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Let \begin{document}$ m $\end{document}, \begin{document}$ n $\end{document} be positive integers, and \begin{document}$ K $\end{document} a set of positive integers with size greater than 2. An \begin{document}$ (m,n,K,1) $\end{document} optical orthogonal signature pattern code, \begin{document}$ (m,n,K,1) $\end{document}-OOSPC, was introduced by Kwong and Yang for 2-D image transmission in multicore-fiber optical code-division multiple-access (OCDMA) networks with multiple quality of services (QoS) requirement. Let \begin{document}$ G $\end{document} be an additive group, a balanced \begin{document}$ (G, K, 1) $\end{document} difference packing, \begin{document}$ (G, K, 1) $\end{document}-BDP, can be used to construct a balanced \begin{document}$ (m,n,K,1) $\end{document}-OOSPC when \begin{document}$ G = {\mathbb{Z}}_m\times {\mathbb{Z}}_n $\end{document}. In this paper, the existences of optimal \begin{document}$ ( {\mathbb{Z}}_{2u}\times {\mathbb{Z}}_{38v}, \{3,4,5\},1) $\end{document}-BDPs are completely solved with \begin{document}$ u, \ v\equiv 1\pmod2 $\end{document}, and the corresponding optimal balanced \begin{document}$ (2u, 38v,\{3,4,5\},1) $\end{document}-OOSPCs are also obtained.

The effects of twinning on the dynamic recrystallization (DRX) of the Mg-6.58Gd-5.7Y-0.55Zr alloy were investigated through uniaxial compression tests at 400 °C and 450 °C with an initial strain rate of 0.001–0.1 s −1 and were characterized through optical microscopy (OM), scanning electron microscopy (SEM), electron back-scattering diffraction (EBSD) and transmission electron microscopy (TEM). At a relatively high strain rate, extension twins can mainly be activated during the initial stage of hot compression, and the subsequent DRX is closely related to the twin-twin and twin-dislocation interactions. The intersected twin boundaries of the extension twins from multiple twin variants may provide effective DRX sites and promote microstructure refinement. However, extension twins from a single twin variant pair may quickly grow, soon swallowing the matrix and decreasing the DRX efficiency.

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} .

Let \begin{document}$ W = \{w_1, w_2, \cdots, w_r\} $\end{document} be a set of \begin{document}$ r $\end{document} integers greater than 1, \begin{document}$ \Lambda_a = (\lambda_a^{(1)}, \lambda_a^{(2)}, \cdots, \lambda_a^{(r)}) $\end{document} be an \begin{document}$ r $\end{document} -tuple of positive integers, \begin{document}$ \lambda_c $\end{document} be a positive integer, and \begin{document}$ Q = (q_1, q_2, \cdots, q_r) $\end{document} be an \begin{document}$ r $\end{document} -tuple of positive rational numbers whose sum is 1. Variable-weight optical orthogonal code ( \begin{document}$ (n, W, \Lambda_a, \lambda_c, Q) $\end{document} -OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service requirements. In this paper, tight upper bounds on the maximum code size of \begin{document}$ (n, \{3, 4, 5\}, \Lambda_a, 1, Q) $\end{document} -OOCs are obtained, and infinite classes of optimal \begin{document}$ (n, \{3, 4, 5\}, \Lambda_a, 1, Q) $\end{document} -OOCs are constructed.

The loading rate significantly affects the mechanical properties of Mg alloys. However, the effects of strain rate on newly developed Mg–Gd–Y alloys at room temperature (RT) have rarely been investigated. Here, uniaxial tensile tests were conducted on a cast Mg-10Gd-3Y-0.5Zr (wt. %) alloy (GW103) in both as-aged and as-solutionized states with different grains sizes at RT and various strain rates. The results showed an anomalous positive strain-rate dependence of elongation in all the GW103 alloys investigated from 1 × 10−5 to 1 × 10−1 s−1. Careful microstructural characterization by slip trace analysis revealed that the exclusive deformation mechanism of pyramidal slip, non-Schmid behavior of basal slip, and slip transfer at 1 × 10−1 s−1; these induced the high RT ductility. The occurrence of pyramidal slip activity at high strain rates was suggested to be associated with rare earth solutes, fine precipitates, and local high stresses at grain boundaries due to dislocation pile-up. It was proposed that the increasing strain rate was approximately equivalent to decreases in grain size, enabling the generation of local stresses at grain boundaries, thus causing this anomalous phenomenon.

