMENSIONAL BALANCED SAMPLING PLANS EXCLUDING CONTIGUOUS UNITS

Abstract.External difference families (EDFs) are a type of new combinatorial designs originated from cryptography. In this paper, some earlier ideas of recursive and cyclotomic constructions of combinatorial designs are extended, and a number of classes of EDFs and disjoint difference families are presented. A link between a subclass of EDFs and a special type of (almost) difference sets is set up.

Existence of (v; {5 ;w ∗ }; 1)-PBDs

ABSTRACT A balanced sampling plan excluding contiguous units (or BSEC for short) was first introduced by Hedayat, Rao and Stufken in 1988. These designs can be used for survey sampling when the units are arranged in one-dimensional ordering and the contiguous units in this ordering provide similar information. In this paper, we generalize the concept of a BSEC to the two-dimensional situation and give constructions of two-dimensional BSECs with block size 3. The existence problem is completely solved in the case where λ = 1.

Intersection Numbers of Kirkman Triple Systems

Abstract In this paper, we investigate the existence of pairwise balanced designs on v points having blocks of size five, with a distinguished block of size w, briefly (v,{5,w ∗ },1) -PBDs. The necessary conditions for the existence of a (v,{5,w ∗ },1) -PBD with a distinguished block of size w are that v⩾4w+1, v≡w≡1 ( mod 4) and either v≡w ( mod 20) or v+w≡6 ( mod 20) . Previously, w⩽33 has been studied, and the necessary conditions are known to be sufficient for w=1, 5, 13 and 21, with 8 possible exceptions when w⩽33. In this article, we eliminate 3 of these possible exceptions, showing sufficiency for w=25 and 33. Our main objective is the study of 37⩽w⩽97, where we establish sufficiency for w=73, 81, 85 and 93, with 67 possible exceptions with 37⩽w⩽97. For w≡13 ( mod 20) , we show that the necessary existence conditions are sufficient except possibly for w=53,133,293 and 453. For w≡1,5 ( mod 20) , we show the necessary existence conditions are sufficient for w⩾1281,1505, and for w≡9,17 ( mod 20) , we show that w⩾2029,2477 is sufficient with one possible exceptional series, namely v=4w+9 when w≡17 ( mod 20) . We know of no example where v=4w+9. In this article, we also study the 4-RBIBD embedding problem for small subdesigns (up to 52 points) and update some results of Bennett et al. on PBDs containing a 5-line. As an application of our results for w=33 and 97, we establish the smallest number of blocks in a pair covering design with k=5 when v≡1 ( mod 4) with 37 open cases, the largest being for v=489; hitherto, there were 104 open cases, the largest being v=2249.

The flower intersection problem for Kirkman triple systems

LetJR(v) denote the set of all integersksuch that there exists a pair ofKTS(v) with preciselyktriples in common. In this article we determine the setJR(v) forv?3(mod6) (only 10 cases are left undecided forv=15, 21, 27, 33, 39) and establish thatJR(v)=I(v) forv?3(mod6) andv?45, whereI(v)={0, 1, ?, tv?6, tv?4, tv} andtv=16v(v?1).

Constructions of Difference Systems of Sets and Disjoint Difference Families

The flower at a point x in a Steiner triple system is the set of all triples containing x. Denote by the set of all integers k such that there exists a pair of KTS(2r+1) having k+r triples in common, r of them being the triples of a common flower. In this article we determine the set for any positive integer (only nine cases are left undecided for r=7,13,16,19), and establish that for and r⩾22 where J[r]={0,1,…,2r(r−1)/3−6,2r(r−1)/3−4,2r(r−1)/3}.