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Journal of Statistical Planning and Inference

Journal of Statistical Planning and Inference 141 (2011) 3808–3813

0378-37

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jspi

Some series of proper generalized neighbor designs

Farrukh Shehzad a,n, Mohammad Zafar Yab b, Rashid Ahmed c

a National College of Business Administration & Economics, Lahore, Pakistanb Department of Statistics, University of Gujrat, Pakistanc Government Higher Secondary School Mitroo, Mailsi, Vehari, Pakistan

a r t i c l e i n f o

Article history:

Received 10 February 2011

Received in revised form

9 June 2011

Accepted 15 June 2011Available online 24 June 2011

Keywords:

Neighbor balanced designs

Generalized neighbor designs

GN2-designs

58/$ - see front matter & 2011 Elsevier B.V. A

016/j.jspi.2011.06.016

esponding author.

ail addresses: fshehzad.stat@gmail.com (F. Sh

a b s t r a c t

Neighbor balanced designs are used to remove the neighbor effects but most of these

designs require a large number of blocks. To neutralize the neighbor effects in such

situations, GN2-designs are most desirable. This article deals with the (i) refinement of

some series of GN2-designs constructed by Zafaryab et al. (2010) and (ii) construction of

some new series of GN2-designs in circular blocks of equal size.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Rees (1967) introduced neighbor designs in Serology. Since then considerable research on neighbor designs is availablein the literature. Neighbor balanced designs satisfy fairly restrictive combinatorial conditions; therefore, most of thesedesigns require a large number of blocks. To make these designs more useful, Misra et al. (1991) proposed generalizedneighbor designs by relaxing the condition of constant l0. If l0 takes only two values i.e. l01 and l02 then these designs arecalled GN2-designs, which are closest to the neighbor balanced designs. These designs are most economical if l01 ¼ 1 andl02 ¼ 2. Rees (1967) gave an idea to generate GN2-designs for even v¼k. Preece (1994) also constructed these designs forsome cases. Chaure and Misra (1996) constructed GNm-designs using the methods through systematically, Galois field,partially balanced incomplete designs, Galois domain and duals. Mishra (2007) constructed families of proper generalizedneighbor designs. Kedia and Misra (2008) constructed some series of generalized neighbor designs, which are obtained bydeveloping initial blocks cyclically. Ahmed et al. (2009) proposed GN2-designs for some cases. Akhtar et al. (2010) listedGN2-designs for 3rkr10 with vr50. Zafaryab et al. (2010) constructed GN2-designs for some cases of k¼4, 6, 8 and 10.They also listed these designs for k¼5, 7 and 9 with 51rvr100. In this article most economical GN2-designs areconstructed for v treatments in b circular blocks of k plots. Here, our main objective is to study the neighboring effectsalong with the main effects of the treatments. Section 2 of the article is reserved for refinement of some theoremspresented by Zafaryab et al. (2010).

�

For k¼4: their Theorems 3.2 and 3.3 are modified in series 2.1, Theorems 3.4 and 3.5 in series 2.2 and Theorems 3.6 and3.7 in series 2.3. � For k¼6: their Theorems 3.9, 3.10 and 3.11 are modified for v and vþ1 in series 2.4, 2.5 and 2.6, respectively.

ll rights reserved.

ehzad), zafar_yab2001@hotmail.com (M. Zafar Yab), rashid701@hotmail.com (R. Ahmed).

F. Shehzad et al. / Journal of Statistical Planning and Inference 141 (2011) 3808–3813 3809

�

For k¼8: their Theorems 3.14 and 3.15 are combined in series 2.7, Theorems 3.16 and 3.17 are modified for v and vþ1in series 2.8 and 2.9, respectively. Theorem 3.18 for v¼16tþ10 and v¼16tþ12 is modified for v and vþ1 in series 2.10and 2.11.

Some new series of GN2-designs for k¼6, 8 and 10 are presented in Section 3. GN2-designs for k¼12 are developed inSection 4.

2. Refined series of GN2-designs

Zafaryab et al. (2010) presented some series of minimal GN2-designs. Some of their series are refined in this section.

Series 2.1. If v¼8tþwþ2, where t positive integer and w¼0, 1 then GN2-design for k¼4 can be generated by developing (tþ1)initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�8, n2¼7 for w¼0, (ii) n1¼v�7, n2¼6 for w¼1.

