Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 4s (2024) 

 

392 https://internationalpubls.com 

Association Schemes for Some Finite Group Rings II 

 

Anuradha Sabharwal 1*, Pooja Yadav2 and R. K. Sharma3 

1Department of Mathematics, University of Delhi, Delhi-110007, India. 

2Department of Mathematics, Kamala Nehru College, University of Delhi, New Delhi-110049, India. 

3Department of Mathematics, Indian Institute of Technology Delhi, New Delhi-110016, India. 

*Corresponding Author: Anuradha Sabharwal 

 

Article History: 

Received: 20-04-2024 

Revised: 10-06-2024 

Accepted: 24-06-2024 

Abstract:  

Association schemes have been used in coding theory and other combinatorial problems. In 

this paper, we construct association schemes for the abelian groups ℤ2
𝑟 ,  ℤ𝑛1 × ℤ𝑛2 ×⋯×

ℤ𝑛𝑟, set of 𝑛 × 𝑛 matrices over ℤ𝑚 and for the general linear group of order 2 over ℤ2, ℤ4, 

and ℤ6. We also obtain association schemes for symmetric groups and alternating groups of 

degree 4 and 5 using canonical forms. 

Keywords: Group ring; Association scheme. 

2020 Mathematics Subject Classification. 05E30. 

 

1. Introduction 

Association schemes (AS) introduced by Bose and Shimamoto [1], play a key role in the study of 

algebraic combinatorics. It has applications in graph theory, coding theory, group theory and design 

theory [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Jørgensen’s list of non-symmetric association schemes with classes 

smaller than 96 in vertices in [12], inspires us to research non-symmetric association schemes for 

various finite groups and group rings. In our previous work [13], we have constructed non symmetric 

commutative AS for symmetric groups, dihedral groups, abelian groups ℤ𝑝
𝑟  (where 𝑝 is an odd prime), 

 ℤ𝑝1 × ℤ𝑝2 ×⋯× ℤ𝑝𝑟 (𝑝𝑖
′𝑠 are distinct primes), finite group rings over ℤ𝑛 and circulant matrices over 

ℤ𝑝, for 𝑝 prime. In the present study, we construct association schemes for the abelian groups   ℤ2
𝑟 =

ℤ2 × ℤ2 ×⋯× ℤ2⏟          
𝑟 𝑡𝑖𝑚𝑒𝑠

 and ℤ𝑛1 × ℤ𝑛2 ×⋯× ℤ𝑛𝑟, the general linear group 𝐺𝐿(2, ℤ2),  𝐺𝐿(2, ℤ4), 

𝐺𝐿(2, ℤ6), symmetric group and alternating group of order 4 and 5. 

In this paper, let 𝒫 denote the partition of 𝑌 × 𝑌 , where 𝑌 is a finite set, and let  ℤ𝑝
𝑟 =

ℤ𝑝 × ℤ𝑝 ×⋯× ℤ𝑝⏟            
𝑟 𝑡𝑖𝑚𝑒𝑠

. Some basic literature and preliminaries on association schemes are given below. 

 

1.1. Association Scheme 

Definition 1.  Let 𝒫 be a partition of 𝑌 × 𝑌  where 𝑌 is a finite set and  let 𝒮0, 𝒮1, … , 𝒮𝓃 binary relations 

on 𝒫. Then 𝒜 = (𝑌,𝒫)  forms 𝑛-class association scheme if the subsequent conditions hold: 

(1) Identity relation 𝒮0 = {(𝑎, 𝑎): 𝑎 ∈ 𝑌} ∈ 𝒫. 



Communications on Applied Nonlinear Analysis 

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(2) 𝒮∗ = {(𝑎, 𝑏): (𝑏, 𝑎) ∈ 𝒮} ∈ 𝒫 for any relation 𝒮 ∈ 𝒫. 

(3) If (𝑎, 𝑏) ∈ 𝒮𝑘, the number of elements 𝑐 ∈ 𝑌 such that (𝑎, 𝑐) ∈ 𝑆𝑙, (𝑐, 𝑏) ∈ 𝑆𝑚 is a constant 𝑝𝑙𝑚
𝑘  

not depending on choice of 𝑎 and 𝑏 for all integers 0 ≤ 𝑘, 𝑙,𝑚 ≤ 𝑛. 

The integers {𝑝𝑙𝑚
𝑘 }0≤𝑘,𝑙,𝑚≤𝑛 are called parameters or intersection numbers of 𝒜. If each relation 𝒮 

in 𝒫 is a symmetric relation, that is, 𝒮 = 𝒮∗, then 𝒜 is called symmetric association scheme and if 

𝑝𝑙𝑚
𝑘 = 𝑝𝑚𝑙

𝑘  ∀ 0 ≤ 𝑘, 𝑙,𝑚 ≤ 𝑛, then it is called commutative association scheme. Let the set 𝑎𝒮 =

{𝑏 ∈ 𝑌| (𝑎, 𝑏) ∈ 𝒮 } for 𝑎 ∈ 𝑌 and 𝒮 ∈ 𝒫. The elements 𝑎 and 𝑏 in 𝑌 are called 𝑘𝑡ℎ associates if 

(𝑎, 𝑏) ∈ 𝒮𝑘 with 𝑎 ≠ 𝑏. Note that every symmetric association scheme is commutative. With 

regards to more basic association schemes results, refer [7, 14]. 

Definition 2.  A finite group 𝐺 with the conjugacy classes 𝐶0, 𝐶1, … , 𝐶𝑑 produces a commutative 

association scheme with a class of relations on 𝐺 defined by 𝒮𝑘 = {(𝑎, 𝑏)| 𝑏𝑎
−1 ∈ 𝐶𝑘} ∀ 0 ≤ 𝑘 ≤ 𝑑. 

This scheme is called the group association scheme of 𝐺. 

Association schemes can be determined for all those groups whose conjugacy classes are known. 

Lemma 1.  Let 𝑌 = ℤ𝑛 and 𝒮𝑘 defines relations on 𝒫 by 𝒮𝑘 = (𝑎, 𝑏)|𝑎 = 𝑘 + 𝑏|𝑎, 𝑏 ∈ ℤ𝑛∀𝑘 ∈ ℤ𝑛. 

Then (𝑌, 𝒫) is a non symmetric commutative association scheme with parameters given by:                

𝑝𝑙𝑚
𝑘 = {

1 𝑖𝑓𝑘 = 𝑙 +𝑚,
0 𝑖𝑓𝑘 ≠ 𝑙 + 𝑚

  

where 𝑘, 𝑙, 𝑚 ∈ 𝑍𝑛. 

Proof. Since ℤ𝑛 is an abelian group, (ℤ𝑛, 𝒫) under given relations becomes a commutative association 

scheme. For arbitrary relations 𝒮𝑙 , 𝒮𝑚, 𝒮𝑘 in 𝒫, we find cardinality 𝑝𝑙𝑚
𝑘 = |𝑎𝒮𝑙 ∩ 𝑏𝒮𝑚

∗ | whenever 

(𝑎, 𝑏) ∈ 𝒮𝑘. Let (𝑎, 𝑏) be an arbitrary element of 𝑌 in 𝒮𝑘 and let 𝑎𝒮𝑙 = 𝑎
′ and 𝑏𝒮𝑚

∗ = 𝑏′. This implies, 

𝑎 = 𝑎′ + 𝑙, 𝑏′ = 𝑏 +𝑚 and 𝑎 = 𝑏 + 𝑘 and we get 𝑝𝑙𝑚
𝑘 = 1 if 𝑎′ = 𝑏′ which implies 𝑝𝑙𝑚

𝑘 = 1 if 𝑘 =

𝑙 + 𝑚. Further, 𝑝𝑙𝑚
𝑘 = 0 if 𝑎′ ≠ 𝑏′, equivalently 𝑝𝑙𝑚

𝑘 = 0, whenever 𝑘 ≠ 𝑙 + 𝑚. ◻ 

In the next section of this paper, we work on non symmetric association scheme of the cyclic groups 

ℤ2
𝑟, ℤ𝑛1 × ℤ𝑛2 ×⋯× ℤ𝑛𝑟 and general linear group of order 2 over ℤ2. 

