Optical Orthogonal Signature Pattern Codes Research Articles

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.

On balanced [formula omitted] difference packings

Let K be a set of positive integers and let G be an additive group. A (G,K,1) difference packing is a set of subsets of G with sizes from K whose list of differences covers every element of G at most once. It is balanced if the number of blocks of size k∈K does not depend on k. In this paper, we determine a balanced (Z4u×Z8v,{4,5},1) difference packing of the largest possible size whenever uv is odd. The corresponding optimal balanced (4u,8v,{4,5},1) optical orthogonal signature pattern codes are also obtained.

New Asymptotically Optimal Optical Orthogonal Signature Pattern Codes from Cyclic Codes

New Asymptotically Optimal Optical Orthogonal Signature Pattern Codes from Cyclic Codes

Optimal optical orthogonal signature pattern codes with weight three and cross-correlation constraint one

Optical orthogonal signature pattern codes (OOSPCs) have attracted wide attention as signature patterns of spatial optical code division multiple access networks. In this paper, an improved upper bound on the size of an $$(m,n,3,\lambda _a,1)$$-OOSPC with $$\lambda _a=2,3$$ is established. The exact number of codewords of an optimal $$(m,n,3,\lambda _a,1)$$-OOSPC is determined for any positive integers $$m,n\equiv 2 ({\mathrm{mod }} 4)$$ and $$\lambda _a\in \{2,3\}$$.

Constructions for optimal [formula omitted]-OOSPCs

Multiple-weight optical orthogonal signature pattern codes (OOSPCs) were introduced by Kwong and Yang for 2-D image transmission in multicore-fiber optical code-division multiple-access (OCDMA) networks with multiple quality of services (QoS) requirement. In this paper, an upper bound on the maximum code size of a (u×v,W,λ,Q)-OOSPC is obtained. A link between optimal (u×v,W,λ,Q)-OOSPCs and block designs is developed. Several infinite families of optimal (u×v,{3,4},1,Q)-OOSPCs are presented by means of semi-cyclic group divisible designs ((W,Q)-SCGDDs) and perfect relative difference families.

Constructions of optimal balanced $ (m, n, \{4, 5\}, 1) $-OOSPCs

Kitayama proposed a novel OCDMA (called spatial CDMA) for parallel transmission of 2-D images through multicore fiber. Optical orthogonal signature pattern codes (OOSPCs) play an important role in this CDMA network. Multiple-weight (MW) optical orthogonal signature pattern code (OOSPC) was introduced by Kwong and Yang for 2-D image transmission in multicore-fiber optical code-division multiple-access (OCDMA) networks with multiple quality of services (QoS) requirements. Some results had been done on optimal balanced \begin{document}$ (m, n, \{3, 4\}, 1) $\end{document} -OOSPCs. In this paper, it is proved that there exist optimal balanced \begin{document}$ (2u, 16v, \{4, 5\}, 1) $\end{document} -OOSPCs for odd integers \begin{document}$ u\geq 1 $\end{document} , \begin{document}$ v\geq 1 $\end{document} .

Asymptotically Optimal Optical Orthogonal Signature Pattern Codes

Optical orthogonal signature pattern codes (OOSPCs) have played an important role in a novel type of optical code-division multiple-access network for 2-D image transmission. In this paper, we give four direct constructions for OOSPCs based on polynomials and rational functions over finite fields. We also use $r$ -simple matrices to present a recursive construction for OOSPCs. These constructions yield new families of asymptotically optimal OOSPCs.

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 optical orthogonal signature pattern codes with maximum collision parameter 2 and weight 4