Ijþ1 ¼ ð0,v�ð4jþ1Þ,1,ð4jþ4ÞÞ, j¼ 0,1,2,. . .,t�1:

Itþ1 ¼ ð0,v�ð4tþ1Þ,v�1,ð4t�2ÞÞ

Series 2.2. If v¼8tþw�4, where t (41) integer and w¼0, 1 then GN2-design for k¼4 can be generated by developing t initial

blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�6, n2¼5 for w¼0, (ii) n2¼4 for w¼1.

Ijþ1 ¼ ð0,v�ð4jþ1Þ,1,ð4jþ4ÞÞ, j¼ 0,1,2,. . .,t�2:

It ¼ ð0,v�ð4sþ1Þ,1,2Þ, where s¼ t�1

Series 2.3. If v¼8tþw�2, where t positive integer and w¼0, 1 then GN2-design for k¼4 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�4, n2¼3 for w¼0, (ii) n1¼v�3, n2¼2 for w¼1.

Ijþ1 ¼ ð0,v�ð4jþ1Þ,1,ð4jþ4ÞÞ, j¼ 0,1,2,. . .,t�2:

It ¼ ð0,v�ð4sþ2Þ,1,ð4sþ2ÞÞ, where s¼ t�1

Series 2.4. If v¼12tþwþ2, where t positive integer and w¼0, 1 then GN2-design for k¼6 can be generated by developing

(tþ1) initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�12, n2¼11 for w¼0, (ii) n1¼v�11, n2¼10for w¼1.

Ijþ1 ¼ ð0,v�ð6jþ1Þ,1,v�ð6jþ2Þ,2,ð6jþ7ÞÞ, j¼ 0,2,4,. . .,t�1:

Ijþ1 ¼ ð0,v�6j,2,ð6jþ5Þ,1,ð6jþ6ÞÞ, j¼ 1,3,5,. . .,t�1:

Itþ1 ¼ ð0,v�ð6t�4Þ,2,v�ð6t�2Þ,3,4Þ for t odd :

Itþ1 ¼ ð0,v�ð6t�3Þ,2,v�ð6t�2Þ,3,4Þ for t even :

Series 2.5. If v¼12tþwþ4, where t positive integer and w¼0, 1 then GN2-design for k¼6 can be generated by developing

(tþ1) initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�10, n2¼9 for w¼0, (ii) n1¼v�9, n2¼8 for

w¼1.

Ijþ1 ¼ ð0,v�ð6jþ1Þ,1,v�ð6jþ2Þ,2,ð6jþ7ÞÞ, j¼ 0,2,4,. . .,t�1:

Ijþ1 ¼ ð0,v�6j,2,ð6jþ5Þ,1,ð6jþ6ÞÞ, j¼ 1,3,5,. . .,t�1:

Itþ1 ¼ ð0,v�ð6t�1Þ,1,v�6t,2,3Þ

Series 2.6. If v¼12tþw�6, where t (41) integer and w¼0, 1 then GN2-design for k¼6 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�8, n2¼7 for w¼0, n1¼v�7, n2¼6 for w¼1.

Ijþ1 ¼ ð0,v�ð6jþ1Þ,1,v�ð6jþ2Þ,2,ð6jþ7ÞÞ, j¼ 0,2,4,. . .,t�2:

Ijþ1 ¼ ð0,v�6j,2,ð6jþ5Þ,1,ð6jþ6ÞÞ, j¼ 1,3,5,. . .,t�2:

It ¼ ð0,v�ð6sþ2Þ,1,ð6sþ1Þ,ð6sþ3Þ,ð6sþ2ÞÞ for t even:

It ¼ ð0,v�ð6sþ2Þ,1,ð6sþ2Þ,ð6sþ4Þ,ð6sþ3ÞÞ for t odd:

Series 2.7. If v¼16tþwþ4, where t positive integer and w¼0, 1 then GN2-design for k¼8 can be generated by developing (tþ1)initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�14, n2¼13 for w¼0, (ii) n1¼v�13, n2¼12 for w¼1.