2. Association schemes for some finite Groups  

Theorem 1.  Let 𝑌 = ℤ2
𝑟 = ℤ2 × ℤ2 ×⋯× ℤ2 and 𝑟 ≥ 2. We define relations 𝒮𝑘 on 𝒫 by 

𝒮𝑘 = {(𝑎, 𝑏)| 𝑏𝑠 ≡ (𝑡𝑠 + 𝑎𝑠) 𝑚𝑜𝑑 2 | 𝑡𝑠 ∈ ℤ2 ∀ 1 ≤ 𝑠 ≤ 𝑟,

𝑎 = (𝑎1, 𝑎2, … , 𝑎𝑟), 𝑏 = (𝑏1, 𝑏2, … , 𝑏𝑟) ∈ 𝑌}
 

where 𝑘 = 2𝑟−1𝑡1 + 2
𝑟−2𝑡2 +⋯+ 2𝑡𝑟−1 + 𝑡𝑟 . 

Then 𝒜 = (𝑌,𝒫) is a symmetric and commutative association scheme with parameters  

𝑝𝑙𝑚
𝑘 = {1 if 𝑡𝑠

(𝑘) ≡ 𝑡𝑠
(𝑙) + 𝑡𝑠

(𝑚)𝑚𝑜𝑑 2 ∀ 1 ≤ 𝑠 ≤ 𝑟
0 otherwise

 

where 𝑘 = ∑ 2𝑟−𝑠𝑡𝑠
(𝑘)𝑟

𝑠=1 ; 𝑙 = ∑ 2𝑟−𝑠𝑡𝑠
(𝑙)𝑟

𝑠=1 ; 𝑚 = ∑ 2𝑟−𝑠𝑡𝑠
(𝑚)𝑟

𝑠=1  for some 𝑡𝑠
(𝑘), 𝑡𝑠

(𝑙), 𝑡𝑠
(𝑚) ∈ ℤ2. 



Communications on Applied Nonlinear Analysis 

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Proof. Observe that |𝑌| = 2𝑟 = |𝒮𝑘| for all 0 ≤ 𝑘 ≤ 2𝑟 − 1. The relations 𝒮𝑘 are disjoint and 

⋃𝒮𝑘:0 ≤ 𝑘 ≤ 2𝑟 − 1 = 𝒫. For arbitrary relations 𝒮𝑙 , 𝒮𝑚, 𝒮𝑘 in 𝒫, we prove that the parameters 𝑝𝑙𝑚
𝑘 =

|𝑎𝒮𝑙 ∩ 𝑏𝒮𝑚
∗ | is constant for (𝑎, 𝑏) ∈ 𝒮𝑘. Let (𝑎, 𝑏) be an arbitrary element of 𝑌 in 𝒮𝑘 where 𝑎 =

(𝑎1, 𝑎2, … , 𝑎𝑟), 𝑏 = (𝑏1, 𝑏2, … , 𝑏𝑟). Let 𝑎𝒮𝑙 = 𝑎
′ = (𝑎′1, 𝑎′2, … , 𝑎′𝑟) and 𝑏𝒮𝑚

∗ = 𝑏′ =

(𝑏′1, 𝑏′2, … , 𝑏′𝑟). Now (𝑎, 𝑏) ∈ 𝒮𝑘 implies 𝑏𝑠 ≡ (𝑡𝑠
(𝑘) + 𝑎𝑠) 𝑚𝑜𝑑 2 where 𝑘 = ∑ 2𝑟−𝑠𝑡𝑠

(𝑘)𝑟
𝑠=1  for 

some 𝑡𝑠
(𝑘) ∈ ℤ2 ∀ 1 ≤ 𝑠 ≤ 𝑟. Similarly, (𝑎, 𝑎′) ∈ 𝒮𝑙  and (𝑏′, 𝑏) ∈ 𝒮𝑚, implies 𝑎′𝑠 ≡ (𝑡𝑠

(𝑙) +

𝑎𝑠) 𝑚𝑜𝑑 2 where 𝑙 = ∑ 2𝑟−𝑠𝑡𝑠
(𝑙)𝑟

𝑠=1  for some 𝑡𝑠
(𝑙) ∈ ℤ2 ∀ 1 ≤ 𝑠 ≤ 𝑟, and 𝑏𝑠 ≡ (𝑡𝑠

(𝑙) + 𝑏′𝑠) 𝑚𝑜𝑑 2 

where 𝑚 = ∑ 2𝑟−𝑠𝑡𝑠
(𝑚)𝑟

𝑠=1  for some 𝑡𝑠
(𝑚) ∈ ℤ2 ∀ 1 ≤ 𝑠 ≤ 𝑟. We observe that 𝑝𝑙𝑚

𝑘 = 1 if and only if 

𝑎′ = 𝑏′ that is, if 𝑡𝑠
(𝑘) ≡ (𝑡𝑠

(𝑙) + 𝑡𝑠
(𝑚)) 𝑚𝑜𝑑 2 for all 1 ≤ 𝑠 ≤ 𝑟.  

To show (𝑌, 𝒫) is a symmetric association scheme, we prove that 𝒮𝑘
∗ = 𝒮𝑘 for all 0 ≤ 𝑘 < 2𝑟. Let 

𝒮𝑘
∗ = 𝒮𝐾 where 𝐾 = ∑ 2𝑟−𝑠𝑇𝑠

𝑟
𝑠=1  and 𝑘 = ∑ 2𝑟−𝑠𝑡𝑠

𝑟
𝑠=1  for some 𝑡𝑠, 𝑇𝑠 ∈ ℤ2. If (𝑏, 𝑎) ∈ 𝒮𝐾, then 

further 𝑎𝑠 ≡ 𝑇𝑠 + 𝑏𝑠 𝑚𝑜𝑑 2 and 𝑏𝑠 ≡ 𝑡𝑠 + 𝑎𝑠 𝑚𝑜𝑑 2 which implies that 𝑇𝑠 = 𝑡𝑠 ∀ 1 ≤ 𝑠 ≤ 𝑟. Thus 

𝐾 = 𝑘, and hence, 𝒮𝑘
∗ = 𝒮𝑘. ◻ 

Theorem 2.  Let 𝑌 = ℤ𝑛1 × ℤ𝑛2 ×⋯× ℤ𝑛𝑟, where 𝑛1, 𝑛2, … , 𝑛𝑟 are pairwise co-prime. The relations 

𝒮𝑘 on 𝒫 defined by 

𝒮𝑘 = {(𝑎, 𝑏)| 𝑏𝑠 ≡ (𝑘 + 𝑎𝑠) 𝑚𝑜𝑑 𝑛𝑠 ∀ 1 ≤ 𝑠 ≤ 𝑟|

                                       𝑎 = (𝑎1, 𝑎2, … , 𝑎𝑟), 𝑏 = (𝑏1, 𝑏2, … , 𝑏𝑟) ∈ 𝑌}
 

where 0 ≤ 𝑘 < 𝑛1𝑛2⋯𝑛𝑟,  

is a non-symmetric and commutative association scheme with parameters 

 𝑝𝑙𝑚
𝑘 = {

1 if 𝑘 ≡ (𝑙 + 𝑚) 𝑚𝑜𝑑 𝑛𝑠 ∀ 1 ≤ 𝑠 ≤ 𝑟
0 otherwise

  

where 0 ≤ 𝑘, 𝑙,𝑚 ≤ 𝑛1𝑛2⋯𝑛𝑟 − 1. 

