Constructions for (2×n,{2,2},1)-2D optical orthogonal codes

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Abstract

Optical orthogonal codes (OOCs) have been designed for OCDMA system. 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). 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. A family of two-dimensional optical orthogonal code (2×n,4,1)-2D OOCs with two-pulses per wavelength is proposed in this paper. A tight upper bound on the maximum possible code size and some direct constructions of almost optimal (2×n,{2,2},1)-2D OOCs with n=q,2q,3q,4q, where q is an odd prime, are given.

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 (n,k,λa,λc) one-dimensional optical orthogonal code (1D OOC, or OOC in short) C is a family of (0,1) sequences of length n and Hamming weight k which satisfy the following two properties:

  • (1)

    The autocorrelation property:t=0n1xtxtτλafor any xC and any integer τ, 0<τ<n.

  • (2)

    The cross-correlation property:t=0n1xtytτλcfor any xyC and any integer τ, where “⊕” represents modulo n addition.

The numbers λa and λc are called the auto- and cross-correlation constraints. Obviously, λa and λc 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 |C|, is the number of codewords in it.

A (n,k,λa,λc) optical orthogonal code in which λa=λc=λ, which is called “symmetric”, is briefly denoted by (n,k,λ)-OOC. It is well known that the size of a (n,k,λ)-OOC cannot exceed the so called Johnson bound [1] given byΦ(n,k,λ)=1kn1k1nλkλ.For this reason, a (n,k,λ)-OOC 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 (m×n,k,λa,λc) 2D optical orthogonal code (2D OOC) C 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:i=0m1j=0n1xi,jxi,jτλafor any xC and any integer τ, 0<τ<n.

  • (2)

    The cross-correlation property:i=0m1j=0n1xi,jyi,jτλcfor any xyC and any integer τ, where “⊕” represents modulo n addition.

For general cases when λa=λc, the upper bound on (m×n,k,λ) 2D OOC is given by [6].Φ(m×n,k,λ)m(mn1)(mn2)(mnλ)k(k1)(k2)(kλ).

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 (m×n,k,λ) AM-OPPW 2D OOC is said to be optimal [6] if its size reaches the bound mkn(m1)k1n(mλ)kλ.

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 n=q,2q,3q,4q, 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 (2×n,{2,2},1)-2D 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 10010000000000110000000000,10000000000010000100010000,10000000010000001001000000,While

The case when n=q

If q=ef+1 is an odd prime, then we use θ to represent an arbitrary primitive element of the Galois field GF(q). And C0e={θei:0if1} is the unique multiplicative subgroup of index e in GF(q). The notation Cje stands for its multiplicative coset θjC0e={θie+j:i=0,1,f1}(0je1), which is often known as the cyclotomic class of index e (with respect to GF(q)).

Suppose q is an odd prime. Let θ be an arbitrary primitive element and e=2, then 1=θ(q1)/2 and C02={1,θ2,θ4,,θ2·((q1)/21)}.Let S={{1,θ

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).

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