Optical orthogonal signature pattern codes (OOSPCs) play an important role in a novel type of optical code-division multiple-access 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 a construction of an (m, n, 4, 2)-OOSPC from a one-factor of Kohler graph of $$\mathbb {Z}_m\times \mathbb {Z}_n$$ which contains a unique element of order 2. In this paper, we study the existence of one-factor of Kohler graph of $$\mathbb {Z}_m\times \mathbb {Z}_n$$ having three elements of order 2. It is proved that there is a one-factor in the Kohler graph of $$\mathbb {Z}_{2^{\epsilon }p}\times \mathbb {Z}_{2^{\epsilon '}}$$ relative to the Sylow 2-subgroup if there is an S-cyclic Steiner quadruple system of order 2p, where $$p\equiv 5\pmod {12}$$ is a prime and $$1\le \epsilon ,\epsilon '\le 2$$ . Using this one-factor, we construct a strictly $$\mathbb {Z}_{2^{\epsilon }p}\times \mathbb {Z}_{2^{\epsilon '}}$$ -invariant regular $$G^*(p,2^{\epsilon +\epsilon '},4,3)$$ relative to the Sylow 2-subgroup. By using the known S-cyclic SQS(2p) and a recursive construction for strictly $$\mathbb {Z}_{m}\times \mathbb {Z}_{n}$$ -invariant regular G-designs, we construct more strictly $$\mathbb {Z}_{m}\times \mathbb {Z}_{n}$$ -invariant 3-(mn, 4, 1) packing designs. Consequently, there is an optimal $$(2^{\epsilon }m,2^{\epsilon '}n,4,2)$$ -OOSPC for any $$\epsilon ,\epsilon '\in \{0,1,2\}$$ with $$\epsilon +\epsilon '>0$$ and an optimal (6m, 6n, 4, 2)-OOSPC where m, n are odd integers whose all prime divisors from the set $$\{p\equiv 5\pmod {12}:p$$ is a prime, $$p<$$ 1,500,000}.

Combinatorial constructions for optimal multiple-weight optical orthogonal signature pattern codes

Multiple-weight (MW) optical orthogonal signature pattern code (OOSPC) was introduced by Kwong and Yang for 2-D image transmission in multicore-fiber optical code-division multiple-access (OCDMA) networks with multiple quality of services (QoS) requirement. In this paper, an upper bound on the maximum code size of (u×v,W,1,Q)-OOSPC is improved. By using quadratic residues and skew starters, some direct constructions for balanced (3×gv,3×g,{3,4},1)-OOSPCs are given for some positive integers g and v. Several recursive constructions for balanced (u×v,W,1)-OOSPCs are presented by means of incomplete difference matrices and perfect relative difference families. By using those constructions, a balanced optimal (3u×9v,{3,4},1)-OOSPC is obtained for any pair of positive integers (u,v) with gcd(u,3)=1.

$(m\kern -1pt,n\kern -1pt,3\kern -1pt,1)$ Optical Orthogonal Signature Pattern Codes With Maximum Possible Size

Kitayama proposed a novel code-division multiple-access (CDMA) network for image transmission called spatial CDMA. Optical orthogonal signature pattern codes (OOSPCs) have attracted wide attention as signature patterns of spatial CDMA. An (m,n,k,λ)-OOSPC is a set of m × n (0,1)-matrices with Hamming weight k and maximum correlation value λ. Let Θ (m,n,k,λ) be the largest possible number of codewords among all (m,n,k,λ) -OOSPCs. In this paper, we concentrate on the calculation of the exact value of Θ (m,n,3,1) and the construction of an (m,n,3,1)-OOSPC with Θ (m,n,3,1) codewords. As a consequence, we show that Θ (m,n,3,1)=[ mn-1/6]-1 when mn≡ 14, 20(mod 24), or mn≡ 8, 16(mod 24) and gcd (m,n,4)=2 , or mn≡ 2(mod 6) and gcd (m,n,4)=4 , and Θ (m,n,3,1)= [mn-1/6] otherwise.

Further results on optimal (&amp;lt;italic&amp;gt;m&amp;lt;/italic&amp;gt;, &amp;lt;italic&amp;gt;n&amp;lt;/italic&amp;gt;, 4, 1) optical orthogonal signature pattern codes