Proof. Observe that |𝑌| = 𝑛1𝑛2⋯𝑛𝑟 = |𝒮𝑘| for all 0 ≤ 𝑘 < 𝑛1𝑛2⋯𝑛𝑟. The relations 𝒮𝑘 being 

disjoint, form partition of 𝒫. For arbitrary relations 𝒮𝑙 , 𝒮𝑚, 𝒮𝑘 in 𝒫, we prove that for each pair 𝑎, 𝑏 

with (𝑎, 𝑏) ∈ 𝒮𝑘 , the number of elements in the set {𝑐 ∈ 𝑌| (𝑎, 𝑐) ∈ 𝒮𝑙 , (𝑐, 𝑏) ∈ 𝒮𝑚} is invariant. Let 

(𝑎, 𝑏) ∈ 𝒮𝑘 where 𝑎 = (𝑎1, 𝑎2, … , 𝑎𝑟), 𝑏 = (𝑏1, 𝑏2, … , 𝑏𝑟) ∈ 𝑌. Further, suppose that 𝑎𝒮𝑙 = 𝑎
′ =

(𝑎′1, 𝑎′2, … , 𝑎′𝑟), 𝑏𝒮𝑚
∗ = 𝑏′ = (𝑏′1, 𝑏′2, … , 𝑏′𝑟). Now (𝑎, 𝑏) ∈ 𝒮𝑘 implies 𝑏𝑠 ≡ (𝑘 +

𝑎𝑠) 𝑚𝑜𝑑 𝑛𝑠 ∀ 1 ≤ 𝑠 ≤ 𝑟. Similarly, (𝑎, 𝑎′) ∈ 𝒮𝑙  and (𝑏′, 𝑏) ∈ 𝒮𝑚. We get 𝑎′𝑠 ≡ (𝑙 + 𝑎𝑠) 𝑚𝑜𝑑 𝑛𝑠 and 

𝑏𝑠 ≡ (𝑚 + 𝑏′𝑠) 𝑚𝑜𝑑 𝑛𝑠 ∀ 1 ≤ 𝑠 ≤ 𝑟.  

 

The above equations give that, 𝑝𝑙𝑚
𝑘 = 1 if and only if 𝑎′ = 𝑏′. That is, 𝑝𝑙𝑚

𝑘 = 1 whenever 𝑘 ≡

(𝑙 + 𝑚) 𝑚𝑜𝑑 𝑛𝑠 ∀ 1 ≤ 𝑠 ≤ 𝑟. Hence, (𝑌, 𝒫) is a non-symmetric and commutative association 

scheme. ◻ 

Theorem 3.  Let 𝑌 = ℤ𝑛1 × ℤ𝑛2 ×⋯× ℤ𝑛𝑟 ,  where 2 < 𝑛1 ≤ 𝑛2 ≤ ⋯ ≤ 𝑛𝑟 and 

𝑔𝑐𝑑(𝑛1, 𝑛2, … , 𝑛𝑟) ≠ 1. The relations 𝒮𝑘 on 𝒫 defined by  



Communications on Applied Nonlinear Analysis 

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𝒮𝑘 = {(𝑎, 𝑏)|𝑏𝑟 ≡ (𝑘 + 𝑎𝑟)𝑚𝑜𝑑𝑛𝑟 , 𝑏𝑠 ≡ (𝑘 + 𝑡𝑠 + 𝑎𝑠) 𝑚𝑜𝑑 𝑛𝑠  ∀ 1 ≤ 𝑠 < 𝑟|

𝑡𝑠 ∈ 𝑍𝑛𝑠  , 𝑎 = (𝑎1, 𝑎2, … , 𝑎𝑟), 𝑏 = (𝑏1, 𝑏2, … , 𝑏𝑟) ∈ 𝑌}
 

 where 𝑘 = 𝑛2𝑛3⋯𝑛𝑟𝑡1 + 𝑛3𝑛4⋯𝑛𝑟𝑡2 +⋯+ 𝑛𝑟𝑡𝑟−1 + 𝑡𝑟 ,   

is a non-symmetric and commutative association scheme with parameters 

𝑝𝑙𝑚
𝑘 = {

1 𝑖𝑓𝑘 ≡ (𝑙 + 𝑚) 𝑚𝑜𝑑 𝑛𝑟 , and
0 otherwise

𝑘 + 𝑡𝑠
(𝑘) ≡ (𝑙 + 𝑚 + 𝑡𝑠

(𝑙) + 𝑡𝑠
(𝑚)) 𝑚𝑜𝑑 𝑛𝑠 ∀ 1 ≤ 𝑠 < 𝑟 

where 𝑘 = 𝑛2𝑛3⋯𝑛𝑟𝑡1
(𝑘) + 𝑛3𝑛4⋯𝑛𝑟𝑡2

(𝑘) +⋯+ 𝑛𝑟𝑑𝑟−1
(𝑘) + 𝑡𝑟

(𝑘);  𝑙 = 𝑛2𝑛3⋯𝑛𝑟𝑡1
(𝑙) +

𝑛3𝑛4⋯𝑛𝑟𝑡2
(𝑙) +⋯+ 𝑛𝑟𝑡𝑟−1

(𝑙) + 𝑡𝑟
(𝑙);  𝑚 = 𝑛2𝑛3⋯𝑛𝑟𝑡1

(𝑚) + 𝑛3𝑛4⋯𝑛𝑟𝑡2
(𝑚) +⋯+ 𝑛𝑟𝑡𝑟−1

(𝑚) + 𝑡𝑟
(𝑚)

 for 

some 𝑡𝑠
(𝑘)
, 𝑡𝑠
(𝑙)
, 𝑡𝑠
(𝑚)

∈ ℤ𝑛𝑠 where 1 ≤ 𝑠 ≤ 𝑟. 

For each 0 ≤ 𝑘 < 𝑛1𝑛2…𝑛𝑟, 𝒮𝑘
∗ = 𝒮𝐾 can be calculated by solving the following equations:  

𝐾 + 𝑘 ≡ 0 𝑚𝑜𝑑 𝑛𝑟
𝐾 + 𝑘 + 𝑇𝑠 + 𝑡𝑠 ≡ 0 𝑚𝑜𝑑 𝑛𝑠 ∀ 1 ≤ 𝑠 ≤ 𝑟 − 1

 

where 𝑘 = 𝑛2𝑛3⋯𝑛𝑟𝑡1 + 𝑛3𝑛4⋯𝑛𝑟𝑡2 +⋯+ 𝑛𝑟𝑡𝑟−1 + 𝑡𝑟 and 𝐾 = 𝑛2𝑛3⋯𝑛𝑟𝑇1 + 𝑛3𝑛4⋯𝑛𝑟𝑇2 +

⋯+ 𝑛𝑟𝑇𝑟−1 + 𝑇𝑟  for some 𝑡𝑠, 𝑇𝑠 ∈ ℤ𝑛𝑠. 

Proof. Again recall, |𝑌| = 𝑛1𝑛2⋯𝑛𝑟 = |𝒮𝑘| for all 0 ≤ 𝑘 < 𝑛1𝑛2⋯𝑛𝑟. Above defined relations 𝒮𝑘 

being disjoint, form a partition of 𝒫. For arbitrary relations 𝒮𝑙 , 𝒮𝑚, 𝒮𝑘 in 𝒫, we show that for each 

(𝑎, 𝑏) ∈ 𝒮𝑘 , the number of elements in the set {𝑐 ∈ 𝑌| (𝑎, 𝑐) ∈ 𝒮𝑙 , (𝑐, 𝑏) ∈ 𝒮𝑚} is invariant. Let 

(𝑎, 𝑏) ∈ 𝒮𝑘 where 𝑎 = (𝑎1, 𝑎2, … , 𝑎𝑟), 𝑏 = (𝑏1, 𝑏2, … , 𝑏𝑟) ∈ 𝑌. Suppose 𝑎𝒮𝑙 = 𝑎
′ =

(𝑎′1, 𝑎′2, … , 𝑎′𝑟), 𝑏𝒮𝑚
∗ = 𝑏′ = (𝑏′1, 𝑏′2, … , 𝑏′𝑟). Then (𝑎, 𝑏) ∈ 𝒮𝑘 implies 𝑏𝑟 ≡ (𝑘 +

𝑎𝑟) 𝑚𝑜𝑑 𝑛𝑟 , 𝑏𝑠 ≡ (𝑘 + 𝑡𝑠
(𝑘) + 𝑎𝑠) 𝑚𝑜𝑑 𝑛𝑠 ∀ 1 ≤ 𝑠 ≤ 𝑟 − 1 where 𝑘 = 𝑛2𝑛3⋯𝑛𝑟𝑡1

(𝑘) +

𝑛3𝑛4⋯𝑛𝑟𝑡2
(𝑘) +⋯+ 𝑛𝑟𝑡𝑟−1

(𝑘) + 𝑡𝑟
(𝑘)

 for some 𝑡𝑠 ∈ 𝑍𝑛𝑠  ∀ 1 ≤ 𝑠 ≤ 𝑟.  

