Optimal Variable-Weight Optical Orthogonal Codes via Difference Packings

Abstract

Variable-weight optical orthogonal code (OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. In this paper, the upper bound on the size of variable-weight OOCs is improved, a cyclic <emphasis emphasistype="italic" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"> <tex Notation="TeX">$t\hbox{-}(v, W, \lambda , Q)$</tex></formula></emphasis> packing is introduced to construct a variable-weight OOC, an upper bound for the number of blocks of <emphasis emphasistype="italic" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"> <tex Notation="TeX">$t\hbox{-}(v, W, \lambda , Q)$</tex></formula></emphasis> packings is obtained, and an equivalence between optimal cyclic packing and optimal variable-weight optical orthogonal code is established. Recursive constructions for optimal <emphasis emphasistype="italic" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"> <tex Notation="TeX">$2\hbox{-}{\rm CP}(W, 1, Q;v)$</tex></formula></emphasis> s are also presented. By using skew starters and these constructions, infinite classes of optimal <emphasis emphasistype="italic" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"> <tex Notation="TeX">$(v, W, 1, \{1/2, 1/2\})$</tex></formula></emphasis> -OOCs are obtained for <emphasis emphasistype="italic" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"> <tex Notation="TeX">$W=\{3, 4\}$</tex></formula></emphasis> , and <emphasis emphasistype="italic" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><formula formulatype="inline"><tex Notation="TeX">$\{4, 5\}$</tex></formula></emphasis> .