Ijþ1 ¼ ð0,v�ð8jþ1Þ,1,v�ð8jþ2Þ,2,v�ð8jþ4Þ,3,ð8jþ8ÞÞ, j¼ 0,1,2,. . .,t�1:

Itþ1 ¼ ð0,v�ð8tþ1Þ,1,v�1,2,v�3,3,4Þ

Series 2.8. If v¼16tþwþ6, where t positive integer and w¼0, 1 then GN2-design for k¼8 can be generated by developing

(tþ1) initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�14, n2¼13 for w¼0, (ii) n1¼v�13, n2¼12

F. Shehzad et al. / Journal of Statistical Planning and Inference 141 (2011) 3808–38133810

for w¼1.

Ijþ1 ¼ ð0,v�ð8jþ1Þ,1,v�ð8jþ2Þ,2,v�ð8jþ4Þ,3,ð8jþ8ÞÞ, j¼ 0,1,2,. . .,t�1:

Itþ1 ¼ ð0,v�8t,1,v�ð8tþ1Þ,2,3,6,4Þ

Series 2.9. If v¼16tþw�8, where t (41) integer and w¼0, 1 then GN2-design for k¼8 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�10, n2¼9 for w¼0, (ii) n1¼v�9, n2¼8 for w¼1.

Ijþ1 ¼ ð0,v�ð8jþ1Þ,1,v�ð8jþ2Þ,2,v�ð8jþ4Þ,3,ð8jþ8ÞÞ, j¼ 0,1,2,. . .,t�2:

It ¼ ð0,v�ð8tþ1Þ,1,v�ð8tþ2Þ,2,3,6,4Þ,

Series 2.10. If v¼16tþw-6, where t positive integer and w¼0, 1 then GN2-design for k¼8 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�8, n2¼7 for w¼0, (ii) n1¼v�7, n2¼6 for w¼1.

Ijþ1 ¼ ð0,v�ð8jþ1Þ,1,v�ð8jþ2Þ,2,v�ð8jþ4Þ,3,ð8jþ8ÞÞ, j¼ 0,1,2,. . .,t�2:

It ¼ ð0,v�ð8sþ2Þ,1,v�ð8sþ3Þ,2,4,3,ð8sþ4ÞÞ, where s¼ t�1

Series 2.11. If v¼16tþw-4, where t positive integer and w¼0, 1 then GN2-design for k¼8 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�6, n2¼5 for w¼0, (ii) n1¼v�5, n2¼4 for w¼1.

Ijþ1 ¼ ð0,v�ð8jþ1Þ,1,v�ð8jþ2Þ,2,v�ð8jþ4Þ,3,ð8jþ8ÞÞ, j¼ 0,1,2,. . .,t�2:

It ¼ ð0,v�ð8sþ1Þ,1,v�ð8sþ2Þ,2,v�ð8sþ3Þ,3,4Þ, where s¼ t�1

3. Some new series of GN2-designs

In this section, some new series of GN2-designs are developed.

Series 3.1. If v¼12tþw�4, where t positive integer and w¼0, 1 then GN2-design for k¼6 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (ii) n1¼v�6, n2¼5 for w¼0, (ii) n1¼v�5, n2¼4 for w¼1.

Ijþ1 ¼ ð0,v�ð6jþ1Þ,1,v�ð6jþ2Þ,2,ð6jþ7ÞÞ, j¼ 0,2,4,. . .,t�2:

Ijþ1 ¼ ð0,v�6j,2,ð6jþ5Þ,1,ð6jþ6ÞÞ, j¼ 1,3,5,. . .,t�2:

It ¼ ð0,v�ð6sþ1Þ,1,v�ð6sþ2Þ,2,3Þ for t odd:

It ¼ ð0,v�6s,2,v�ð6sþ1Þ,3,4Þ for t even:

Series 3.2. If v¼12tþw�2, where t positive integer and w¼0, 1 then GN2-design for k¼6 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�4, n2¼3 for w¼0, (ii) n1¼v�3, n2¼2 for w¼1.

Ijþ1 ¼ ð0,v�ð6jþ1Þ,1,v�ð6jþ2Þ,2,ð6jþ7ÞÞ, j¼ 0,2,4,. . .,t�2:

Ijþ1 ¼ ð0,v�6j,2,ð6jþ5Þ,1,ð6jþ6ÞÞ, j¼ 1,3,5,. . .,t�2:

It ¼ ð0,v�ð6sþ2Þ,1,v�ð6sþ3Þ,2,ð6sþ3ÞÞ for t odd

It ¼ ð0,v�ð6sþ2Þ,1,v�ð6sþ3Þ,2,ð6sþ2ÞÞ for t even

Series 3.3. If v¼16tþw�2, where t positive integer and w¼0, 1 then GN2-design for k¼8 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�4, n2¼3 for w¼0, (ii) n1¼v�3, n2¼2 for w¼1.