To support high-speed multiple access network applications, especially image applications, Kitayama rst presented the concept of optical orthogonal signature pattern code (OOSPC). An (<italic>m</italic>, <italic>n</italic>, <italic>k</italic>, λ)-OOSPC is a family of (0, 1)-matrices of Hamming weight <italic>k</italic> satisfying two correlation properties. Let Θ(<italic>m</italic>, <italic>n</italic>, <italic>k</italic>, λ) denote the largest possible number of codewords among all (<italic>m</italic>, <italic>n</italic>, <italic>k</italic>, λ)-OOSPCs. An (<italic>m</italic>, <italic>n</italic>, <italic>k</italic>, λ)-OOSPC with Θ(<italic>m</italic>, <italic>n</italic>, <italic>k</italic>, λ) codewords is said to be optimal. In this paper, we confirm Θ(<italic>m</italic>, <italic>n</italic>, 4, 1) = 「<italic>mn</italic>-1/12」, in other words, we construct an (<italic>m</italic>, <italic>n</italic>, 4, 1)-OOSPC with 「<italic>mn</italic>-1/12」 codewords for any positive integers m and n satisfying one of the following: (1) <italic>mn</italic>≡ 8; 16 (mod 24), each of whose odd prime factor is congruent to 1 modulo 6, with the exception of gcd(<italic>m</italic>, <italic>n</italic>, 2) = 1, or <italic>mn</italic>≡ 16 (mod 32) and gcd(<italic>m</italic>, <italic>n</italic>, 4) = 2; (2) <italic>mn</italic>≡ 0 (mod 24) and gcd(<italic>m</italic>, <italic>n</italic>, 6) = 2.

Combinatorial constructions for maximum optical orthogonal signature pattern codes

An (m,n,k,λa,λc) optical orthogonal signature pattern code (OOSPC) is a family C of m×n (0,1)-matrices of Hamming weight k satisfying two correlation properties. OOSPCs find application in transmitting two-dimensional image through multicore fiber in CDMA networks. Let Θ(m,n,k,λa,λc) denote the largest possible number of codewords among all (m,n,k,λa,λc)-OOSPCs. An (m,n,k,λa,λc)-OOSPC with Θ(m,n,k,λa,λc) codewords is said to be maximum. For the case λa=λc=λ, the notations (m,n,k,λa,λc)-OOSPC and Θ(m,n,k,λa,λc) can be briefly written as (m,n,k,λ)-OOSPC and Θ(m,n,k,λ). In this paper, some direct constructions for (3,n,4,1)-OOSPCs, which are based on skew starters and an application of the Theorem of Weil on multiplicative character sums, are given for some positive integer n. Several recursive constructions for (m,n,k,1)-OOSPCs are presented by means of incomplete different matrices and group divisible designs. By utilizing those constructions, the number of the codewords of a maximum (m,n,4,1)-OOSPC is determined for any positive integers m,n such that gcd(m,18)=3 and n≡0(mod12). It is established that Θ(m,n,4,1)=(mn−12)/12 for any positive integers m,n such that gcd(m,18)=3 and n≡0(mod12).

Determination of the sizes of optimal [formula omitted]-OOSPCs for [formula omitted

Optical orthogonal signature pattern codes (OOSPCs) were first introduced by Kitayama in [K. Kitayama, Novel spatial spread spectrum based fiber optic CDMA networks for image transmission, IEEE J. Sel. Areas Commun. 12 (4) (1994) 762–772]. They find application in transmitting a two-dimensional image through multicore fiber in the CDMA communication system. For given positive integers m, n, k, λa and λc, let Θ(m,n,k,λa,λc) denote the largest number of codewords among all (m,n,k,λa,λc)-OOSPCs. An (m,n,k,λa,λc)-OOSPC with Θ(m,n,k,λa,λc) codewords is said to be optimal. In this paper, the number of codewords in an optimal (m,n,k,λ,k−1)-OOSPC (i.e., the exact value of Θ(m,n,k,λ,k−1)) is determined for any positive integers m, n, k and λ=k−1,k.

Optical Orthogonal Signature Pattern Codes With Maximum Collision Parameter $2$ and Weight $4$

An optical orthogonal signature pattern code (OOSPC) finds application in transmitting 2-D images through multicore fiber in code-division multiple-access (CDMA) communication systems. Observing a one-to-one correspondence between an OOSPC and a certain combinatorial subject, called a packing design, we present a construction of optimal OOSPCs with weight 4 and maximum collision parameter 2, which generalizes a well-known Köhler construction of optimal optical orthogonal codes (OOC) with weight 4 and maximum collision parameter 2. Using this new construction enables one to obtain infinitely many optimal OOSPCs, whose existence was previously unknown. We prove that for a multiple n of 4, there exists no optimal OOSPC of size 6 n with weight 4 and maximum collision parameter 2, together with a report which shows a gap between optimal OOCs and optimal OOSPCs when 6 and n are not coprime. We also present a recursive construction of OOSPCs which are asymptotically optimal with respect to the Johnson bound. As a by-product, we obtain an asymptotically optimal (m, n, 4, 2)-OOSPC for all positive integers m and n.

