Constructions for (2×n,{2,2},1)-2D optical orthogonal codes
Introduction
An optical orthogonal code (OOC), which is initially proposed for the application in optical code-division multiple-access communication systems in 1989 [1], is a family of (0,1) sequences with good auto- and cross-correlation properties. The use of OOCs enables a large number of asynchronous users to transmit information at ultra-high rates safely. The lack of a network synchronization requirement enhances the flexibility of the system. In addition to the optical multiple-access channel, OOCs also find applications in other wide-band code-division multiple-access environments, such as mobile radio, frequency-hopping spread-spectrum communications, and radar and sonar signal design. It receives increasing interest by both the research and industrial communities [2], [3], [4], [5] and their references. Many more potential applications are being actively explored.
An one-dimensional optical orthogonal code (1D OOC, or OOC in short) is a family of (0,1) sequences of length n and Hamming weight k which satisfy the following two properties:
- (1)
The autocorrelation property:for any and any integer τ, .
- (2)
The cross-correlation property:for any and any integer τ, where “⊕” represents modulo n addition.
The numbers and are called the auto- and cross-correlation constraints. Obviously, and cannot exceed k. The high weight of codewords facilitates the detection of the desired signal, and the low auto- and cross-correlations reduce the interference from unwanted signals of synchronization in a network. The size of an optical orthogonal code, denoted by , is the number of codewords in it.
A optical orthogonal code in which , which is called “symmetric”, is briefly denoted by . It is well known that the size of a cannot exceed the so called Johnson bound [1] given byFor this reason, a is said to be optimal when its size reaches this bound.
However, the use of 1D code leads to several drawbacks. First, the low cardinality of 1D code limits the number of users in the network. And the code length of a conventional 1D OOC is always large in order to achieve good bit error rate performance, which occupies a large bandwidth and reduce the bandwidth utilization.
To overcome these shortcomings, two-dimensional (2D) coding techniques, which combine wavelength hopping and time spreading in optical codes, have been proposed [6]. Recently many researchers are working on constructions and designs of 2D OOCs, see [7], [8], [9], [10], [11], [12] and their references.
A 2D optical orthogonal code (2D OOC) is a collection of binary {0,1} m×n matrices, each of Hamming weight k, which satisfy the following two properties:
- (1)
The autocorrelation property:for any and any integer τ, .
- (2)
The cross-correlation property:for any and any integer τ, where “⊕” represents modulo n addition.
For general cases when , the upper bound on 2D OOC is given by [6].
Most researches on 2D OOCs have concentrated on the codes with one-pulse per wavelength (OPPW) or at most one-pulse per wavelength (AM-OPPW). A AM-OPPW 2D OOC is said to be optimal [6] if its size reaches the bound
However, the OPPW or AM-OPPW 2D OOCs are so sparse that reduce the bandwidth utilization, so the more pulses per wavelength cases should be used. In this paper, we consider the (2×n,4,1) 2D OOCs with two-pulses per wavelength, (2×n,{2,2}, 1)-2D OOCs in short), where , and q is an odd prime. They do use the bandwidth more efficiently. The bound is studied in Section 2. The direct construction of optimal such 2D OOCs is presented by using cyclotomy classes in Section 3.
Section snippets
Bound for the maximum possible code size
It is convenient for us to view optical orthogonal codes from a set-theoretical perspective. A OOC can be alternatively considered as a family of 2×2 matrices of integers modulo n, in which each matrix corresponds to a codeword and the integers within each matrix specify to nonzero bits of the 2D codeword. For example, suppose n=13, the three codewords in term of binary {0,1} 2×n matrices are While
The case when n=q
If is an odd prime, then we use θ to represent an arbitrary primitive element of the Galois field . And is the unique multiplicative subgroup of index e in . The notation Cje stands for its multiplicative coset , which is often known as the cyclotomic class of index e (with respect to ).
Suppose q is an odd prime. Let θ be an arbitrary primitive element and e=2, then and Let
Conclusion
In this paper, we have constructed some two-dimensional optical orthogonal codes with two-pulses per wavelength, where n=q, 2q, 3q, 4q and q is an odd prime. They do use the bandwidth more efficiently.
However, there are many possible areas for future study. For instance, whether the parameter n in 2×n can be generalized rather than q, 2q, 3q, 4q? whether the parameter 2×n can be generalized to m×n? And since the codewords with just two wavelengths have limited applications, constructing 2D OOCs
Acknowledgments
The authors would like to thank the anonymous referees for their careful reading of the manuscript and their helpful comments. The work is supported by National Natural Science Foundation of China under Grant (No. 11126291) and NUIST Grant (No. 20110386, No. 2012x021).
References (13)
- et al.
Optical orthogonal codesdesign, analysis and applications
IEEE Transactions on Information Theory
(1989) - et al.
Optimal (4up, 5, 1) optical orthogonal codes
Journal of Combinatorial Designs
(2004) - et al.
Recursive constructions for optimal (n, 4, 2)-OOCS
Journal of Combinatorial Designs
(2004) - et al.
Families of optimal oocs with
IEEE Transactions on Information Theory
(2008) - et al.
New results on optimal (v, 4, 2, 1) optical orthogonal codes
Designs, Codes and Cryptography
(2011) - et al.
Performance comparison of multiwavelength CDMA and WDMA+ CDMA for fiber-optic networks
IEEE Transactions on Communications
(1997)