On the Existence of Optimal (v, 5, 1) and (v, 6, 1) Binary Cyclically Permutable Constant-Weight Codes

Abstract

The problem of the existence of optimal $(v,k,1)$ binary cyclically permutable constant-weight (CPCW) codes has been completely solved for codeword weights $k<5$. We consider the smallest open cases, namely $k=5$ and $k=6$. We present such codes for small values of the code length v and derive necessary conditions for the existence of optimal $\left(k\right(k-1)t+2,k,1)$ CPCW codes. These necessary conditions can be used to construct such codes, as well as to show that optimal codes with some parameters do not exist. In particular, we use them to prove that an optimal $(92,6,1)$ CPCW code does not exist.

1. Introduction

1.1. Definitions

Denote by ${\mathbb{Z}}_v$ the additive group of integers modulo v. Let $C=\{c_0,c_1,\dots ,c_{k-1}\}$ be a k-element subset of ${\mathbb{Z}}_v$. Then, $▵C=\{c_i-c_j|i,j=0,1,\dots ,k-1;i\ne j\}$ is the multiset of differences of C, and $C+t$ denotes a t-translate of C, where $C+t=\{c_0+t,c_1+t,\dots ,c_{k-1}+t\}$ for $t\in {\mathbb{Z}}_v$.

Definition 1.

A $(v,k,1)$ binary cyclically permutable constant-weight (CPCW) code of size s can be defined as a collection $\mathcal{C}=\{C_1,\dots ,C_s\}$ of k-subsets of ${\mathbb{Z}}_v$ (codewords) such that any two distinct translates of a codeword share at most one element, and any two translates of two distinct codewords also share at most one element:

$$\begin{array}{c}\hfill |C_i\cap (C_i+t)|\le 1,1\le i\le s,1\le t\le v-1\end{array}$$

(1)

$$\begin{array}{c}\hfill |C_i\cap (C_j+t)|\le 1,1\le i<j\le s,0\le t\le v-1\end{array}$$

(2)

Everywhere in the text that follows, the abbreviation CPCW means “binary cyclically permutable constant weight”. Originally, a CPCW code was defined as a collection of $\{0,1\}$ sequences, but, for some purposes, it is easier to present the codewords as subsets of ${\mathbb{Z}}_v$, and that is why the above definition is widely used too. Condition (1) is called the auto-correlation property and (2) the cross-correlation property. We denote by s the size of $\mathcal{C}$, namely the number of its codewords.

Consider a codeword $C=\{c_0,c_1,\dots ,c_{k-1}\}$. The auto-correlation property means that all the differences of a codeword of a $(v,k,1)$ CPCW code are different, and the cross-correlation property means that $▵C_1\bigcap ▵C_2=\varnothing$ for two codewords $C_1$ and $C_2$. Since $|▵C|=k(k-1)$, the size of a $(v,k,1)$ CPCW code cannot exceed

$$⌊{\displaystyle \frac{v-1}{k(k-1)}}⌋.$$

CPCW codes that reach this bound are called optimal [1,2].

If the code size is exactly $(v-1)/k(k-1)$, the optimal $(v,k,1)$ CPCW code is perfect; that is, all nonzero differences are covered.

CPCW codes are related to many other combinatorial structures. Definition 1, for instance, holds for a $(v,k,1)$ optical orthogonal code (OOC) too [1]. In this paper, we will use the relation to partial designs.

Definition 2.

Let $V={\left\{P_i\right\}}_{i=1}^v$ be a finite set of points and $\mathcal{B}={\left\{B_j\right\}}_{j=1}^b$ be a finite collection of k-element subsets of V, called blocks. $D=(V,\mathcal{B})$ is a design (partial design) with parameters 2-(v,k,1) if any 2-subset of V is contained in exactly (at most) one block of $\mathcal{B}$. A 2-(v,k,1) design is also called a Steiner system and denoted by $S(2,k,v)$.

Partial designs are also known as packings [3] or packing designs [4]. We call them partial designs following [5]. An automorphism of a 2-$(v,k,1)$ partial design D is a permutation of its points that maps each block of D to a block of D.

Definition 3.

A 2-(v,k,1) design (partial design) is cyclic if it has an automorphism α permuting its points in one cycle, and it is strictly cyclic if each block orbit under this automorphism is of length v (no short orbits).

A cyclic partial design with the maximum possible number of blocks is optimal.

Definition 4.