The microstructure evolution and plastic deformation mechanism of Mg-6.58Gd-5.7Y-0.55Zr alloy were investigated by uniaxial compression at temperatures ranging from 300 °C to 450 °C and with an initial strain rate of 0.1 s−1, and characterized by optical microscope (OM), scanning electron microscope (SEM) and electron back-scattering diffraction (EBSD). The effect of temperature on twinning behavior was focused in detail. At relatively low temperature, such as 300 °C, multiple twin variant pairs can be activated in one grain, and the twining process was dominated by nucleation. However, at high temperature, most of the twins activated in one grain belong to the single twin variant pair with the highest Schmid Factor (SF), and twin growth seems to represent the twinning behavior at 450 °C. The twinning behavior of Mg-6.58Gd-5.7Y-0.55Zr alloy at temperatures ranging from 300 °C to 450 °C basically follows Schmid law, and can be well understood in terms of variation of stress state and activated deformation modes with compression temperature.

Let $W=\{w_{1}, \ldots , w_{r}\}$ be a set of positive integers, $\lambda _{c}$ a positive integer, $\Lambda _{a}=(\lambda _{a}^{(1)}, \ldots \lambda _{a}^{(r)})$ an $r$ -tuple of positive integers, and $Q=(q_{1}, \ldots q_{r})$ an $r$ -tuple of positive rational numbers whose sum is 1. In 1996, Yang introduced variable-weight optical orthogonal code, $(n, W, \Lambda _{a}, \lambda _{c}, Q)$ -OOC, for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. Some work had been done on the constructions of optimal $(n, W, \Lambda _{a}, 1, Q)$ -OOCs with unequal auto-correlation constraints for $W=\{3, 4\}$ and {3, 5}, while little is known on optimal $(n, W, \Lambda _{a}, 1, Q)$ -OOCs for $|W|\geq 3$ . In this paper, we focus our main attentions on $(n, \{3, 4, 5\}, \Lambda _{a}, 1, Q)$ -OOCs with $\Lambda _{a}\in \{(2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)\}$ . Tight upper bounds on the maximum code size of $(n, \{3, 4, 5\}, \Lambda _{a}, 1, Q)$ -OOCs are obtained, and infinite classes of optimal $(n, \{3, 4, 5\}, \Lambda _{a}, 1, Q)$ -OOCs are constructed.

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 ( Z 4 u × Z 8 v , { 4 , 5 } , 1 ) difference packing of the largest possible size whenever uv is odd. The corresponding optimal balanced ( 4 u , 8 v , { 4 , 5 } , 1 ) optical orthogonal signature pattern codes are also obtained.

A multi-direction impact forging (MDIF) has been applied to a solutionized Mg-Gd-Y-Zr alloy in this work. An inhomogeneous microstructure, mostly fine grains, some twins and a few of coarse deformed grains were achieved with a weak and non-basal texture after 50 forging passes. Due to retarded DRX behavior by RE solute drag effect and the sharp drop of forging temperature, the alloy did not obtain a full DRX microstructure until 100 forging passes. During MDIF process, density of extension twins initially were induced in microstructure, which effectively refined the coarse grains by dividing and intersecting each other. In turn, those twining regions provided nucleation sites for DRX, and a continuous dynamic recrystallization converted most of the twin-matrix laminates to fine equi-axial grains. The ultimate tensile strength and elongation to failure of GW94 alloy were continuously improved during MDIF process, from less than 200 MPa and 2% of the initial solutionized specimen to more than 300 MPa and 7% of the one by 50 forging passes respectively.

The concept of a ( q , k , λ , t ) almost difference family (ADF for short) was introduced by Ding and Yin as a useful generalization of the concept of an almost difference set. Some results had been obtained for the existences of ( q , K , λ ) -ADFs, where K = { k 1 , k 2 , ź , k r } is a set of positive integers. In this note, new cyclic ( q , K , λ ) -ADFs are obtained via cyclotomy.