Ijþ1 ¼ ð0,v�ð8jþ1Þ,1,v�ð8jþ2Þ,2,v� 8jþ4ð Þ,3,ð8jþ8ÞÞ, j¼ 0,1,2,. . .,t�2:

It ¼ ð0,v�ð8sþ2Þ,1,v�ð8sþ3Þ,2,v�ð8sþ4Þ,3,ð8sþ4ÞÞ, where s¼ t�1

Series 3.4. If v¼16tþwþ2, where t positive integer and w¼0, 1 then GN2-design for k¼8 can be generated by developing

(tþ1) initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�16, n2¼15 for w¼0, (ii) n1¼v�15, n2¼14for w¼1.

Ijþ1 ¼ ð0,v�ð8jþ1Þ,1,v�ð8jþ2Þ,2,v�ð8jþ4Þ,3,ð8jþ8ÞÞ; j¼ 0,1,2,. . .,t�1:

Itþ1 ¼ ð0,v�2,1,v�3,v�2,ð8t�3Þ,2ð8t�1Þ,ð8t�2ÞÞ

Series 3.5. If v¼20tþ w�10, where t (41) integer and w¼0, 1 then GN2-design for k¼10 can be generated by developing

t initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�12, n2¼11 for w¼0, (ii) n1¼v�11, n2¼10

F. Shehzad et al. / Journal of Statistical Planning and Inference 141 (2011) 3808–3813 3811

for w¼1.

Ijþ1 ¼ ð0,v�ð10jþ1Þ,1,v�ð10jþ2Þ,2,v�ð10jþ3Þ,3,v�ð10jþ5Þ,4,10jþ11Þ; j¼ 0,2,4,. . .,t�2:

Ijþ1 ¼ ð0,v�10j,2,v�ð10jþ1Þ,3,v�ð10jþ3Þ,4,v�ð10jþ4Þ,5,ð10jþ10ÞÞ; j¼ 1,3,5,. . .,t�2:

It ¼ ð0,v�ð10sþ2Þ,1,v�ð10sþ3Þ,2,4,3,ð10sþ3Þ,ð10sþ6Þ,ð10sþ2ÞÞ for t even :

It ¼ ð0,v�ð10sþ2Þ,1,v�ð10sþ3Þ,2,4,3,ð10sþ4Þ,ð10sþ7Þ,ð10sþ3ÞÞ for t odd:

Series 3.6. If v¼20tþw�8, where t positive integer and w¼0, 1 then GN2-design for k¼10 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�10, n2¼9 for w¼0, (ii) n1¼v�9, n2¼8 for w¼1.

Ijþ1 ¼ ð0,v�ð10jþ1Þ,1,v�ð10jþ2Þ,2,v�ð10jþ3Þ,3,v�ð10jþ5Þ,4,10jþ11Þ j¼ 0,2,4,. . .,t�2:

Ijþ1 ¼ ð0,v�10j,2,v�ð10jþ1Þ,3,v�ð10jþ3Þ,4,v�ð10jþ4Þ,5,ð10jþ10ÞÞ j¼ 1,3,5,. . .,t�2:

It ¼ ð0,v�ð10sþ1Þ,1,v�ð10sþ2Þ,2,v�ð10sþ3Þ,3,4,7,5Þ for t odd:

It ¼ ð0,v�10s,2,v�ð10sþ1Þ,3,v�ð10sþ2Þ,4,5,8,6Þ for t even:

Series 3.7. If v¼20tþw�6, where t positive integer and w¼0, 1 then GN2-design for k¼10 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�8, n2¼7 for w¼0, (ii) n1¼v�7, n2¼6 for w¼1.

