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

Abstract

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 -OOCs with . So far, there is no explicit construction of optimal -OOCs with . 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 explicitly, for any prime and . In addition, for , by breaking blocks of size 7 into 3 of -OOCs and -OOCs, we obtain new infinite classes of optimal -OOCs and -OOCs, respectively.

1. Introduction

In 1989, Salehi [1] defined the constant-weight optical orthogonal codes (OOCs), and applied them in the optical code division multiple access (OCDMA) network. Later on, Yang et al. [2] introduced the variable-weight optical orthogonal codes (variable-weight OOCs) in order to meet the requirements of differential quality-of-service (QoS). That is, the codewords of low code weight are assigned to the low-QoS (quality-of-services) requirement applications and high code weight codewords are assigned to high-QoS requirement applications. Because of its desirable features, OOCs have found wide range of applications in many communications fields, such as mobile radio, frequency-hopping spread-spectrum communications, radar, sonar signal design, and so on [3–7].

In [8], suppose is an ordering of a set of integers greater than 1 and is an -tuple (autocorrelation sequence) of positive integers. Moreover, suppose that is a positive integer (cross-correlation parameter) and is an -tuple (weight distribution sequence) of positive rational numbers whose sum is 1. Then, an optical orthogonal code (OOC) (briefly, -OOC) is a set of subsets (called codeword-sets) of with sizes (weights) from satisfying the following three properties:(1)Weight distribution property: the number of codewords with weight is exactly , , and , where (2)The autocorrelation property: for any with weight and (3)The cross-correlation property: for any , with and

If , then the notation -OOC is used to denote -OOC. is normalized if it is written in the form with . And, an -OOC is said to be balanced if .

If and , an -OOC is a conflict-avoiding code [9].

It is well known that constructing the optimal OOCs with and minimal is preferred due to its correlation property. Namely, the smaller the value of the correlation is, the smaller the interference is.

The following lemma provides an upper bound on the size of an -OOC.

Lemma 1. ([8]) If is an -OOC with and normalized weight distribution sequence , then we have

An -OOC is optimal if reaches the upper bound.

2-CP is a generalization of 2- packing. It is a useful tool for constructing -OOC. The details of the concept of 2-CP were introduced in [10].

Let be an Abelian group, and assume that , , and . Define , , and , where , , and are multisets.

Let be a family of subsets of , , . A 2-CP (W,1,n) is (base blocks set), such that the differences in , i.e., , cover each element of at most once. A 2-CP is a 2-CP with the property that the number of blocks of size is , where , .

A 2-CP is g-regular if covers each element of exactly once, and each element of is not covered.

The following results are the relationships between a 2-CP and an -OOC.

Lemma 2. An optimal 2- is equivalent to an optimal -OOC.

Lemma 3. Let , where . If , then a -regular 2-CP is optimal.

Recently, many researchers have tried to construct OOCs via various methods, such as the search methods using the greedy strategy, the outer-product matrix algorithm [11], projective geometry [12], the finite field theory [13, 14], the combinational design theory [15–21], and so on. In particular, researchers have obtained many optimal OOCs with and by constructing 2-CP s. For the case of constant-weight OOCs, see [22–24] for some of the examples. For the case of double-weight OOCs, some works have been done on , see [10, 25–27] for some of the examples.

For the case of double-weight optimal OOCs with and , little work has been done. The representative articles are [28–31]. So far, as the authors are aware, no explicit result of OOCs with has been obtained. Based on the results of performance analysis in and [32], it is easy to see that heavier-weight codewords have better code performance than lighter-weight codewords due to higher autocorrelation peaks. So, in this paper, our aim is to present explicit results of optimal OOCs with .

In [8], Buratti et al. presented many explicit results of optimal OOCs with and by using elementary facts of the quadratic residues and cyclic packing. In this paper, continue using the construction method in [8] and further elementary conclusions of the quadratic residues, two new infinite classes of OOCs with are presented. The main results are as follows.

Theorem 4. For any prime and , there exist an optimal 2-CP and an optimal -OOC.

Theorem 5. For any prime and , there exist an optimal 2-CP and an optimal -OOC.

An outline of this paper is as follows. In Section 2, the basic knowledge of the quadratic residues is introduced. Theorem 4 and Theorem 5 are proved in Section 3 and Section 4, respectively, Conclusions are presented in Section 5.

2. Preliminaries

For a fixed prime , let be a primitive element of , and denote the quadratic residues and the quadratic nonresidues of , respectively.

For any , , , define(1)(2)

Suppose that and each element of is greater than 1. Assume that . The difference lists are defined as follows:

, .

In order to prove the main results, we need to present the elementary facts of numbers of in . The interested readers may refer to [33] for the details.

Lemma 6. Suppose that is a prime, we have(1), if and only if (2), if and only if (3), if and only if (4), if and only if (5), if and only if (6), if and only if (7), if and only if (8), if and only if (9), if and only if (10), if and only if (11), if and only if (12), if and only if (13), if and only if (14), if and only if (15), if and only if (16), if and only if

3. Proof of Theorem 4

In this section, Theorem 4 will be proved by considering the prime in fourteen cases, where explicit constructions are presented, respectively. According to the Chinese Remainder Theorem, for any and , since , there exists a one-to-one and onto mapping from to .

3.1. The Cases of ≡ 71, 191, 239, 359, 431, 599, 311, 479, 551, 671, 719, 839

Lemma 7. Suppose that ≡ 71, 191, 239, 359, 431, 599, 311, 479, 551, 671, 719, 839 is a prime, and . Then, forms a 2-CP , where

Proof. We compute the difference lists , , from and . It is not difficult to see that , . So, we only need to compute the difference lists for as follows:
, , , , , , , , , , and .
According to (1) and (2) of Lemma 6, we know that , , , and . It is easy to check that , , . Hence, covers every element in exactly once, and each element of is not covered. So, forms a 2-CP .

3.2. The Cases of ≡ 43, 67, 163, 403, 547, 667, 187, 283, 307, 523, 643, 787

Lemma 8. Suppose that 43, 67, 163, 403, 547, 667, 187, 283, 307, 523, 643, 787 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , and .
According to (3) and (4) of Lemma 6, we know that , , and . It is easy to check that forms a 2-CP .

3.3. The Cases of ≡ 23, 263, 407, 527, 743, 767, 47, 143, 167, 383, 503, 647

Lemma 9. Suppose that 23, 263, 407, 527, 743, 767, 47, 143, 167, 383, 503, 647 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , and .
According to (5) and (6) of Lemma 6, we know that , , and . It is easy to check that forms a 2-CP .

3.4. The Cases of ≡ 211, 331, 379, 499, 571, 739

Lemma 10. Suppose that ≡ 211, 331, 379, 499, 571, 739 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , , and .
According to (7) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

3.5. The Cases of ≡ 19, 139, 451, 619, 691, 811

Lemma 11. Suppose that ≡ 19, 139, 451, 619, 691, 811 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , and .
According to (8) of Lemma 6, we know that , , , and . It is easy to check that forms a 2-CP .

3.6. The Cases of ≡ 11, 179, 491, 611, 659, 779

Lemma 12. Suppose that ≡ 11, 179, 491, 611, 659, 779 is a prime. Then, forms a 2-CP , where

Proof. We compute the differences of , , from and . It is not difficult to see that , . So, we only need to compute the differences of for as follows:
, , , , , , , , , , , , and .
According to (13) of Lemma 6, we know that , , , and