Since (𝑎, 𝑎′) ∈ 𝒮𝑙 and (𝑏′, 𝑏) ∈ 𝑆𝑚,  

we have 𝑎′𝑟 ≡ (𝑙 + 𝑎𝑟) 𝑚𝑜𝑑 𝑛𝑟 , 𝑎′𝑠 ≡ (𝑙 + 𝑡𝑠
(𝑙) + 𝑎𝑠) 𝑚𝑜𝑑 𝑛𝑠 ∀ 1 ≤ 𝑠 ≤ 𝑟 − 1 where 𝑙 =

𝑛2𝑛3⋯𝑛𝑟𝑡1
(𝑙) + 𝑛3𝑛4⋯𝑛𝑟𝑡2

(𝑙) +⋯+ 𝑛𝑟𝑡𝑟−1
(𝑙) + 𝑡𝑟

(𝑙)
 for some 𝑡𝑠

(𝑙) ∈ ℤ𝑛𝑠∀1 ≤ 𝑠 ≤ 𝑟, and 

 𝑏𝑟 ≡ (𝑚 + 𝑏′𝑟) 𝑚𝑜𝑑 𝑛𝑟 , 𝑏𝑠 ≡ (𝑚 + 𝑡𝑠
(𝑚) + 𝑏′𝑠) 𝑚𝑜𝑑 𝑛𝑠  ∀ 1 ≤ 𝑠 ≤ 𝑟 − 1 where 𝑚 =

𝑛2𝑛3⋯𝑛𝑟𝑡1
(𝑚)

+ 𝑛3𝑛4⋯𝑛𝑟𝑡2
(𝑚)

+⋯+ 𝑛𝑟𝑡𝑟−1
(𝑚)

+ 𝑡𝑟
(𝑚)

 for some 𝑡𝑠
(𝑚)

∈ ℤ𝑛𝑠  ∀ 1 ≤ 𝑠 ≤ 𝑟.  

We observe that 𝑝𝑙𝑚
𝑘 = 1 if and only if 𝑎′ = 𝑏′. That is, 𝑝𝑙𝑚

𝑘 = 1 if 𝑘 ≡ (𝑙 + 𝑚) 𝑚𝑜𝑑 𝑛𝑟 and 𝑘 +

𝑡𝑠
(𝑘) ≡ (𝑙 + 𝑚 + 𝑡𝑠

(𝑙) + 𝑡𝑠
(𝑚)) 𝑚𝑜𝑑 𝑛𝑠 ∀ 1 ≤ 𝑠 ≤ 𝑟 − 1 Hence, (𝑌, 𝒫) is a non-symmetric and 

commutative association scheme. 

Next, we find 𝒮𝑘
∗. Let 𝒮𝑘

∗ = 𝒮𝐾 where 𝐾 = 𝑛2𝑛3⋯𝑛𝑟𝑇1 + 𝑛3𝑛4⋯𝑛𝑟𝑇2 +⋯+ 𝑛𝑟𝑇𝑟−1 + 𝑇𝑟 for some 

𝑇𝑠 ∈ ℤ𝑛𝑠. If (𝑏, 𝑎) ∈ 𝒮𝐾, then 𝑎𝑟 ≡ 𝐾 + 𝑏𝑟 𝑚𝑜𝑑 𝑛𝑟; 𝑎𝑠 ≡ 𝐾 + 𝑇𝑠 + 𝑏𝑠 𝑚𝑜𝑑 𝑛𝑠 ∀ 1 ≤ 𝑠 ≤ 𝑟 − 1. 

Also, since (𝑎, 𝑏) ∈ 𝒮𝑘, we get 𝑏𝑟 ≡ 𝑘 + 𝑎𝑟 𝑚𝑜𝑑 𝑛𝑟; 𝑏𝑠 ≡ 𝑘 + 𝑡𝑠 + 𝑎𝑠 𝑚𝑜𝑑 𝑛𝑠 ∀ 1 ≤ 𝑠 ≤ 𝑟 − 1. 



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This implies that 𝐾 + 𝑘 ≡ 0 𝑚𝑜𝑑 𝑛𝑟 and 𝐾 + 𝑘 + 𝑇𝑠 + 𝑡𝑠 ≡ 0 𝑚𝑜𝑑 𝑛𝑠 ∀ 1 ≤ 𝑠 ≤ 𝑟 − 1. Solving 

these equations, we get the value of 𝐾 as claimed. ◻ 

Corollary 1.  Let 𝑌 = ℳ𝑛(ℤ𝑚) be the set of all 𝑛 × 𝑛 matrices over ℤ𝑚 where 𝑚 ≥ 2 and 𝑛 ≥ 2. Any 

element of ℳ𝑛(ℤ𝑚) can be written as 𝑛2-tuple in ℤ𝑚
𝑛2. Therefore, relations defined on ℤ𝑚

𝑛2 will form 

an association scheme over ℳ𝑛(ℤ𝑚). 

Proof. Follows from Theorem 1 when 𝑚 = 2 , and from Theorem 3 when 𝑚 > 2. ◻ 

Presentations of some general linear groups 𝐺𝐿(2, ℤ𝑛) are given in [15] for 𝑛 = 4, 6, 8, 10. With the 

help of these presentations, we can compute the canonical form for these groups (provided in Table 1 

Canonical forms of some general linear groups 

The the canonical form of the presentation of 𝐺𝐿(2, ℤ2) is 𝐴𝑎𝐵𝑏: 0 ≤ 𝑎 ≤ 1,0 ≤ 𝑏 ≤ 2. The group is 

isomorphic onto 𝑆3, symmetric group on 3-symbols. Hence, another familiar presentation can be given 

as 𝐺𝐿(2, ℤ2) = ⟨𝐴, 𝐵|𝐴
2 = 𝐵3 = 𝐼, 𝐴𝐵 = 𝐵−1𝐴⟩. Using this presentation, we get a non symmetric 

association scheme. 

Table 1 Canonical forms of some general linear groups 

Group Generators Presentation Canonical form 

𝐺𝐿(2, ℤ2) 𝐴, 𝐵 τ2 = σ3 = (τσ)3 = 1 𝐴𝑎𝐵𝑏: 

0 ≤ 𝑎 ≤ 1, 0 ≤ 𝑏 ≤ 2 

𝐺𝐿(2, ℤ4) 𝐴, 𝐵, 𝐶 𝐴2 = 𝐵2 = 𝐶4 = 𝑋3 = 𝑌4  

= (𝐶𝐵𝐴)4 = 𝐼, 

𝐶2𝐴 = 𝐴𝐶2, 𝐶2𝐵 = 𝐵𝐶2, 𝐵𝐶 = 𝐶−1𝐵 

where 𝑋 = 𝐴𝐵 and  𝑌 = 𝑋𝐶𝑋 

𝐴𝑎𝐶𝑏𝑌𝑐𝑋𝑑: 

0 ≤ 𝑎 ≤ 1,0 ≤ 𝑏, 𝑐 ≤ 3, 

0 ≤ 𝑑 ≤ 2 

𝐺𝐿(2, ℤ6) 𝐴, 𝐵, 𝐶 𝐴2 = 𝐵2 = 𝐶4 = 𝑋12 = 𝑌6                

= (𝐴𝐵)3 = (𝐵𝐶)2 = 𝐼, 

𝐶𝐴 = 𝐴𝐶2, 𝐶2𝐵 = 𝐵𝐶2,   

(𝐴𝐶)12 = 𝐶2, 

(𝐶𝐴)2𝐵(𝐶𝐴)2 = (𝐴𝐶)2𝐵(𝐴𝐶)2 

where 𝑋 = 𝐵(𝐶𝐴)2𝐵 and 𝑌 = 𝐶𝐴𝐵 

𝐶𝑎𝑋𝑏𝑌𝑐𝐵𝑑: 

 0 ≤ 𝑎, 𝑑 ≤ 1,  

0 ≤ 𝑏 ≤ 11,   0 ≤ 𝑐 ≤ 5 

Theorem 4.  Let 𝑌 = 𝐺𝐿(2, ℤ2). Define relations 𝒮𝑘 on 𝒫 by 

𝒮𝑘 = {(𝐴
𝑎𝐵𝑏, 𝐴𝑘+𝑎𝐵𝑘+𝑏)| 0 ≤ 𝑎 ≤ 1, 0 ≤ 𝑏 ≤ 2}  for all 0 ≤ 𝑘 ≤ 5. 