Variable-weight optical orthogonal code (OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. It is proved that optimal $(v, \{3, 5\}, 1, (1/2, 1/2))$-OOCs exist for some complete congruence classes of $v$. In this paper, for $Q\in \{(2/3,1/3), (3/4,1/4)\}$, by using skew starters, it is also proved that optimal $(v, \{3,5\}, 1, Q)$-OOCs exist for some complete congruence classes of $v$.

A novel multi-directional impact forging (MDIF) process was successfully applied to a Mg–Gd–Y-Zr alloy and the superplasticity of the MDIFed alloy was investigated. It was shown that the initial microstructure after MDIF had a unique characteristic of coarse grains decorated by profuse tension twins. Tensile tests at various temperatures and initial strain rates displayed that the MDIFed alloy exhibited superplasticity at temperatures higher than 400 °C, especially the maximum elongation of about 300% was achieved at 450 °C and 5 × 10 −4 s −1 . It was observed that the initial twin lamellas diminished and even vanished with the occurrence of static recrystallization (SRX) during preheating. The detailed microstructural analysis revealed that SRX during preheating and dynamic recrystallization (DRX) during tensile testing conduced to the superplasticity. With strain rate sensitivity exponent m -value of about 0.32, the superplastic deformation mechanism was supposed to be predominated by glide-controlled dislocation creep assisted by grain boundaries sliding.

Let $$M=(m_{i,j})$$M=(mi,j) be a magic square, where $$0\le m_{i,j}\le n^2-1$$0≤mi,j≤n2-1, $$0\le i, j\le n-1$$0≤i,j≤n-1. M is called diagonally ordered if both the main diagonal and the back diagonal, when traversed from left to right, have strictly increasing values. Let $$M=nA+B$$M=nA+B, where $$A=(a_{i,j})$$A=(ai,j), $$B=(b_{i,j})$$B=(bi,j), $$0\le a_{i,j}, \ b_{i,j}\le n-1$$0≤ai,j,bi,j≤n-1 for $$0\le i, j\le n-1$$0≤i,j≤n-1. M is called rational if both A and B possess the property that the sums of the n numbers in every row and every column are the same; otherwise, M is said to be irrational. In this paper, a pair of weakly diagonally ordered irrational orthogonal matrices (WDOIOM for short) is introduced to construct an irrational diagonally ordered magic square (IDOMS). It is proved that there exists a WDOIOM(n) for each positive integer $$n\ge 5$$nź5, and there does not exist a WDOIOM(n) for $$n\in \{2,3,4\}$$nź{2,3,4}. Consequently, it is proved that there exists an IDOMS(n) for each positive integer $$n\ge 5$$nź5, and there does not exist an IDOMS(n) for $$n\in \{2,3,4\}$$nź{2,3,4}.

In present research the serrated flow has been observed successfully after a critical amount of strain. Two relationships between the critical strain and temperature i.e. normal and inverse, corresponding to each serration type were studied. In order to investigate systematically the onset of serrated flow and serration type in NZ31 alloy, samples in solutionized condition were tensile tested at the temperature ranging from 100 °C to 300 °C with the strain rate ranging from 1×10−4 s−1 to 1×10−2 s−1. Results showed that normal critical strain appeared with type A and B serrated flow at temperature from 150°C to 250 °C, and inverse critical strain appeared with type C at temperature from 275 °C to 300 °C. Through analyzing the mechanism of three serration types, we found that the production of serration required improvement in diffusion for solute atoms for pinning process at low temperature, and enhance the moving ability of dislocations for unpinning process at high temperature, which need the assistance of the strain and stress respectively. So, in this work, the critical strain for pinning and the critical stress for unpinning processes were defined, which give a better explanation to the variation tendency of two definitions in accordance with temperature. Furthermore, this relationship results in the critical strain for onset of serrated flow changing from normal to inverse and corresponding different serrations.