A circulant matrix (circulant) of order v is a (0,1) square matrix $M={\left(m_{i,j}\right)}_{v\times v}$ with v rows and columns such that $m_{i+1,j+1}=m_{i,j}$, $i,j\in {\mathbb{Z}}_v$.

Perfect $(v,k,1)$ CPCW codes are equivalent to strictly cyclic 2-$(v,k,1)$ designs. An optimal $(v,k,1)$ CPCW code is equivalent to an optimal 2-$(v,k,1)$ strictly cyclic partial design. The incidence matrix of a 2-$(v,k,1)$ strictly cyclic partial design contains submatrices, which are circulant matrices of order v. Each such circulant matrix corresponds to the translates of one of the codewords of the corresponding optimal $(v,k,1)$ CPCW code. An example illustrating this equivalence is presented in Figure 1.

We are not going to define more related structures here. We shall only mention that optimal $(v,k,1)$ CPCW codes correspond to $(v,k;⌊(v-1)/k(k-1)⌋)$ difference packings, and that perfect $(v,k,1)$ CPCW codes are equivalent to $(v,k,1)$ cyclic difference families. An interested reader can find more information, for instance, in [6,7].

1.2. Known Results and the Contribution of the Present Paper

Optimal $(v,k,1)$ CPCW codes are studied in [2,8,9,10]. There are also many results on CPCW codes that were obtained for related combinatorial objects, like optical orthogonal codes [11,12,13,14,15,16,17,18,19,20,21], difference families [6,22,23,24,25,26], and combinatorial designs [27,28,29,30].

It was proved in [12] that an optimal $(v,3,1)$ CPCW code exists for all v except for $v=6t+2$ and $t\equiv 2$ or $3 (\mathrm{mod} 4)$. Direct constructions with explicit codewords are presented in [21] to show the existence of an optimal (v,4,1) CPCW (OOC) code for any positive integer $v\ne 25$. The existence of an optimal $(v,k,1)$ CPCW code for $k\ge 5$ is still an open problem.

In this paper, we consider $k=5$ and 6.

(a)

Optimal (43,5,1) CPCW code

$\mathcal{C}$	$▵\mathrm{C}$
{0, 1, 3, 7, 19}	{1, 2, 3, 4, 6, 7, 12, 16, 18, 19, 24, 25, 27, 31, 36, 37, 39, 40, 41, 42}
{0, 5, 13, 22, 33}	{5, 8, 9, 10, 11, 13, 15, 17, 20, 21, 22, 23, 26, 28, 30, 32, 33, 34, 35, 38}

Uncovered differences: 0, 14, 29

(b)

Related optimal partial cyclic 2-(43,5,1) design

Figure 1. The corresponding partial design.

Figure 1. The corresponding partial design.

First, we will summarize the known results for optimal CPCW codes or related combinatorial objects for $k=5$ and 6.

• An optimal $(15p,5,1)$ OOC with p a product of primes congruent to 1 modulo 4 and greater than 5 exists according to [18].
• It is shown in [14] that an optimal $(4p,5,1)$ OOC exists for prime $p\equiv 1 (\mathrm{mod} 10)$, and that there is an optimal $(4up,5,1)$ OOC for $u=2,3$ and prime $p\equiv 11 (\mathrm{mod} 20)$.
• A proof of the existence of a $(3^{s}5v,5,1)$ OOC if v is a product of primes congruent to 1 modulo 4 and s is a non-negative integer is given in [16,17].
• There are also several direct constructions for optimal $(gv,5,1)$ OOCs where $60\le g\le 180$, $g\equiv 0 (\mathrm{mod} 20)$, and v is a product of primes greater than 5.
• A class of optimal $(gv,6,1)$ optical orthogonal codes with v a product of primes congruent to 7 modulo 12 and greater than 7 and $g=15,20,105,140$ is obtained in [19].
• Some combinatorial constructions for optimal $(v,k,1)$ OOCs using g-regular cyclic packings are derived in [4,20].

Some existence results about CPCW codes with $k=5$ and 6 follow from results about cyclic difference families and combinatorial designs.