Then (𝑌, 𝒫) is a non symmetric association scheme with parameters as follows: 

𝑝𝑙𝑚
𝑘 = {

1 if 𝑘 = 𝑙 + 𝑚,
0 if 𝑘 ≠ 𝑙 + 𝑚

 

where 0 ≤ 𝑘, 𝑙,𝑚 ≤ 5. 

Proof. Observe that 𝒮0 = {(𝑀,𝑀)| 𝑀 ∈ 𝐺𝐿(2, ℤ2)} is an identity relation. For arbitrary relations 

𝒮𝑙 , 𝒮𝑚, 𝒮𝑘 ∈ 𝒫, we find the cardinality 𝑝𝑙𝑚
𝑘  such that |𝑀𝒮𝑙 ∩ 𝑁𝒮𝑚

∗ | = 𝑝𝑙𝑚
𝑘  for all (𝑀,𝑁) ∈ 𝒮𝑘. Let 

(𝑀,𝑁) ∈ 𝒮𝑘 and let 𝑀𝒮𝑙 = 𝑀
′ and 𝑁𝒮𝑚

∗ = 𝑁′. That is, 𝑀 = 𝐴𝑎𝐵𝑏, 𝑁 = 𝐴𝑘+𝑎𝐵𝑘+𝑏;𝑀 =

𝐴𝑎1𝐵𝑏1 , 𝑀′ = 𝐴𝑙+𝑎1𝐵𝑙+𝑏1; 𝑁′ = 𝐴𝑎2𝐵𝑏2 , 𝑁 = 𝐴𝑚+𝑎2𝐵𝑚+𝑏2 where 0 ≤ 𝑎, 𝑎1, 𝑎2 ≤ 1 and 0 ≤



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𝑏, 𝑏1, 𝑏2 ≤ 2. Since every pair (𝑀,𝑁) in 𝑌 are 𝑘𝑡ℎ associates for exactly one 𝑘, we find that 𝑝𝑙𝑚
𝑘  can 

be either 0 or 1. Hence, with the help of the above equations, we obtain 𝑝𝑙𝑚
𝑘 = 1 if 𝑀′ = 𝑁′ 

equivalently, if 𝑘 = 𝑙 + 𝑚. Moreover, 𝑝𝑙𝑚
𝑘 = 0 if 𝑀′ ≠ 𝑁′ that is, whenever 𝑘 ≠ 𝑙 + 𝑚. Also it can 

be easily proved that the relations are not symmetric and hence (𝑌, 𝒫) is a non symmetric association 

scheme. ◻ 

Theorem 5.  Let 𝑌 = 𝐺𝐿(2, ℤ4) and 𝒫 be a partition of 𝑌 × 𝑌. For all 𝑘 written in form of 48𝑠 +

16𝑛 + 4𝑡 + 𝑟 (𝑠 ∈ ℤ2; 𝑛 ∈ ℤ3; 𝑟, 𝑡 ∈ ℤ4), the relations 𝒮𝑘 in 𝒫 defined by 

 𝒮𝑘 = {(𝐴
𝑎𝐶𝑏𝑌𝑐𝑋𝑑 , 𝐴𝑎+𝑠𝐶𝑏+𝑡𝑌𝑐+𝑟𝑋𝑑+𝑛)| 𝑎 ∈ ℤ2; 𝑏, 𝑐 ∈ ℤ4; 𝑑 ∈ ℤ3}  

is a non-symmetric association scheme with parameters  

𝑝𝑙𝑚
𝑘 =

{
 
 

 
 
1 𝑖𝑓 𝑠(𝑘) ≡ (𝑠(𝑙) + 𝑠(𝑚)) 𝑚𝑜𝑑 2,

 𝑛(𝑘) ≡ (𝑛(𝑙) + 𝑛(𝑚)) 𝑚𝑜𝑑 2,

 𝑡(𝑘) ≡ (𝑡(𝑙) + 𝑡(𝑚)) 𝑚𝑜𝑑 4, 𝑎𝑛𝑑

 𝑟(𝑘) ≡ (𝑟(𝑙) + 𝑟(𝑚)) 𝑚𝑜𝑑 4

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒  

 

 

 

Proof. Observe that |𝒮𝑘| = |𝑌| for all 0 ≤ 𝑘 < 96. The relations 𝒮𝑘 are disjoint and ∪ 𝒮𝑘:0 ≤ 𝑘 <

96 = 𝒫. 

𝒮0 = {(𝑀,𝑀):𝑀 ∈ 𝐺𝐿(2, ℤ4)} is an identity relation. Let 𝒮𝑙 , 𝒮𝑚, 𝒮𝑘 be arbitrary relations in 𝒫 and 

(𝑀,𝑁) be any element of 𝒮𝑘. Since 𝑀 ∈ 𝑌, 𝑀 is of the form 𝐴𝑎𝐶𝑏𝑌𝑐𝑋𝑑 for some 𝑎, 𝑏, 𝑐, 𝑑 (from Table 

[table:canonical forms An,Sn]). Suppose 𝑀𝒮𝑙 = 𝑀
′ and 𝑁𝒮𝑚

∗ = 𝑁′ where 𝑀′, 𝑁′ ∈ 𝑌. 

Now (𝑀, 𝑁) ∈ 𝒮𝑘, (𝑀,𝑀′) ∈ 𝒮𝑙 and (𝑁′, 𝑁) ∈ 𝒮𝑚, implies  

𝑁 = 𝐴(𝑎+𝑠
(𝑘))

𝑚𝑜𝑑 2  𝐶(𝑏+𝑡
(𝑘))

𝑚𝑜𝑑 4  𝑌(𝑐+𝑟
(𝑘))

𝑚𝑜𝑑 4 𝑋(𝑑+𝑛
(𝑘))

𝑚𝑜𝑑 3, 

M′ =  A(a+s
(l))

mod 2  C(b+t
(l))

mod 4  Y(c+r
(l))

mod 4  X(d+n
(l))

mod 3,  and 

N′ = A(a+s
(k)−s(m))

mod 2 C(b+t
(k)−t(m))

mod 4  Y(c+r
(k)−r(m))

mod 4  X(d+n
(k)−n(m))

mod 3  

respectively, where 𝑘 = 48𝑠(𝑘) + 16𝑛(𝑘) + 4𝑡(𝑘) + 𝑟(𝑘), 𝑙 = 48𝑠(𝑙) + 16𝑛(𝑙) + 4𝑡(𝑙) + 𝑟(𝑙) and 𝑚 =

48𝑠(𝑚) + 16𝑛(𝑚) + 4𝑡(𝑚) + 𝑟(𝑚), for some 𝑠(𝑘), 𝑠(𝑙), 𝑠(𝑚) ∈ ℤ2; 𝑛
(𝑘), 𝑛(𝑙), 𝑛(𝑚) ∈ ℤ3 and 

𝑡(𝑘), 𝑡(𝑙), 𝑡(𝑚), 𝑟(𝑘), 𝑟(𝑙), 𝑟(𝑚) ∈ ℤ4. 