Let W = { w 1 , w 2 , ? , w r } be an ordering of a set of r integers greater than 1, ? a = ( λ a ( 1 ) , λ a ( 2 ) , ? , λ a ( r ) ) be an r -tuple of positive integers, λ c be a positive integer, and Q = ( q 1 , q 2 , ? , q r ) be an r -tuple of positive rational numbers whose sum is 1. In 1996, Yang introduced variable-weight optical orthogonal code ( ( n , W , ? a , λ c , Q ) -OOC) for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. Some work had been done on the constructions of optimal ( n , { 3 , 4 } , ? a , 1 , Q ) -OOCs with unequal auto- and cross-correlation constraints. In this paper, we focus our main attention on ( n , { 3 , 5 } , ? a , 1 , Q ) -OOCs, where ? a ? { ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) } . Tight upper bounds on the maximum code size of an ( n , { 3 , 5 } , ? a , 1 , Q ) -OOC are obtained, and infinite classes of optimal balanced ( n , { 3 , 5 } , ? a , 1 ) -OOCs are constructed.

A pair of binary sequences is generalized from the concept of a two-level autocorrelation function of single binary sequence. In this paper, we describe two classes of binary sequence pairs of period $N=2q$, where $q=4f+1$ is an odd prime and $f$ is an even integer. Those classes of binary sequence pairs are based on cyclic almost difference set pairs. They have optimal three-level cross-correlation, and either balanced or almost balanced.

Let W = { w 1 , ? , w r } be an ordering of a set of r integers greater than 1, ? a = ( λ a ( 1 ) , ? , λ a ( r ) ) be an r -tuple of positive integers, λ c be a positive integer, and Q = ( q 1 , ? , q r ) be an r -tuple of positive rational numbers whose sum is 1. In 1996, Yang introduced variable-weight optical orthogonal code ( ( n , W , ? a , λ c , Q ) -OOC) for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. Most existing works on variable-weight optical orthogonal codes assume that λ a ( 1 ) , ? , λ a ( r ) = λ c = 1 . In this paper, new balanced ( n , { 3 , 4 } , ? a , 1 ) -OOCs are constructed, where ? a ? { ( 1 , 2 ) , ( 2 , 1 ) , ( 2 , 2 ) } .

In 1996, Yang introduced variable-weight optical orthogonal code for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. Let W = { w 1 , … , w r } be an ordering of a set of r integers greater than 1 , λ be a positive integer ( auto- and cross-correlation parameter ), and Q = ( q 1 , … , q r ) be an r -tuple ( weight distribution sequence ) of positive rational numbers whose sum is 1 . A ( v , W , λ , Q ) variable-weight optical orthogonal code ( ( v , W , λ , Q ) -OOC) is a collection of ( 0 , 1 ) sequences with weights in W , auto- and cross-correlation parameter λ . Some work has been done on the construction of optimal ( v , W , 1 , Q ) -OOCs, while little is known on the construction of ( v , W , λ , Q ) -OOCs with λ ≥ 2 . It is well known that ( v , W , λ , Q ) -OOCs with λ ≥ 2 have much bigger cardinality than those of ( v , W , 1 , Q ) -OOCs for the same v , W , Q . In this paper, a new upper bound on the number of codewords of ( v , W , λ , Q ) -OOCs is given, and infinite classes of optimal ( v , { 3 , 4 } , 2 , Q ) -OOCs are constructed.

The microstructures and mechanical properties of a sand-cast Mg–Nd–Zn alloy in the as-cast, solution-treated and peak-aged conditions were investigated. The as-cast alloy was comprised of α magnesium matrix and Mg12Nd eutectic compounds. The eutectic compounds dissolved into the matrix and small Zr-containing particles precipitated at grain interiors, due to the solution treatment. After the solution treatment, two kinds of cooling manner, either cooling in air or quenching in water, were employed. It was worth noting that some basal precipitates formed in the matrix during the in-air cooling process after solution treatment, which led to the succedent weak ageing hardening response and low strength in peak-aged condition. The hardness, yield strength, ultimate tensile strength and elongation at room temperature, of the samples in the T61 condition, were HV81, 191 MPa, 258 MPa and 4.2%, respectively. When tensile tested at high temperature, they exhibited serrated flow. Moreover, the casting surface of the tensile testing bar also had a great influence on its mechanical properties.