Ijþ1 ¼ ð0,v�ð10jþ1Þ,1,v�ð10jþ2Þ,2,v�ð10jþ3Þ,3,v�ð10jþ5Þ,4,10jþ11Þ; j¼ 0,2,4,. . .,t�2:

Ijþ1 ¼ ð0,v�10j,2,v�ð10jþ1Þ,3,v�ð10jþ3Þ,4,v�ð10jþ4Þ,5,ð10jþ10ÞÞ; j¼ 1,3,5,. . .,t�2:

It ¼ ð0,v�ð10sþ2Þ,1,v�ð10sþ3Þ,2,v�ð10sþ4Þ,3,ð10sþ4Þ,ð10sþ7Þ,ð10sþ6ÞÞ for t odd:

It ¼ ð0,v�ð10sþ2Þ,1,v�ð10sþ3Þ,2,v�ð10sþ4Þ,3,ð10sþ3Þ,ð10sþ6Þ,ð10sþ5ÞÞ for t even:

Series 3.8. If v¼20tþw�4, where t positive integer and w¼0, 1 then GN2-design for k¼10 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�6, n2¼5 for w¼0, (ii) n1¼v�5, n2¼4 for w¼1.

Ijþ1 ¼ ð0,v�ð10jþ1Þ,1,v�ð10jþ2Þ,2,v�ð10jþ3Þ,3,v�ð10jþ5Þ,4,10jþ11Þ; j¼ 0,2,4,. . .,t�2:

Ijþ1 ¼ ð0,v�10j,2,v�ð10jþ1Þ,3,v�ð10jþ3Þ,4,v�ð10jþ4Þ,5,ð10jþ10ÞÞ; j¼ 1,3,5,. . .,t�2:

It ¼ ð0,v�ð10sþ1Þ,1,v�ð10sþ2Þ,2,v�ð10sþ3Þ,3,v�ð10sþ4Þ,4,5Þ; for t odd:

It ¼ ð0,v�10s,2,v�ð10sþ1Þ,3,v�ð10sþ2Þ,4,v�ð10sþ3Þ,5,6Þ; for t even:

Series 3.9. If v¼20tþw�2, where t positive integer and w¼0, 1 then GN2-design for k¼10 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�4, n2¼3 for w¼0, (ii) n1¼v�3, n2¼2 for w¼1.

Ijþ1 ¼ ð0,v�ð10jþ1Þ,1,v�ð10jþ2Þ,2,v�ð10jþ3Þ,3,v�ð10jþ5Þ,4,10jþ11Þ; j¼ 0,2,4,. . .,t�2:

Ijþ1 ¼ ð0,v�10j,2,v�ð10jþ1Þ,3,v�ð10jþ3Þ,4,v�ð10jþ4Þ,5,ð10jþ10ÞÞ; j¼ 1,3,5,. . .,t�2:

It ¼ ð0,v�ð10sþ1Þ,1,v�ð10sþ3Þ,2,v�ð10sþ4Þ,3,v�ð10sþ5Þ,4,ð10sþ7ÞÞ; for t odd:

It ¼ ð0,v�ð10sþ2Þ,1,v�ð10sþ3Þ,2,v�ð10sþ4Þ,3,v�ð10sþ5Þ,4,ð10sþ4ÞÞ; for t even:

Series 3.10. If v¼20tþwþ2, where t positive integer and w¼0, 1 then GN2-design for k¼10 can be generated by developing

(tþ1) initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�20, n2¼19 for w¼0, (ii) n1¼v�19, n2¼18for w¼1.

Ijþ1 ¼ ð0,v�ð10jþ1Þ,1,v�ð10jþ2Þ,2,v�ð10jþ3Þ,3,v�ð10jþ5Þ,4,10jþ11Þ; j¼ 0,2,4,. . .,t�1:

Ijþ1 ¼ ð0,v�10j,2,v�ð10jþ1Þ,3,v�ð10jþ3Þ,4,v�ð10jþ4Þ,5,ð10jþ10ÞÞ; j¼ 1,3,5,. . .,t�1:

Itþ1 ¼ ð0,v�ð10t�3Þ,1,v�ð10t�2Þ,2,4,3,7,13,10Þ; for t odd:

Itþ1 ¼ ð0,v�ð10t�4Þ,1,v�ð10t�3Þ,2,v�ð10t�2Þ,3,5,4,7Þ; for t even:

Series 3.11. If v¼20tþwþ4, where t positive integer and w¼0, 1 then GN2-design for k¼10 can be generated by developing

(tþ1) initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�18, n2¼17 for w¼0, (ii) n1¼v�1i, n2¼16for w¼1.