Since every pair (𝑀,𝑁) in 𝒫 are 𝑘𝑡ℎ associates for exactly one 𝑘, we find that 𝑝𝑙𝑚
𝑘  can be either 0 or 

1. Hence we obtain that 𝑝𝑙𝑚
𝑘 = 1 if and only if 𝑀′ = 𝑁′ that is, if 𝑠(𝑘) ≡ (𝑠(𝑙) + 𝑠(𝑚)) 𝑚𝑜𝑑 2, 𝑡(𝑘) ≡

(𝑡(𝑙) + 𝑡(𝑚)) 𝑚𝑜𝑑 4, 𝑟(𝑘) ≡ (𝑟(𝑙) + 𝑟(𝑚)) 𝑚𝑜𝑑 4 and 𝑛(𝑘) ≡ (𝑛(𝑙) + 𝑛(𝑚)) 𝑚𝑜𝑑 3. ◻ 

Theorem 6.  Let 𝑌 = 𝐺𝐿(2, ℤ6) and 𝒫 be a partition of 𝑌 × 𝑌. For all 𝑘 written in form of 144𝑠 +

72𝑛 + 12𝑡 + 𝑟 (𝑠, 𝑛 ∈ ℤ2, 𝑡 ∈ ℤ6, 𝑟 ∈ ℤ12), the relations 𝒮𝑘 in 𝒫 defined by 

 𝒮𝑘 = {(𝐶
𝑎𝑋𝑏𝑌𝑐𝐵𝑑, 𝐶𝑎+𝑠𝑋𝑏+𝑟𝑌𝑐+𝑡𝐵𝑑+𝑛) | 𝑎, 𝑑 ∈ ℤ2; 𝑏 ∈ ℤ12; 𝑐 ∈ ℤ6} 

is a non-symmetric association scheme with parameters  



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𝑝𝑙𝑚
𝑘 =

{
 
 

 
 
1 𝑖𝑓 𝑠(𝑘) ≡ (𝑠(𝑙) + 𝑠(𝑚)) 𝑚𝑜𝑑 2,

 𝑛(𝑘) ≡ (𝑛(𝑙) + 𝑛(𝑚)) 𝑚𝑜𝑑 2,

 𝑡(𝑘) ≡ (𝑡(𝑙) + 𝑡(𝑚)) 𝑚𝑜𝑑 6, 𝑎𝑛𝑑

 𝑟(𝑘) ≡ (𝑟(𝑙) + 𝑟(𝑚)) 𝑚𝑜𝑑 12

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒  

 

 

Proof. Here, |𝑐𝑘| = |𝑌| for all 0 ≤ 𝑘 < 288. The relations 𝒮𝑘 are disjoint and ∪ 𝒮𝑘:0 ≤ 𝑘 < 288 = 𝒫. 

Proceeding as in proof of theorem 5, we can find cardinality 𝑝𝑙𝑚
𝑘  such that for all (𝑥, 𝑦) ∈ 𝒮𝑘, 

|𝑥𝒮𝑙 ∩ 𝑦𝒮𝑚
∗ | = 𝑝𝑙𝑚

𝑘  is a constant. ◻ 

Note: Let 𝑆𝐿(2, ℤ𝑝) be the special linear group over ℤ𝑝 and 𝐺𝐿(2, ℤ𝑝) be general linear group over 

ℤ𝑝. The structure of these conjugacy classes are worked out in detail in [16]. Using these classes one 

can compute group association scheme for 𝑆𝐿(2, ℤ𝑝) and 𝐺𝐿(2, ℤ𝑝) using Definition 2. 

Let us represent 𝑆𝑛 as the symmetric group of degree 𝑛. Group association scheme for symmetric 

groups have been described by Tomiyama and Yamazaki in [17]. In [13], Sabharwal et al. identified 

the association schemes for the symmetric groups 𝑆3 and 𝑆4 without using the conjugacy classes. In 

next theorem, we have determined non symmetric commutative association scheme for the symmetric 

groups 𝑆4 and 𝑆5 and alternating groups 𝐴3, 𝐴4 and 𝐴5 without using conjugacy classes. 

Lemma 2.  Let 𝑋 = 𝐴3 = 𝜎
𝑖:0 ≤ 𝑖 ≤ 2 where 𝜎3 = 1. Then the relations 𝒮𝑘 on 𝒫 defined by 

𝒮𝑘 = {(𝜎
𝑖 , 𝜎𝑘+𝑖) |  0 ≤ 𝑖 ≤ 2} for all 0 ≤ 𝑘 ≤ 2 

is a non-symmetric commutative association scheme and intersection numbers of this association 

scheme are as follows:  

𝑝𝑙𝑚
𝑘 = {

1 if  𝑘 = 𝑙 + 𝑚,
0 if  𝑘 ≠ 𝑙 + 𝑚

 

Proof. Since 𝐴3 is isomorphic to ℤ3 by mapping 𝜎𝑖 ↦ 𝑖, and the set of relations {(𝑖, 𝑘 + 𝑖) | 𝑖 = 0,1,2} 

forms AS for ℤ3 , we can conclude the result. ◻ 

Presentations of alternating and symmetric groups of degree less than 8 are given in [18]. With the 

help of these presentations, we can compute the canonical form for these groups (provided in Table 2 

Canonical forms of alternating and symmetric groups of degree 4 and 5). In [13], canonical form of 𝑆4 

is discussed. 

Table 2 Canonical forms of alternating and symmetric groups of degree 4 and 5 

Group Generators Presentation Canonical form 

𝐴4 𝜏, 𝜎 𝐴2 = 𝐵3 = 𝐼, 𝐴𝐵 = 𝐵−1𝐴 𝜏𝑎𝜎𝜏𝑏𝜎𝑐: 

0 ≤ 𝑎, 𝑏 ≤ 1,  

0 ≤ 𝑐 ≤ 2 

𝑆4 𝜏, 𝜎 𝜏2 = 𝜎3 = (𝜏𝜎)3 = 1 𝐴2𝜏𝑎𝜎𝑏𝜏𝜎𝑐: 

 0 ≤ 𝑎 ≤ 1, 0 ≤ 𝑏 ≤ 2,  

 0 ≤ 𝑐 ≤ 3 



Communications on Applied Nonlinear Analysis 

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𝐴5 𝜏, 𝜎 𝜏2 = 𝜎3 = 𝛾5 = 1 

where 𝛾 = 𝜏𝜎 

 

𝜏𝑎𝜎𝜏𝜎2𝛾𝑏𝜏𝑐𝜎𝜏𝜎𝑑: 

 0 ≤ 𝑎, 𝑐 ≤ 1, 

 0 ≤ 𝑏 ≤ 4, 0 ≤ 𝑑 ≤ 2 

𝑆5 𝜏, 𝜎 𝜏5 = 𝜎6 = (𝛾)2 = (𝛿)2 = 1 

where 𝛾 = 𝜏𝜎 and 𝛿 = 𝜏2𝜎2 

𝜏𝑎𝜎𝛾𝑏𝜏4𝜎5𝛿𝑐𝜏𝜎𝑑: 

 0 ≤ 𝑎 ≤ 4, 0 ≤ 𝑏, 𝑐 ≤ 1, 

  0 ≤ 𝑑 ≤ 5 

Theorem 7.  Let 𝑌 = 𝐴4 and 𝒫 be a partition of 𝑌 × 𝑌. For all 𝑘 written in form of 3𝑛 +

𝑡 (𝑛 ∈ ℤ4, 𝑡 ∈ ℤ3), the relations 𝒮𝑘 on 𝒫 defined by  

𝒮𝑘 = {(𝛼𝑖
(𝑗)
, 𝛼(𝑖+𝑡)𝑚𝑜𝑑 3
(𝑗+𝑛)𝑚𝑜𝑑 4) | 0 ≤ 𝑖 ≤ 2, 0 ≤ 𝑗 ≤ 3}  

where 𝛼𝑖
(0) = 𝜎𝑖+1,  𝛼𝑖

(1) = 𝜎𝜏𝜎𝑖 , 𝛼𝑖
(2) = 𝜏𝜎𝑖+1, 𝛼𝑖

(3) = 𝜏𝜎𝜏𝜎𝑖 

is a non-symmetric association scheme with parameters  

𝑝𝑙𝑚
𝑘 = {

1 𝑖𝑓  𝑛(𝑘) ≡ (𝑛(𝑙) + 𝑛(𝑚)) 𝑚𝑜𝑑 4, 𝑎𝑛𝑑

 𝑡(𝑘) ≡ (𝑡(𝑙) + 𝑡(𝑚)) 𝑚𝑜𝑑 3

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

 

 where 𝑘 = 3𝑛(𝑘) + 𝑡(𝑘); 𝑙 = 3𝑛(𝑙) + 𝑡(𝑙); 𝑚 = 3𝑛(𝑚) + 𝑡(𝑚); for some 𝑡(𝑘), 𝑡(𝑙), 𝑡(𝑚) ∈ ℤ3 and 

𝑛(𝑘), 𝑛(𝑙), 𝑛(𝑚) ∈ ℤ4. 

