Optimal Optical Orthogonal Codes Research Articles

Retracted: A Family of Optimal Multiple-Weight Optical Orthogonal Codes for Fiber-Optic Networks.

[This retracts the article DOI: 10.1155/2022/2499606.].

New Constructions of Optimal Optical Orthogonal Codes Based on Partitionable Sets and Almost Partitionable Sets

Variable-weight optical orthogonal codes (OOCs) were introduced by Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. In this paper, partitionable sets and almost partitionable sets are used to construct variable-weight OOCs. Four new recursive constructions for optimal cyclic packing are presented. Consequently, new infinite classes of optimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(v,W,1,Q)$ </tex-math></inline-formula> -OOCs are obtained.

Constructions of optimal two-dimensional optical orthogonal codes with AM-OPPW restriction for λ = 2

Two-dimensional optical orthogonal codes (2-D OOCs) are of current practical interest as they enable optical communication at a lower chip rate. In this paper, we are concerned about optimal 2-D (u×v,k,2)-OOCs with at most one-pulse per wavelength (AM-OPPW) restriction. An improved upper bound on the size of optimal AM-OPPW 2-D (u×v,k,2)-OOCs is derived, some combinatorial constructions are presented and many new infinite families of optimal AM-OPPW 2-D (u×v,4,2)-OOCs are obtained.

A few more optimal optical orthogonal codes with non-constant auto-correlation function

A few more optimal optical orthogonal codes with non-constant auto-correlation function

A Family of Optimal Multiple-Weight Optical Orthogonal Codes for Fiber-Optic Networks.

Optical orthogonal codes (OOCs) were designed for multimedia optical CDMA systems with quality of service requirements in optical fiber networks. Two-dimensional (2-D) multiple-weight optical orthogonal codes have been invested as they can overcome the drawbacks of nonlinear effects in large spreading sequences. In this paper, we reveal the combinatorial properties of optimal 2-D OOCs and focus our attention on the constructions for a family of optimal 2-D multiple-weight optical orthogonal codes by combinatorial methods, such as incomplete difference matrix, h-perfect cyclic packing, and skew starter. In particular, an improved construction of skew starters with multiple weights is also proposed to solve the existence of optimal multiple-weight optical orthogonal codes. Our numerical examples demonstrate that the proposed construction is very helpful for optimizing the utilization of optical network effectively.

Constructions of Optimal Optical Orthogonal Codes with Weights Set 5,7 via Cyclic Packing

Double-weight optical orthogonal codes are variable-weight optical orthogonal codes (OOCs), which have been widely applied in optical networks and systems. Some works have been devoted to optimal n , W , 1 , Q -OOCs with max w : w ∈ W ≤ 6 . So far, there is no explicit construction of optimal n , W , 1 , Q -OOCs with W = 5,7 . It is known that heavier-weight codewords have better code performance than lighter-weight codewords. So, in this paper, we use cyclic packing to construct two infinite classes of optimal OOCs with weights set 5,7 explicitly, for any prime p ≡ 3 mod 4 and p ≥ 7 . In addition, for 1 ≤ t &lt; p − 1 / 2 , by breaking t blocks of size 7 into 3 of 31 p , 5,7 , 1 , 1 / 2 , 1 / 2 -OOCs and 41 p , 5,7 , 1 , 2 / 3 , 1 / 3 -OOCs, we obtain new infinite classes of optimal 31 p , 3,5,7 , 1 , 7 t / p − 1 + 6 t , p − 1 / 2 p − 1 + 6 t , p − 1 − 2 t / 2 p − 1 + 6 t -OOCs and 41 p , 3,5,7 , 1 , 14 t / 3 p − 1 + 4 t , 2 p − 1 / 3 p − 1 + 4 t , p − 1 − 2 t / 3 p − 1 + 4 t -OOCs, respectively.

On optimal (Z6m×Z6n,4,1) and (Z2m×Z18n,4,1) difference packings and their related codes

AbstractWe give a direct construction of a relative difference family for and every prime with . These allow us to construct an optimal difference packing and an optimal difference packing for every pair of positive integers . The corresponding optimal optical orthogonal signature pattern codes are also obtained.

A mixed construction of variable-weight optical orthogonal code for optical code division multiple access networks

Optical orthogonal codes play an important role in optical code-division multiple access systems. In order to support heterogeneous quality of service, it is of interests to employ optical orthogonal codes in which different codewords may have different Hamming weights. In this paper a construction method of variable-weight optical orthogonal codes combining meta-heuristic search algorithm and combinatorial block designs is studied. This method is capable of producing variable-weight optical orthogonal codes with large and flexible parameters. We construct three infinite families of optimal variable-weight optical orthogonal codes using the proposed method.

A New Construction of Optimal Optical Orthogonal Codes From Sidon Sets

Two new constructions for families of optical orthogonal codes are presented. The first is a generalization of the well-known construction of Sidon sets given by I. Z. Ruzsa. The second construction is optimal with respect to the Johnson bound, and its parameters (n, w, λ) are respectively (p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h+1</sup> - p, p, 1), where p is any prime, h is an integer greater than 1 and the family size is p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h-1</sup> + p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h-2</sup> + ⋯ + p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> + p.

Strong difference families of special types

Strong difference families of special types are introduced to produce new relative difference families from the point of view of both asymptotic existences and concrete examples. As applications, group divisible designs of type 30u with block size 6 are discussed, r-rotational balanced incomplete block designs with block size 6 are derived for r∈{6,10}, and several classes of optimal optical orthogonal codes with weight 5, 6, 7, or 8 are obtained.