The concept of a (q, k, λ, t) almost difference family (ADF) has been introduced and studied by C. Ding and J. Yin as a useful generalization of the concept of an almost difference set. In this paper, we consider, more generally, (q, K, λ, t, Q)-ADFs, where K = {k1, k2, …, kr} is a set of positive integers and Q = (q1, q2, …, qr) is a given block-size distribution sequence. A necessary condition for the existence of a (q, K, λ, t, Q)-ADF is given, and several infinite classes of (q, K, λ, t, Q)-ADFs are constructed.

Optical orthogonal codes (1D constant-weight OOCs or 1D CWOOCs) were first introduced by Salehi as signature sequences to facilitate multiple access in optical fibre networks. In fiber optic communications, a principal drawback of 1D CWOOCs is that large bandwidth expansion is required if a big number of codewords is needed. To overcome this problem, a two-dimensional (2D) (constant-weight) coding was introduced. Many optimal 2D CWOOCs were obtained recently. A 2D CWOOC can only support a single QoS (quality of service) class. A 2D variable-weight OOC (2D VWOOC) was introduced to meet multiple QoS requirements. A 2D VWOOC is a set of 0, 1 matrices with variable weight, good auto, and cross-correlations. Little is known on the construction of optimal 2D VWOOCs. In this paper, new upper bound on the size of a 2D VWOOC is obtained, and several new infinite classes of optimal 2D VWOOCs are obtained.

Variable-weight optical orthogonal code (OOC) was introduced by G. C. Yang [IEEE Trans. Commun., 1996, 44: 47–55] for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. In this paper, seven new infinite classes of optimal (v, {3, 4, 6}, 1,Q)-OOCs are constructed.

It was found that the rolled Mg-2.74Gd-1.06Zn (wt.%) alloy sheets could exhibit excellent room-temperature ductility and formability by our previous studies. In this work, the Mg-2.74Gd-1.06Zn alloy was processed using hot rolling, forging and annealing treatments in order to control the texture and the grain size. Emphasis was laid upon the influence of these factors on the room-temperature ductility and the tensile behavior. The results showed that the excellent tensile ductility of the Mg-2.74Gd-1.06Zn alloy sheets should be mainly attributed to its low intensity of (0002) pole figure, instead of the non-basal texture type. The Mg-2.74Gd-1.06Zn alloy sheets with a grain size of 10-30 mu m all exhibited a large elongation-to-failure (>38%); comparing with the weak texture, the grain size had a negligible effect on their room-temperature ductility. As grain size increased further (>50 mu m), the twining activity increased substantially during the post-uniform deformation, leading to its premature failure and low post-uniform elongation. Crown Copyright (C) 2012 Published by Elsevier Ltd. All rights reserved.

Variable-weight optical orthogonal code (OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service requirements. It is proved that optimal (v, {3, 4}, 1, (1/2, 1/2))-OOCs exist for some complete congruence classes of v. In this paper, for $${Q \in \{(1/3, 2/3), (2/3, 1/3)\}}$$ , by using skew starters, it is also proved that optimal (v, {3, 4}, 1, Q)-OOCs exist for some complete congruence classes of v.

The tensile deformation behavior of Mg-3Gd-Zn (wt.%) alloy sheets annealed and aged at different temperatures has been studied. Samples annealed at 400 degrees C for 1 h exhibit serrated flow when they are tensile tested at temperatures ranging from 150 degrees C to 250 degrees C and at strain rates of 1 x 10(-4) s(-1) to 1 x 10(-2) s(-1). The critical strain for onset of the serrated flow is observed to increase with increasing strain rate but decrease with increasing temperature. The temperature dependences of yield strength and ultimate strength exhibit a plateau, and the expected ductility dips are also observed in the temperature regime of serrated flow. For samples annealed at 480 degrees C for 1 h, the precipitates and serration both disappear. However, subsequent aging treatments result in the reappearance of a large amount of precipitates and the serration phenomenon, proving that the serrated flow in the Mg-3Gd-Zn sheets is mainly controlled by the shearing of precipitates by gliding dislocations. In addition, the observed static strain aging effect, the negative strain rate sensitivity and the calculated activation energy, combined with observations made by scanning electron microscopy, suggest that dynamic strain aging may also contribute to the serrations in the present alloy. Crown Copyright (C) 2011 Published by Elsevier B.V. All rights reserved.

