Department of Mathematics, University of Kalyani, Kalyani, West Bengal-741235, India
Received January 18, 2013; Accepted March 25, 2013; Published March 29, 2013
Citation: Basu M, Ghosh DK (2013) On n-dimensional Uniform t-(v, k, λ)_{n} Designs. J Appl Computat Math 2:125. doi: 10.4172/2168-9679.1000125
Copyright: © 2013 Basu M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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In this paper, we define a new unifom t-(v, k, λ)n design on n-dimension. We illustrate with examples this design for n=2 and n=3. For n=1, we show that this is a t-(v, k, λ) design. We consider the cases of symmetric and Steiner system of uniform t-(v, k, λ)n design.
t-(v, k, λ) design; Symmetric design; Steiner design
A t-(v, k, λ) design is an ordered pair (X, B) where X is a v-set of points and B, called block set of b blocks such that each point lies on exactly r blocks, each block contains k points of X with the property that every t-subset of X is contained in exactly λ blocks where t ≤ k ≤ v [1,2].
The necessary conditions for holding a t-(v, k, λ) design are as follows:
1. vr=bk
2.
Corollary 1: For any t-(v, k, λ) design if i ≤ t then number of blocks containing a given i-subset of the points is a constant
Corollary 2: Any t-(v, k, λ) design holds design, where 1 ≤ v_{2} ≤ v_{1}< .
Definition 1: A t-(v, k, λ) design is said to be a symmetric design if v=b.
In this paper, we define an n-dimensional t-(v, k, λ)_{n} design. We describe this design with illustrative examples for n=2 and n=3. We also show that it is a t-(v, k, λ) design for n=1.
Definition 3: Let X={X_{1}, X_{2}, . . . , X_{i}, . . . , X_{n}}, where X_{i}={x_{i1}, x_{i2}, . . . , x_{ip}, . . . , x_{iv}} be an n-dimensional set, where X_{i}={x_{i1}, x_{i2}, . . . , x_{ip}, . . . , x_{iv}} of cardinality v and {x_{1l}, x_{2m}, . . . , x_{ip}, . . . , x_{nu}} is defined a node of X where 1≤ l, m, . . . , p, . . . , u ≤ v so that the total number of nodes of X is v. We define X is an n-dimension of order v. Let B={B_{1}, B_{2}, . . . , B_{j} , . . . , B_{b}} of cardinality b, called block set, of order k (≤ v) the block B_{j} ⊆ X ∀ j, B_{j} contains total k^{n} nodes in which every element of B_{j} contains k^{n-1} nodes and occurs in exactly r blocks. Also let T={T_{1}, T_{2}, . . . , T_{i}, . . . , T_{n}} is an n-dimension of order where T_{i} ⊆ X_{i} ∀ i and t ≤ k ≤ v, each element of T_{i} contains t^{n-1} nodes and T_{i} contains total t^{n} nodes such that every T_{i} occurs in exactly λ blocks. Then the ordered pair (X, B) is defined to be an n-dimensional uniform design and is denoted by t-(v, k, λ)_{n} design.
The necessary conditions for holding t-(v, k, λ)_{n} design are as follows:
1.
2.
Corollary 3: If t-(v, k, λ)_{n} design is a t-design, i ≤ t, then number of blocks containing a given i-subset of the points is a constant
Corollary 4: Any t-(v, k, λ)_{n} design holds design, where 1 ≤ v_{2} ≤ v_{1}<v.
It shows that t-(v, k, λ)n design holds all the necessary conditions and the corollaries of t-(v, k, λ) design for n=1.
Definition 4: A t-(v, k, λ)_{n} design is said to be a symmetric designif
Definition 5: A t-(v, k, λ)_{n} design is defined as a Steiner system if λ=1 and denoted by S(t, k, v)_{n}.
Theorem 1: If t=k, then t-(v, k, λ)_{n} design is Steiner.
Example 1: Let X={X_{1}, X_{2}} Where X_{1}={1, 2, 3, 4, 5}, X_{2}={a, b, c, d, e} i.e., X is 2-dimensional of order 5 i.e., n=2, v=5. We write a node of X as (i, j) where i ∈ X_{1} and j ∈ X_{2}. Therefore we have the following nodes:
a | b | c | d | e | |
1 | (1,a) | (1,b) | (1,c) | (1,d) | (1,e) |
2 | (2,a) | (2,b) | (2,c) | (2,d) | (2,e) |
3 | (3,a) | (3,b) | (3,c) | (3,d) | (3,e) |
4 | (4,a) | (4,b) | (4,c) | (4,d) | (5,e) |
5 | (5,a) | (5,b) | (5,c) | (5,d) | (5,e) |
1. Now we construct the block set B of order 4 and are given below:
Each row or column of B_{j}, 1 ≤ j ≤ 25 is called the element of B_{j}. Therefore, we have v=5, k=4, b=25 and r=4. Hence it holds 4-(5, 4, 1)_{2}, 3-(5, 4, 4)_{2}, 2-(5, 4, 9)_{2}, 1-(5, 4, 16)2 designs. Also it satisfies all the necessary conditions and the corollaries of the n-dimensional Uniform t-(v, k, λ)_{n} Design.
Each row or column of B_{j}, 1 ≤ j 100 is called the element of B_{j}. Therefore, we have v=5, k=3, b=100, r=6. Hence it holds 3-(5, 3, 1)_{2}, 2-(5, 3, 9)_{2}, 1-(5, 3, 36)_{2} designs. Also it satisfies all the necessary conditions and the corollaries of the n-dimensional Uniform t-(v, k, λ)_{n} Design.
Example 2: Let X={X_{1}, X_{2}, X_{3}} Where X_{1}={1, 2, 3}, X_{2}={1, 2, 3}, X_{3}={1, 2, 3} i.e., X is 3-dimensional of order 3. We write {X_{1}, X_{2}, X_{3}} as (i, j, l) where i ∈ X_{1}, j ∈ X_{2} and l ∈ X_{3}. Therefore we have the following nodes (Figure 1).
3. Now we construct the block set B of order 2 and are given in Figure 2.
Each plane of B_{j}, 1 ≤ j ≤ 27 is called the element of B_{j}. Therefore, we have v=3, k=2, b=27, r=2. Hence it holds 2-(3, 2, 1)_{3}, 1-(3, 2, 8)_{3} designs. Also it satisfies necessary conditions and the corollaries of the n-dimensional Uniform t-(v, k, λ)_{n} Design.
In this paper, we introduce a new design on n-dimension. The existing one dimensional t-(v, k, λ) design has many applications in code authentication, optical orthogonal codes, erasure codes and information dispersal, group testing and superimposed codes software testing, game scheduling, disc layout and interconnection network, threshold and ramp schemes etc. But in practical, code authentication (Fibonacci coding/ decoding) where Fibonacci coding is defined by:
Game scheduling (multi-criterion), multi drop networks, software (multi-purposes) testing, threshold and ramp schemes etc. are in more than one dimension. Hence the n-dimensional design has more applications in real world problems disk layout and striping, partial match queries of files etc.
The authors would like to thank the anonymous reviewers for their careful reading of the paper and also for their thoughtful, constructive comments and suggestions that greatly improve the content and presentation of the paper. The second author thanks UGC-JRF for financial support of his research work.
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