Construction of Some New Quasi Rees Neighbor Designs Using Cyclic Shifts

Many popular neighbor designs are used in serology, agriculture, and forestry which manifest neighbor effects very much. If every treatment appears as a neighbor with other ( v -2) treatments once but emerges twice with only one treatment, such designs are called Quasi Rees neighbor designs (QRNDs) in k size of circular blocks. These designs were used for counterbalancing the neighboring effects for the cases for which minimal neighbor designs cannot be constructed. In this article, various generators are constructed to obtain circular binary NDs, using cyclic shifts.


Introduction
In serology, initially Rees (1967) used neighbor designs (NDs) and applied this technique for the research on viruses. Then Hwang (1973), Cheng (1983), Azais et al. (1993), Ahmed and Akhtar (2008), Akhtar et al. (2010),  and ZafarYab et al. (2010) generated NDs for several cases. Shahid et al. (2017) presented some classes of NDs in linear blocks. Preece (1994) defined circular QRNDs in following manners: Every block has k (>2) distinct treatments from S, (ii) Formation of blocks should be circular such that every treatment has one left-and one right-neighbor.
(iii) Each treatment from S comes exactly r times.
(iv) No treatment from S has itself as a neighbor.
(v) Every treatment from S appears once as neighbors with (v-2) other elements and appears twice with only one of the other treatments.
In this work to obtain circular binary QRNDs (CBQRNDs), some generators are either searched out from the existing generalized neighbor designs or newly developed. Following generators are searched out from the existing generalized NDs.
• ZafarYab et al. (2010) expressed some series of generalized NDs in which v/2 pairs of treatments appear two times for k = 8 and ν = 8t, 16t, and 24t.
The layout of this paper is as follows: Method of construction of CBQRNDs is explained in Section 2. Using cyclic shifts, some generators are developed in Section 3 and 4 to obtain CBQRNDs for even and odd v, respectively. In Section 5, a catalogue of CBQRNDs is presented for odd v and k, where v < 50 and 3 ≤ k ≤ 9. A brief conclusion is provided in Section 6.

Construction of CBQRNDs
The method of cyclic shifts introduced by Iqbal (1991), is explained here only to obtain circular QRNDs. The rules of construction are: Rule I: Let Sj = [sj1, sj2, …, sj(k-1)] be the set of shifts, where 1≤ sij ≤ v-1. S* must contain (i) every element from Sj with its complement, and (ii) sum of elements of Sj (mod v) along with its complement for each set. In Rule I, complement of 'a' is 'v-a'.
▪ If 1, 2 … (v-1) appear exactly once in S* except one value which appears twice along with its complement then it will be QRND.
▪ Design will be binary through Rule I, if aggregate of any two, three… or (k-1) successive elements of S is not 0 (mod v), if so, reorder the values.
▪ If 1, 2 … (v-2) appear exactly once in S* except one value which comes twice along with its complement then it will be QRND.
▪ Design will be binary through Rule II, if aggregate of any two, three …, or (k-2) successive values of S is not 0 mod (v-1), if so, reorder the values.

Generators to obtain CBQRNDs for v even Generator 3.1
Following i sets provide the CBQRNDs for ν = 8i-2 and k = 4. Here (v-1) treatment appears two times as neighbors with each of other treatments while remaining other pairs appear only once as neighbors.  Where w = 0 for i odd and w = 1 for i even.

Generator 3.3
Following i sets provide CBQRNDs for ν = 16i−2 and k = 8. Where w = 1 for i odd and w = 0 for i even.

Generator 3.5
Following set of shifts provide CBQRNDs for ν = 2k-2, here k is size of the block.

CBQRNDs for odd v and k
CBQRNDs are presented below for odd v and k, where v < 50 and 3 ≤ k ≤ 9.