Variable-weight optical orthogonal codes (OOCs) were introduced by Yang for multi-media optical CDMA systems with multiple quality of service (QoS) requirements. Some works had been done on the existence of optimal (v,W, 1, Q)-OOCs for W ∈ {{3, 4}, {3, 5}, {4, 5}}. In this paper, by using quadratic residues, seven new infinite classes of optimal (v, {k, 6}, 1, Q)-OOCs for k ∈ {3, 4} are constructed. key words: cyclic packing, optical orthogonal code, quadratic residue, relative difference family

In this note, a doubly magic rectangle is introduced to construct a doubly pandiagonal magic square. A product construction for doubly magic rectangles is also presented. Infinite classes of doubly pandiagonal magic squares are then obtained, and an answer to problem 22 of [G. Abe, Unsolved problems on magic squares, Discrete Math. 127 (1994) 3] is given.

Zero-difference balanced (ZDB) functions were introduced recently by Ding for the construction of optimal constant-composition codes, and optimal and perfect difference systems of sets. They are closely related to partitioned difference families. In this paper, we present generic constructions of ZDB functions from functions with difference-balanced property. In particular, two classes of ZDB functions with new and flexible parameters are reported. Employing these new ZDB functions, we obtain at the same time optimal (1) constant-composition codes, (2) constant-weight codes, and (3) perfect difference systems of sets, all with new and flexible parameters.

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, four general constructions for optimal cyclic packings and optimal variable-weight OOCs are presented. Many new infinite classes of optimal (v, W,1,Q)-OOCs are obtained. New infinite classes of optimal (v, W,1,Q)-OOCs for |W| ≥ 4 are easily obtained by the constructions.

In order to develop new magnesium alloy sheets with high formability at room temperature, the microstructure, texture, ductility, and stretch formability of rolled Mg–2%Gd–1%Zn and Mg–3%Gd–1%Zn sheets were investigated. The microstructures of these rolled sheets consist of fine recrystallized grains with a large amount of homogeneously distributed tiny particles in the matrix. The basal plane texture intensity is quite low and the basal pole is tilted by about 30° from the normal direction toward both the rolling direction and the transverse direction. The sheets exhibit an excellent ultimate elongation of ∼50% and a uniform elongation greater than 30%, and the Erichsen values reach ∼8 at room temperature. The flow curves of the two Mg–Gd–Zn alloys sheets display a remarkable linear hardening after an obvious yield point. The majority of the grains in the tilted texture have an orientation favorable for both basal slip and tensile twinning because of a high Schmid factor. The excellent stretch formability at room temperature can be attributed to the non-basal texture and low texture intensity, which led to the following characteristics: a lower 0.2% proof stress, a larger uniform elongation, a smaller Lankford value and a larger strain hardening exponent.

Almost difference families ( ( q , k , λ , t ) -ADFs ) were introduced by Ding and Yin as an useful generalization of almost difference sets (ADSs), constructions and infinite classes of almost difference families were also presented. In this paper, several new infinite classes of ( q , k , λ , t ) -ADFs for k=4, 5 and 6 are constructed.

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 t-(v, W, λ, Q) packing is introduced to construct a variable-weight OOC, an upper bound for the number of blocks of t-(v, W, λ, Q) packings is obtained, and an equivalence between optimal cyclic packing and optimal variable-weight optical orthogonal code is established. Recursive constructions for optimal 2-CP(W, 1, Q;v)s are also presented. By using skew starters and these constructions, infinite classes of optimal (v, W, 1, {1/2, 1/2})-OOCs are obtained for W = {3, 4}, and {4, 5}.