New Asymptotically Optimal Optical Orthogonal Signature Pattern Codes from Cyclic Codes

New Asymptotically Optimal Optical Orthogonal Signature Pattern Codes from Cyclic Codes

Bounds and constructions of optimal optical orthogonal codes with low correlation zone

Bounds and constructions of optimal optical orthogonal codes with low correlation zone

On silver and golden optical orthogonal codes

It is several years that no theoretical construction for optimal (v, k, 1) optical orthogonal codes (OOCs) with new parameters has been discovered. In particular, the literature almost completely lacks optimal (v, k, 1)-OOCs with k > 3 which are not regular. In this paper we will show how some elementary difference multisets allow to obtain three new classes of optimal but not regular (3p, 4, 1)-, (5p, 5, 1)-, and (2p, 4, 1)-OOCs which are describable in terms of Pell and Fibonacci numbers. The OOCs of the first two classes (resp. third class) will be called silver (resp. golden) since they exist provided that the square of a silver element (resp. golden element) of ℤp is a primitive square of ℤp.

Geometric Orthogonal Codes of Size Larger Than Optical Orthogonal Codes

The class of geometric orthogonal codes (GOCs) was introduced by Doty and Winslow (2016) for more robust macrobonding 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 has never been demonstrated. This paper gives the first infinite family of GOCs of size greater than optimal OOCs.

Combinatorial constructions of optimal (m, n, 4, 2) optical orthogonal signature pattern codes

Optical orthogonal signature pattern codes (OOSPCs) play an important role in a novel type of optical code-division multiple-access (CDMA) network for 2-dimensional image transmission. There is a one-to-one correspondence between an $(m, n, w, \lambda)$-OOSPC and a $(\lambda+1)$-$(mn,w,1)$ packing design admitting an automorphism group isomorphic to $\mathbb{Z}_m\times \mathbb{Z}_n$. In 2010, Sawa gave the first infinite class of $(m, n, 4, 2)$-OOSPCs by using $S$-cyclic Steiner quadruple systems. In this paper, we use various combinatorial designs such as strictly $\mathbb{Z}_m\times \mathbb{Z}_n$-invariant $s$-fan designs, strictly $\mathbb{Z}_m\times \mathbb{Z}_n$-invariant $G$-designs and rotational Steiner quadruple systems to present some constructions for $(m, n, 4, 2)$-OOSPCs. As a consequence, our new constructions yield more infinite families of optimal $(m, n, 4, 2)$-OOSPCs. Especially, we shall see that in some cases an optimal $(m, n, 4, 2)$-OOSPC can not achieve the Johnson bound.

New Families of Optimal Variable-Weight Optical Orthogonal Codes With High 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.

Combinatorial Constructions of Optimal Three-Dimensional Optical Orthogonal Codes

In this paper, we study three-dimensional (u x v x w, k, lambda) optical orthogonal codes (OOCs) with at most one optical pulse per wavelength/time plane (AM-OPP) restriction, which is denoted by AM-OPP 3-D (u x v x w, k, lambda)-OOC. We build an equivalence relation between such an OOC and a certain combinatorial subject, called a w-cyclic group divisible packing of type (vw)(u). By this link, the upper bound of the number of codewords is improved and some new combinatorial constructions are presented. As an application, the exact number of codewords of an optimal AM-OPP 3-D (uxvxw, 3, 1)-OOC is determined for any positive integers v, w, and u (sic) 2 (mod 6) with some possible exceptions.

Semi-cyclic Holey Group Divisible Designs and Applications to Sampling Designs and Optical Orthogonal Codes

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.

Optimal three-dimensional optical orthogonal codes of weight three

Using polarization technique in optical code division multiple access, we can schedule the transmission of optical pulses in spatial domain, in addition to the frequency domain and time domain. An optical orthogonal code (OOC) which spreads in these dimensions is called a three-dimensional (3-D) OOC. In this paper, we study 3-D OOC with at most one optical pulse per wavelength/time plane, which have the favorable property that the Hamming auto-correlation is identically equal to 0. An upper bound on the number of codewords for general Hamming cross-correlation requirement is given. A 3-D OCC with at most one pulse per wavelength/time plane and Hamming cross-correlation no more than 1 is shown to be equivalent to a generalized Bhaskar Rao group divisible design (GBRGDD), signed over a cyclic group. Through this equivalence, necessary and sufficient conditions for the existence of GBRGDD of weighted 3, signed over a cyclic group, are derived.

Semi-cyclic holey group divisible designs with block size three

In this paper we discuss the existence problem for a semi-cyclic holey group divisible design of type (n,m^t) with block size 3, which is denoted by a 3-SCHGDD of type (n,m^t). When n=3, a 3-SCHGDD of type (3,m^t) is equivalent to a (3,mt;m)-cyclic holey difference matrix, denoted by a (3,mt;m)-CHDM. It is shown that there is a (3,mt;m)-CHDM if and only if (t-1)m\equiv 0 (mod 2) and t\geq 3 with the exception of m\equiv 0 (mod 2) and t=3. When n\geq 4, the case of t odd is considered. It is established that if t\equiv 1 (mod 2) and n\geq 4, then there exists a 3-SCHGDD of type (n,m^t) if and only if t\geq 3 and (t-1)n(n-1)m\equiv 0 (mod 6) with some possible exceptions of n=6 and 8. The main results in this paper have been used to construct optimal two-dimensional optical orthogonal codes with weight 3 and different auto- and cross-correlation constraints by the authors recently.