Ijþ1 ¼ ð0,v�ð10jþ1Þ,1,v�ð10jþ2Þ,2,v�ð10jþ3Þ,3,v�ð10jþ5Þ,4,10 jþ11Þ; j¼ 0,2,4,. . .,t�1:

Ijþ1 ¼ ð0,v�10j,2,v�ð10jþ1Þ,3,v�ð10jþ3Þ,4,v�ð10jþ4Þ,5,ð10jþ10ÞÞ; j¼ 1,3,5,. . .,t�1:

Itþ1 ¼ ð0,v�ð10t�1Þ,1,v�10t,2,3,6,4,11,5Þ,

Series 3.12. If v¼20tþwþ6, where t positive integer and w¼0, 1 then GN2-design for k¼10 can be generated by developing

(tþ1) initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�16, n2¼15 for w¼0, (ii) n1¼v�15, n2¼14

F. Shehzad et al. / Journal of Statistical Planning and Inference 141 (2011) 3808–38133812

for w¼1.

Ijþ1 ¼ ð0,v�ð10jþ1Þ,1,v�ð10jþ2Þ,2,v�ð10jþ3Þ,3,v�ð10jþ5Þ,4,10jþ11Þ; j¼ 0,2,4,. . .,t�1:

Ijþ1 ¼ ð0,v�10j,2,v�ð10jþ1Þ,3,v�ð10jþ3Þ,4,v�ð10jþ4Þ,5,ð10jþ10ÞÞ; j¼ 1,3,5,. . .,t�1:

Itþ1 ¼ ð0,v�ð10t�2Þ,1,v�ð10t�1Þ,2,v�10t,3,5,4,8Þ,

Series 3.13. If v¼20tþwþ8, where t positive integer and w¼0, 1 then GN2-design for k¼10 can be generated by developing

(tþ1) initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�14, n2¼13 for w¼0, (ii) n1¼v�13, n2¼12for w¼1.

Ijþ1 ¼ ð0,v�ð10jþ1Þ,1,v�ð10jþ2Þ,2,v�ð10jþ3Þ,3,v�ð10jþ5Þ,4,10jþ11Þ; j¼ 0,2,4,. . .,t�1:

Ijþ1 ¼ ð0,v�10j,2,v�ð10jþ1Þ,3,v�ð10jþ3Þ,4,v�ð10jþ4Þ,5,ð10jþ10ÞÞ; j¼ 1,3,5,. . .,t�1:

Itþ1 ¼ ð0,v�ð10t�1Þ,1,v�10t,2,v�ð10t�1Þ,3,5,4,7Þ,

4. GN2-designs for k¼12

Series 4.1. If v¼24tþw�12, where t (41) integer and w¼0, 1 then GN2-design for k¼12 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�14, n2¼13 for w¼0, (ii) n1¼v�13, n2¼12 for

w¼1.

Ijþ1 ¼ ð0,v�ð12jþ1Þ,1,v�ð12jþ2Þ,2,v�ð12jþ3Þ,3,v�ð12jþ5Þ,4,v�ð12jþ6Þ,5,ð12jþ12ÞÞ, j¼ 0,1,2,. . .,t�2:

It ¼ ð0,v�ð12sþ1Þ,1,v�ð12sþ2Þ,2,v�ð12sþ3Þ,3,4,7,5,11,7Þ, where s¼ t�1:

Series 4.2. If v¼24tþw�10, where t positive integer and w¼0, 1 then GN2-design for k¼12 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�12, n2¼11 for w¼0, (ii) n1¼v�11, n2¼10 for

w¼1.

Ijþ1 ¼ ð0,v�ð12jþ1Þ,1,v�ð12jþ2Þ,2,v�ð12jþ3Þ,3,v�ð12jþ5Þ,4,v�ð12jþ6Þ,5,ð12jþ12ÞÞ, j¼ 0,1,2,. . .,t�2:

It ¼ ð0,v�1,1,v�2,2,ð12sþ5Þ,3,v�ð12sþ1Þ,4,v�ð12sþ2Þ,5,ð12sþ6ÞÞ, where s¼ t�1:

Series 4.3. If v¼24tþw�8, where t positive integer and w¼0, 1 then GN2-design for k¼12 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�10, n2¼9 for w¼0, (ii) n1¼v�9, n2¼8 for w¼1.