Proof. Observe that |𝒮𝑘| = |𝑌| for all 0 ≤ 𝑘 < 12. The relations 𝒮𝑘 are disjoint and ∪ 𝒮𝑘: 0 ≤ 𝑘 <

12 = 𝒫. 

𝒮0 = {(𝛼𝑖
(𝑗)
, 𝛼𝑖
(𝑗)
) : 0 ≤ 𝑖 ≤ 2, 0 ≤ 𝑗 ≤ 3} is an identity relation. For arbitrary relations 𝒮𝑙 , 𝒮𝑚, 𝒮𝑘 in 

𝒫, we prove that for all (𝑥, 𝑦) ∈ 𝒮𝑘, |𝑥𝒮𝑙 ∩ 𝑦𝒮𝑚
∗ | is constant. Now let (𝑥, 𝑦) be an arbitrary element 

of 𝒮𝑘. Since 𝑥 ∈ 𝑌, 𝑥 is of the form 𝜏𝑎𝜎𝜏𝑏𝜎𝑐  for some 𝑎, 𝑏, 𝑐 (from Table [table:canonical forms 

An,Sn]) and further 𝑥 = 𝛼𝑖
(𝑗)

 for some 𝑖, 𝑗. Suppose 𝑥𝒮𝑙 = 𝑥
′ and 𝑦𝒮𝑚

∗ = 𝑦′ where 𝑥′, 𝑦′ ∈ 𝑌. 

Now (𝑥, 𝑦) ∈ 𝒮𝑘 implies 𝑦 = 𝛼
(𝑖+𝑡(𝑘))

𝑚𝑜𝑑 3

(𝑗+𝑛(𝑘))
𝑚𝑜𝑑 4 where 𝑘 = 3𝑛(𝑘) + 𝑡(𝑘) for some 𝑡(𝑘) ∈ ℤ3, 𝑛

(𝑘) ∈ ℤ4. 

Similarly, (𝑥, 𝑥′) ∈ 𝒮𝑙 and (𝑦′, 𝑦) ∈ 𝒮𝑚, implies 𝑥′ = 𝛼
(𝑖+𝑡(𝑙))

𝑚𝑜𝑑 3

(𝑗+𝑛(𝑙))
𝑚𝑜𝑑 4 where 𝑙 = 3𝑛(𝑙) + 𝑡(𝑙) for some 

𝑡(𝑙) ∈ ℤ3, 𝑛
(𝑙) ∈ ℤ4, and 𝑦′ = 𝛼

(𝑖+𝑡(𝑘)−𝑡(𝑚))
𝑚𝑜𝑑 3

(𝑗+𝑛(𝑘)−𝑛(𝑚))
𝑚𝑜𝑑 4 where 𝑚 = 3𝑛(𝑚) + 𝑡(𝑚) for some 𝑡(𝑚) ∈

ℤ3, 𝑛
(𝑚) ∈ ℤ4. Since every pair (𝑥, 𝑦) in 𝒫 are 𝑘𝑡ℎ associates for exactly one k, we find that 𝑝𝑙𝑚

𝑘  can 

be either 0 or 1. Hence we obtain that 𝑝𝑙𝑚
𝑘 = 1 if and only if 𝑥′ = 𝑦′ that is, if 𝑡(𝑘) ≡

(𝑡(𝑙) + 𝑡(𝑚)) 𝑚𝑜𝑑 3 and 𝑛(𝑘) ≡ (𝑛(𝑙) + 𝑛(𝑚)) 𝑚𝑜𝑑 4. ◻ 

Theorem 8.  Let 𝑌 = 𝐴5 and 𝒫 be a partition of 𝑌 × 𝑌. For all 𝑘 written in form of 12𝑛 + 4𝑡 +

𝑟 (𝑛 ∈ ℤ5, 𝑡 ∈ ℤ3, 𝑟 ∈ ℤ4), the relations 𝒮𝑘 on 𝒫 defined by 

 𝒮𝑘 = {(𝛼𝑖𝑗
(𝑠), 𝛼(𝑖+𝑛)𝑚𝑜𝑑 5(𝑗+𝑡)𝑚𝑜𝑑 3

(𝑠+𝑟)𝑚𝑜𝑑 4 ) | 𝑖 ∈ ℤ5, 𝑗 ∈ ℤ3, 𝑠 ∈ ℤ4}  



Communications on Applied Nonlinear Analysis 

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where α𝑖𝑗
(0) = στσ2γ𝑖στσ𝑗 , α𝑖𝑗

(1) = στσ2γ𝑖τστσ𝑗 , α𝑖𝑗
(2) = τστσ2γ𝑖στσ𝑗 , α𝑖𝑗

(3) = τστσ2γ𝑖τστσ𝑗 

is a non-symmetric association scheme with parameters 

 𝑝𝑙𝑚
𝑘 =

{
 
 

 
 1 if 𝑛(𝑘) ≡ (𝑛(𝑙) + 𝑛(𝑚)) 𝑚𝑜𝑑 5,

 𝑡(𝑘) ≡ (𝑡(𝑙) + 𝑡(𝑚)) 𝑚𝑜𝑑 3, 𝑎𝑛𝑑

 𝑟(𝑘) ≡ (𝑟(𝑙) + 𝑟(𝑚)) 𝑚𝑜𝑑 4

0 otherwise }
 
 

 
 

 

Proof. Observe that |𝒮𝑘| = |𝑌| for all 0 ≤ 𝑘 < 60. The relations 𝒮𝑘 are disjoint and ∪ {𝒮𝑘:0 ≤ 𝑘 <

60} = 𝒫. 

𝒮0 = {(𝛼𝑖𝑗
(𝑠), 𝛼𝑖𝑗

(𝑠)) : 0 ≤ 𝑖 ≤ 4, 0 ≤ 𝑗 ≤ 2, 0 ≤ 𝑠 ≤ 3} is an identity relation. Let 𝒮𝑙 , 𝒮𝑚, 𝒮𝑘 be arbitrary 

relations in 𝒫 and (𝑥, 𝑦) be any element of 𝒮𝑘. Since 𝑥 ∈ 𝑌, 𝑥 is of the form 𝜏𝑎𝜎𝜏𝜎2𝛾𝑏𝜏𝑐𝜎𝜏𝜎𝑑 for 

some 𝑎, 𝑏, 𝑐, 𝑑 (from Table [table:canonical forms An,Sn]) and further 𝑥 = 𝛼𝑖𝑗
(𝑠)

 for some 𝑖, 𝑗, 𝑠. 

Suppose 𝑥𝒮𝑙 = 𝑥
′ and 𝑦𝒮𝑚

∗ = 𝑦′ where 𝑥′, 𝑦′ ∈ 𝑌. 

Now (𝑥, 𝑦) ∈ 𝒮𝑘 implies 𝑦 = 𝛼
(𝑖+𝑛(𝑘))

𝑚𝑜𝑑 5
(𝑗+𝑡(𝑘))

𝑚𝑜𝑑 3

(𝑠+𝑟(𝑘))
𝑚𝑜𝑑 4  where 𝑘 = 12𝑛(𝑘) + 4𝑡(𝑘) + 𝑟(𝑘) for some 

𝑡(𝑘) ∈ ℤ3,  𝑛
(𝑘) ∈ ℤ5,  𝑟

(𝑘) ∈ ℤ4. Similarly, (𝑥, 𝑥′) ∈ 𝒮𝑙 and (𝑦′, 𝑦) ∈ 𝒮𝑚, implies 𝑥′ =

𝛼
(𝑖+𝑛(𝑙))

𝑚𝑜𝑑 5
(𝑗+𝑡(𝑙))

𝑚𝑜𝑑 3

(𝑠+𝑟(𝑙))
𝑚𝑜𝑑 4  where 𝑙 = 12𝑛(𝑙) + 4𝑡(𝑙) + 𝑟(𝑙) for some 𝑡(𝑙) ∈ ℤ3, 𝑛

(𝑙) ∈ ℤ5, 𝑟
(𝑙) ∈ ℤ4;  and 

𝑦′ = 𝛼
(𝑖+𝑛(𝑘)−𝑛(𝑚))