Construction of Wavelength/Time Codes for Fiber-Optic CDMA Networks

Fiber-optic code-division multiple-access (FO-CDMA) technology for high-speed access networks has attracted a lot of attention recently. Constructions of 2-D codes, suitable for incoherent wavelength/time (W/T) FO-CDMA, have been proposed to reduce the time spread of the 1-D sequences. The W/T construction of codes can be broadly classified as 1) hybrid sequences and 2) matrix codes. Earlier, we reported a new family of unipolar wavelength/time multiple-pulses-per-row (W/T MPR) matrix codes that have good cardinality, spectral efficiency and at the same time have the lowest off-peak autocorrelation and cross-correlation values equal to unity. In this paper, we propose an algorithm to construct W/T MPR codes, starting with distinct 1-D optical orthogonal codes (OOCs) of a family as the row vectors of the code. The 2-D optical orthogonal signature pattern codes (OOSPCs), which are also constructed from 1-D OOCs, need much higher temporal length than that of W/T MPR codes, to construct codes of given cardinality and weight. Representation of the results of the codes generated using the algorithm are presented and the correlation properties are verified. We have analyzed the performance of the W/T MPR codes considering multiple access interference as the main source of noise. Also, we have compared cardinality and spectral efficiency of the codes with other W/T codes.

Double-weight signature pattern codes for multicore-fiber code-division multiple-access networks

To transmit digitized image pixels in optical code-division multiple-access (CDMA) networks with multicore fiber, classes of two-dimensional (2-D) patterns, so-called optical orthogonal signature pattern codes (OOSPCs), have previously been proposed. The new technology enables parallel transmission and simultaneous access of multiple 2-D images in multiple-access environments. In this letter, we construct two new families of OOSPCs without the assumption of identical weight for all signature patterns in a code set. Since the performance of a signature pattern varies with its weight, these new double-weight OOSPCs are especially useful for optical CDMA networks with multiple performance requirements.

Image transmission in multicore-fiber code-division multiple-access networks

Two-dimensional (2-D) signature patterns, so-called optical orthogonal signature pattern codes (OOSPCs), were previously proposed to encode binary digitized image pixels in optical code-division multiple-access (CDMA) networks with "multicore" fiber. The new technology enables parallel transmission and simultaneous access of 2-D images in a multiple-access environment. However, previous work on OOSPCs assumed that the weight of every signature pattern in a code set was the same. In this paper, we construct and analyze a new family of OOSPCs without this assumption. Since the performance of a signature pattern varies with its code weight, our approach is useful for optical CDMA networks with a multiple performance requirement.

Two-dimensional spatial signature patterns

An optical orthogonal signature pattern code (OOSPC) is a collection of (0,1) two-dimensional (2-D) patterns with good correlation properties (i.e., high autocorrelation peaks with low sidelobes, and low cross-correlation functions). Such codes find applications, for example, to parallelly transmit and access images in "multicore-fiber" code-division multiple-access (CDMA) networks. Up to now all work on OOSPCs has been based on an assumption that at most one pulse per column or one pulse per row and column is allowed in each two-dimensional pattern. However, this restriction may not be required in such multiple-access networks if timing information can be extracted from other means, rather than from the autocorrelation function. A new class of OOSPCs is constructed without the restriction. The relationships between two-dimensional binary discrete auto- and cross-correlation arrays and their corresponding "sets" for OOSPCs are first developed. In addition, new bounds on the size of this special class of OOSPCs are derived. Afterwards, four algebraic techniques for constructing these new codes are investigated. Among these constructions, some of them achieve the upper bounds with equality and are thus optimal. Finally, the codes generated from some constructions satisfy the restriction of at most one pulse per row or column and hence can be used in applications requiring, for example, frequency-hopping patterns.

Two-dimensional spatial codes for image transmissioninmulticore-fibre CDMA networks

Spatial optical orthogonal signature pattern codes (OOSPCs) which are collections of (0,1) two-dimensional patterns with good correlation properties, are constructed. Such codes find applications, for example, to parallelly transmit and access images in multicore-fibre code-division multiple-access (CDMA) networks.