Ijþ1 ¼ ð0,v�ð12jþ1Þ,1,v�ð12jþ2Þ,2,v�ð12jþ3Þ,3,v�ð12jþ5Þ,4,v�ð12jþ6Þ,5,ð12jþ12ÞÞ, j¼ 0,1,2,. . .,t�2:

It ¼ 0,v�ð12sþ1Þ,1,v�ð12sþ2Þ,2,v�ð12sþ3Þ,3,v�ð12sþ4Þ,4,7,6,8ð Þ, where s¼ t�1:

Series 4.4. If v¼24tþw�6 where t positive integer and w¼0, 1 then GN2-design for k¼12 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�8, n2¼7 if w¼0, (ii) n1¼v�7, n2¼6 for w¼1.

Ijþ1 ¼ ð0,v�ð12jþ1Þ,1,v�ð12jþ2Þ,2,v�ð12jþ3Þ,3,v�ð12jþ5Þ,4,v�ð12jþ6Þ,5,ð12jþ12ÞÞ, j¼ 0,1,2,. . .,t�2:

It ¼ ð0,v�ð12sþ2Þ,1,v�ð12sþ3Þ,2,v�ð12sþ4Þ,3,v�ð12sþ5Þ,4,6,5,ð12sþ6ÞÞ, where s¼ t�1:

Series 4.5. If v¼24tþw�4, where t positive integer and w¼0, 1 then GN2-design for k¼12 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�8, n2¼7 for w¼0, (ii) n1¼v�7, n2¼6 for w¼1.

Ijþ1 ¼ ð0,v�ð12jþ1Þ,1,v�ð12jþ2Þ,2,v�ð12jþ3Þ,3,v�ð12jþ5Þ,4,v�ð12jþ6Þ,5,ð12jþ12ÞÞ, j¼ 0,1,2,. . .,t�2:

It ¼ ð0,v�ð12sþ1Þ,1,v�ð12sþ2Þ,2,v�ð12sþ3Þ,3,v�ð12sþ4Þ,4,v�ð12sþ5Þ,5,6Þ, where s¼ t�1:

Series 4.6. If v¼24tþw�2 where t positive integer and w¼0, 1 then GN2-design for k¼12 can be generated by developing t

initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�4, n2¼3 for w¼0, (ii) n1¼v�3, n2¼2 for w¼1.

Ijþ1 ¼ ð0,v�ð12jþ1Þ,1,v�ð12jþ2Þ,2,v�ð12jþ3Þ,3,v�ð12jþ5Þ,4,v�ð12jþ6Þ,5,ð12jþ12ÞÞ, j¼ 0,1,2,. . .,t�2:

It ¼ ð0,v�ð12sþ2Þ,1,v�ð12sþ3Þ,2,v�ð12sþ4Þ,3,v�ð12sþ5Þ,4,v�ð12sþ6Þ,5,ð12sþ6ÞÞ, where s¼ t�1:

Series 4.7. If v¼24t, where t positive integer then GN2-design for k¼12 can be generated by developing t initial blocks cyclically

mod v with parameters l01 ¼ 1, l02 ¼ 2 , n1¼v�2, n2¼1.

Ijþ1 ¼ ð0,v�ð12jþ1Þ,1,v�ð12jþ2Þ,2,v�ð12jþ3Þ,3,v�ð12jþ5Þ,4,v�ð12jþ6Þ,5,ð12jþ12ÞÞ, j¼ 0,1,2,. . .,t�1:

Series 4.8. If v¼24tþwþ2, where t positive integer and w¼0, 1 then GN2-design for k¼12 can be generated by developing

(tþ1) initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�24, n2¼23 for w¼0, (ii) n1¼v�23, n2¼22

F. Shehzad et al. / Journal of Statistical Planning and Inference 141 (2011) 3808–3813 3813

for w¼1.

Ijþ1 ¼ ð0,v�ð12jþ1Þ,1,v�ð12jþ2Þ,2,v�ð12jþ3Þ,3,v�ð12jþ5Þ,4,v�ð12jþ6Þ,5,ð12jþ12ÞÞ, j¼ 0,1,2,. . .,t�1:

Itþ1 ¼ ð0,v�2,1,v�3,3,v�ð12t�7Þ,4,v�ð12t�6Þ,5,v�ð12t�5Þ,6,7Þ

Series 4.9. If v¼24tþwþ4, where t positive integer and w¼0, 1 then GN2-design for k¼12 can be generated by developing

(tþ1) initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�22, n2¼21 for w¼0, (ii) n1¼v�21, n2¼20for w¼1.