Optimal frequency-hopping sequences can be constructed from cyclic (υ, K, 1)-PBDs. A (υ, K, 1) difference family can be used to construct a (υ, K, 1)-PBD. A lot of work has been done on the existence of (υ, k, 1) difference families when K = {k} is a singleton. For K = {3, 4}, it is proved that for each prime power υ ≡ 1 (mod 18) and υ ⩾ 19, there exists a balanced (υ, K, 1) difference family. In this paper, it is proved that there exists a balanced (υ, {3, 6}, 1) difference family for each prime power υ ≡ 1 (mod 36) and υ ⩾ 37. An infinite class of optimal frequency-hopping sequence is then obtained.

Variable-weight optical orthogonal code (OOC) was introduced by G-C Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirement. In this article, new infinite classes of optimal (u, W, 1, {1/2, 1/2})-OOCs are obtained for W={3, 4}, {3, 5} and {3, 6}. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 274–291, 2010

SUMMARY Variable-weight optical orthogonal code (OOC) was introduced by G-C Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirement. In this paper, a construction for optimal (v, {3, 4}, 1, {s/(s + 1), 1/(s + 1)} )-OOCs is given. Fors = 2, it is proved that for each prime v ≡ 1 (mod 24), there exists a (v, {3, 4}, 1, {2/3, 1/3})OOC. A recursive construction for cyclic difference family is also presented. By using these constructions, a number of new infinite classes of optimal (v, {3, 4}, 1,Q)-OOCs for Q = {1/2, 1/2} and {2/3, 1/3} are constructed.

Generalized Steiner systems GS(2,k,v,g) were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size g+1 with minimum Hamming distance 2k−3, in which each codeword has length v and weight k. As to the existence of a GS(2,k,v,g), a lot of work has been done for k=3, while not so much is known for k=4. The notion k-∗GDD was first introduced by Chen et al. and used to construct GS(2,3,v,6). The necessary condition for the existence of a 4-∗GDD(6v) is v≥14. In this paper, it is proved that there exists a 4-∗GDD(6v) for any prime power v≡3,5,7(mod8) and v≥19. By using this result, the known results on the existence of optimal quaternary constant weight codes are then extended.

External difference families (EDFs) are a type of new combinatorial designs originated from cryptography. Some results had been obtained by Chang and Ding, the connection between EDFs and disjoint difference families (DDFs) was also established. In this paper, further cyclotomic constructions of EDFs and DDFs are presented, and several classes of EDFs and DDFs are obtained. Answers to problems 1 and 4 by Chang and Ding are also given. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 333–341, 2009

An (n,M,d;q) code is called equidistant code if the Hamming distance between any two codewords is d. It was proved that for any equidistant (n,M,d;q) code, d⩽nM(q-1)/(M-1)q(=dopt, say). A necessary condition for the existence of an optimal equidistant code is that dopt be an integer. If dopt is not an integer, i.e. the equidistant code is not optimal, then the code with d=⌊dopt⌋ is called good equidistant code, which is obviously the best possible one among equidistant codes with parameters n,M and q. In this paper, some constructions of good equidistant codes from balanced arrays and nested BIB designs are described.

A (k, n; w) multi-receiver multi-message authentication code allows a transmitter to broadcast up to w − 1 distinct authenticated messages to n receivers in such a way that (1) not only an opponent but also any up to k − 1 receivers cannot cheat any other receivers, and (2) all the receivers can independently verify the authenticity of the messages. Obana and Kurosawa [Designs, Codes and Cryptography, 22 (2001), pp. 47–63] used a special pair of orthogonal arrays, called TWOOA, to construct a (k, n; 2) multi-receiver single-message authentication code. In this paper, we generalize the notion of a TWOOA, and then use this generalized TWOOA to construct a (k, n; w) multi-receiver multi-message authentication code, which exceeds that of Safavi-Naini and Wang [Proc. of Eurocrypt'98, LNCS 1403, Springer (1998), pp. 527–541] at least in the numbers of receivers and authenticated messages. The structures of TWOOAs are investigated. Two constructions for TWOOAs are also provided.

Variable-weight optical orthogonal code (OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirement. In this paper, it is proved that for each prime p ≡ 5 (mod 8), there exists an optimal (14p, {4, 5}, 1, {3/4, 1/4})-OOC