𝑚𝑜𝑑 5
(𝑗+𝑡(𝑘)−𝑡(𝑚))

𝑚𝑜𝑑 3

(𝑠+𝑟(𝑘)−𝑟(𝑚))
𝑚𝑜𝑑 4  where 𝑚 = 12𝑛(𝑚) + 4𝑡(𝑚) + 𝑟(𝑚) for some 𝑡(𝑚) ∈

ℤ3, 𝑛
(𝑚) ∈ ℤ5, 𝑟

(𝑚) ∈ ℤ4. Since every pair (𝑥, 𝑦) in 𝒫 are 𝑘𝑡ℎ associates for exactly one k, we find that 

𝑝𝑙𝑚
𝑘  can be either 0 or 1. Hence we obtain that 𝑝𝑙𝑚

𝑘 = 1 if and only if 𝑥′ = 𝑦′ that is, if 𝑡(𝑘) ≡

(𝑡(𝑙) + 𝑡(𝑚)) 𝑚𝑜𝑑 3, 𝑟(𝑘) ≡ (𝑟(𝑙) + 𝑟(𝑚)) 𝑚𝑜𝑑 4 and 𝑛(𝑘) ≡ (𝑛(𝑙) + 𝑛(𝑚)) 𝑚𝑜𝑑 5. ◻ 

Theorem 9.  Let 𝑌 = 𝑆4 and 𝒫 be a partition of 𝑌 × 𝑌. For all 𝑘 written in form of 4𝑛 +

𝑡 (𝑛 ∈ ℤ6, 𝑡 ∈ ℤ4), the relations 𝒮𝑘 on 𝒫 defined by 

 𝒮𝑘 = {(𝛼𝑖
(𝑗)
, 𝛼(𝑖+𝑡)𝑚𝑜𝑑 4
(𝑗+𝑛)𝑚𝑜𝑑 6) | 𝑖 ∈ ℤ4, 𝑗 ∈ ℤ6}  

where α𝑖
(0) = τσ𝑖,  α𝑖

(1) = στσ𝑖 ,  α𝑖
(2) = σ2τσ𝑖,  α𝑖

(3) = σ𝑖 ,  α𝑖
(4) = τστσ𝑖 ,  α𝑖

(5) = τσ2τσ𝑖 

is a non-symmetric association scheme with parameters 

𝑝𝑙𝑚
𝑘 = {

1 𝑖𝑓  𝑛(𝑘) ≡ (𝑛(𝑙) + 𝑛(𝑚)) 𝑚𝑜𝑑 6, 𝑎𝑛𝑑

 𝑡(𝑘) ≡ (𝑡(𝑙) + 𝑡(𝑚)) 𝑚𝑜𝑑 4 

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

 

Proof. 𝒮0 = {(𝛼𝑖
(𝑗)
, 𝛼𝑖
(𝑗)
)  : 0 ≤ 𝑖 ≤ 2, 0 ≤ 𝑗 ≤ 3} is an identity relation. It can be verified that (𝑌, 𝒫) 

is an association scheme and the relations are non-symmetric.  

Let 𝒮𝑙 , 𝒮𝑚 , 𝒮𝑘 ∈ 𝒫. To find cardinality 𝑝𝑙𝑚
𝑘  such that for all (𝑥, 𝑦) ∈ 𝒮𝑘,  |𝑥𝒮𝑙 ∩ 𝑦𝒮𝑚

∗ | = 𝑝𝑙𝑚
𝑘 . 



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Let (𝑥, 𝑦) ∈ 𝒮𝑘, 𝑥𝒮𝑙 = 𝑥
′ and 𝑦𝒮𝑚

∗ = 𝑦′ where 𝑥′, 𝑦′ ∈ 𝑌. Since 𝑥 ∈ 𝑌, 𝑥 is of the form 𝜏𝑎𝜎𝑏𝜏𝜎𝑐 for 

some 𝑎, 𝑏, 𝑐 (from Table [table:canonical forms An,Sn]) and further 𝑥 = 𝛼𝑖
(𝑗)

 for some 𝑖, 𝑗. Proceeding 

as in proof of previous theorem, we get 𝑦 = 𝛼
(𝑖+𝑡(𝑘))

𝑚𝑜𝑑 4

(𝑗+𝑛(𝑘))
𝑚𝑜𝑑 6 where 𝑘 = 4𝑛(𝑘) + 𝑡(𝑘) for some 𝑡(𝑘) ∈

ℤ4, 𝑛
(𝑘) ∈ ℤ6; 𝑥′ = 𝛼

(𝑖+𝑡(𝑙))
𝑚𝑜𝑑 4

(𝑗+𝑛(𝑙))
𝑚𝑜𝑑 6 where 𝑙 = 4𝑛(𝑙) + 𝑡(𝑙) for some 𝑡(𝑙) ∈ ℤ4, 𝑛

(𝑙) ∈ ℤ6, and 𝑦′ =

𝛼
(𝑖+𝑡(𝑘)−𝑡(𝑚))

𝑚𝑜𝑑 4

(𝑗+𝑛(𝑘)−𝑛(𝑚))
𝑚𝑜𝑑 6 where 𝑚 = 4𝑛(𝑚) + 𝑡(𝑚) for some 𝑡(𝑚) ∈ ℤ4, 𝑛

(𝑚) ∈ ℤ6. Using these equations, 

we have 𝑝𝑙𝑚
𝑘 = 1 if and only if 𝑥′ = 𝑦′ that is, if 𝑡(𝑘) ≡ (𝑡(𝑙) + 𝑡(𝑚)) 𝑚𝑜𝑑 4 and 𝑛(𝑘) ≡

(𝑛(𝑙) + 𝑛(𝑚)) 𝑚𝑜𝑑 6. ◻ 

Theorem 10.  Let 𝑌 = 𝑆5 and 𝒫 be a partition of 𝑌 × 𝑌. For all 𝑘 written in form of 24𝑡 + 4𝑛 +

𝑟 (𝑡 ∈ ℤ5, 𝑛 ∈ ℤ6, 𝑟 ∈ ℤ4), the relations 𝒮𝑘 on 𝒫 defined by 

 𝒮𝑘 = {(𝛼𝑖𝑗
(𝑠), 𝛼(𝑖+𝑡)𝑚𝑜𝑑 5(𝑗+𝑛)𝑚𝑜𝑑 6

(𝑠+𝑟)𝑚𝑜𝑑 4 ) | 𝑖 ∈ ℤ5, 𝑗 ∈ ℤ6, 𝑠 ∈ ℤ4} 

 where α𝑖𝑗
(0) = τ𝑖στ4σ5τσ𝑗 ,  α𝑖𝑗

(1) = τ𝑖σγτ4σ5τσ𝑗 ,  α𝑖𝑗
(2) = τ𝑖στ4σ5δτσ𝑗 ,  α𝑖𝑗

(3) = τ𝑖σγτ4σ5δτσ𝑗 

is a non-symmetric association scheme with parameters 

𝑝𝑙𝑚
𝑘 =

{
 
 

 
 1 if 𝑛(𝑘) ≡ (𝑛(𝑙) + 𝑛(𝑚)) 𝑚𝑜𝑑 6,

 𝑡(𝑘) ≡ (𝑡(𝑙) + 𝑡(𝑚)) 𝑚𝑜𝑑 5, 𝑎𝑛𝑑

 𝑟(𝑘) ≡ (𝑟(𝑙) + 𝑟(𝑚)) 𝑚𝑜𝑑 4

0 otherwise }
 
 

 
 

 

Proof. Here, |𝒮𝑘| = |𝑌| for all 0 ≤ 𝑘 < 120. The relations 𝒮𝑘 are disjoint and ∪ 𝒮𝑘 : 0 ≤ 𝑘 < 120 =

𝒫. Proceeding as in proof of theorem 8, we can find cardinality 𝑝𝑙𝑚
𝑘  such that for all (𝑥, 𝑦) ∈ 𝒮𝑘,  

|𝑥𝒮𝑙 ∩ 𝑦𝒮𝑚
∗ | = 𝑝𝑙𝑚

𝑘  is a constant. ◻ 

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