Ijþ1 ¼ ð0,v�ð12jþ1Þ,1,v�ð12jþ2Þ,2,v�ð12jþ3Þ,3,v�ð12jþ5Þ,4,v�ð12jþ6Þ,5,ð12jþ12ÞÞ, j¼ 0,1,2,. . .,t�1:

Itþ1 ¼ ð0,v�1,1,v�2,2,v�4,3,v�ð12t�4Þ,4,v�ð12t�3Þ,5,10Þ

Series 4.10. If v¼24tþwþ6, where t positive integer and w¼0, 1 then GN2-design for k¼12 can be generated by developing

(tþ1) initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�20, n2¼19 for w¼0, (ii) n1¼v�19, n2¼18for w¼1.

Ijþ1 ¼ ð0,v�ð12jþ1Þ,1,v�ð12jþ2Þ,2,v�ð12jþ3Þ,3,v�ð12jþ5Þ,4,v�ð12jþ6Þ,5,ð12jþ12ÞÞ, j¼ 0,1,2,. . .,t�1:

Itþ1 ¼ ð0,v�2,2,v�3,3,v�4,5,v�ð12t�5Þ,6,v�ð12t�4Þ,7,8Þ

Series 4.11. If v¼24tþwþ8, where t positive integer and w¼0, 1 then GN2-design for k¼12 can be generated by developing

(tþ1) initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�18, n2¼17 for w¼0, (ii) n1¼v�17, n2¼16for w¼1.

Ijþ1 ¼ ð0,v�ð12jþ1Þ,1,v�ð12jþ2Þ,2,v�ð12jþ3Þ,3,v�ð12jþ5Þ,4,v�ð12jþ6Þ,5,ð12jþ12ÞÞ, j¼ 0,1,2,. . .,t�1:

Itþ1 ¼ ð0,v�1,1,v�3,3,v�2,5,v�ð12t�4Þ,6,v�ð12t�3Þ,7,10Þ,

Series 4.12. If v¼24tþwþ10, where t positive integer and w¼0, 1 then GN2-design for k¼12 can be generated by developing

(tþ1) initial blocks cyclically mod v with parameters l01 ¼ 1, l02 ¼ 2 and (i) n1¼v�16, n2¼15 for w¼0, (ii) n1¼v�15, n2¼14for w¼1.

Ijþ1 ¼ ð0,v�ð12jþ1Þ,1,v�ð12jþ2Þ,2,v�ð12jþ3Þ,3,v�ð12jþ5Þ,4,v�ð12jþ6Þ,5,ð12jþ12ÞÞ, j¼ 0,1,2,. . .,t�1:

Itþ1 ¼ ð0,v�1,1,v�3,3,v�ð12t�3Þ,4,v�ð12t�2Þ,5,v�ð12t�1Þ,6,9Þ,

References

Ahmed, R., Akhtar, M., Tahir, M.H., 2009. Economical generalized neighbor designs for use in Serology. Computational Statistics and Data Analysis 53,4584–4589.

Akhtar, M., Ahmed, R., Shehzad, F., 2010. Generalized neighbor designs in circular blocks. World Applied Science Journal 8 (2), 161–166.Chaure, N.K., Misra, B.L., 1996. On construction of generalized neighbor design. Sankhya B 58, 245–253.Kedia, R.G., Misra, B.L., 2008. On construction of generalized neighbor design of use in serology. Statistics and Probability Letters 18, 254–256.Mishra, N.S., 2007. Families of proper generalized neighbor designs. Journal of Statistical Planning and Inference 137, 1681–1686.Misra, B.L., Bhagwandas, Nutan, 1991. Families of neighbor designs and their analysis. Communications in Statistics—Simulation and Computation 20 (2

and 3), 427–436.Preece, D.A., 1994. Balanced ouchterlony neighbor designs. Journal of Combinatorial Mathematics and Combinatorial Computing 15, 197–219.Rees, D.H., 1967. Some designs of use in serology. Biometrics 23, 779–791.Zafaryab, M., Shehzad, F., Ahmed, R., 2010. Proper generalized neighbor designs in circular blocks. Journal of Statistical Planning and Inference 140 (11),

3498–3504.