• It is shown in [23] that a $(p,5,1)$ difference family exists for $p\equiv 1 (\mathrm{mod} 20)$ for prime p.
• According to [26], a $(20t+1,5,1)$ difference family exists for $1\le t\le 50$ except possibly for $t=16,25,31,34,40,45$.
• The existence of a cyclic $(pq,5,1)$ difference family whenever p and q are primes congruent to $11 (\mathrm{mod} 20)$ is proved in [6].
• Existence of a $(q,6,1)$ cyclic difference family for any prime power $q\equiv 1 (\mathrm{mod} 30)$ with the exception of $q=61$ is shown in [22].
• It was proved in [29] that a cyclic $(5p,5,1)$ combinatorial design exists for all $p\equiv 1 (\mathrm{mod} 4)$ and $p\ne 5$.

Nonexistence results for optimal $(v,5,1)$ OOCs with $v\equiv 22 (\mathrm{mod} 40)$ and for an optimal $(62,6,1)$ OOC are presented in [7].

Classification results about $(v,k,1)$ CPCW codes for $k=5$ and 6 or about related combinatorial objects can be found in [25,27,28,30,31]. The results obtained in these works show that, when v grows, the number of nonequivalent codes becomes extremely large. This makes the classifications for the next larger lengths practically unusable.

In this work, we consider $k=5$ and 6 and the first values of v for which existence or nonexistence results are not known and try to find an example of an optimal $(v,k,1)$ CPCW code for these parameters. We conduct this for $k=5$ and $v\le 181$ and $k=6$ and $v\le 151$. We cannot find an optimal code for 9 of the considered parameter sets. We notice that two of these gaps are for $v=k(k-1)t+2$. This motivates us to consider the properties of such codes, and we derive necessary conditions for their existence and prove that an optimal $(92,6,1)$ CPCW code does not exist.

The paper is organized as follows: Section 2 demonstrates the basis of our approach. Section 3 presents both the computer-aided and analytical results that we obtain. Section 4 contains final remarks and open problems.

2. Methods

The explicit examples of new optimal $(v,5,1)$ and $(v,6,1)$ CPCW codes are obtained by the algorithm for classification of such codes, which we have used and described in [31]. It implies the construction of an array L with all possible codewords, namely with k-subsets of ${\mathbb{Z}}_v$ for which the auto-correlation property holds. We then construct codes by choosing their codewords from L in all possible ways. For that purpose, backtrack search on the elements of L is applied, and some partial solutions are filtered away if there exists an automorphism of ${\mathbb{Z}}_v$ that maps the partial solution to a partial solution that has already been constructed.

Our aim in the present work is not to obtain all the codes with these parameters but to find one such code. That is why we let the computation stop after having constructed one code, or after having worked for several days on a personal computer. The fact that we have not succeeded to construct a code with definite parameters does not mean that such a code does not exist.

The necessary conditions for the existence of optimal $\left(k\right(k-1)t+2,k,1)$ CPCW codes are derived using a method that is very popular in Design Theory, namely the counting of points and pairs of points in two different ways. The nonexistence of an optimal $(92,6,1)$ CPCW code is proved by using these necessary conditions and applying counting of the pairs of points in one more way.

3. Results

3.1. Examples of Optimal $(v,5,1)$ and $(v,6,1)$ CPCW Codes

We present in

Table 1. Optimal $(v,5,1)$ CPCW codes before the present study.

41	$\sout{\mathsf{42}}$	43	44	45	46–60	61	$\sout{62}$	63–64	65	66–80
81	82–87	88	89	?	101	$\sout{102}$	?	105	?	121
?	124	125	?	132	?	135	?	141	$\sout{142}$	?
145	?	161	?	164	165	?	181	$\sout{182}$	?	185

Normal font for theoretical results, italic for computer-aided, strikethrough for nonexistence and symbol ? when nothing is known.

and

Table 2. Optimal $(v,6,1)$ CPCW codes before the present study.

$\sout{61}$

$\sout{62}$

63

$\sout{\mathsf{64}}$

$\sout{\mathsf{65}}$

66–91

$\sout{\mathsf{92}}$

93–105

?

121

?

151

Normal font for theoretical results, italic for computer-aided, strikethrough for nonexistence and symbol ? when nothing is known.

all the previously known existence and nonexistence results regarding optimal $(v,5,1)$ and $(v,6,1)$ CPCW codes with $v\le 181$ and 151, respectively. Normal font is used for the v with theoretical results from [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,26,29] and italics for the results from the computer classifications in [25,27,28,30,31]. A normal or an italic v is struck through if the corresponding result (theoretic or computer-aided) shows nonexistence of optimal $(v,5,1)$ or $(v,6,1)$ CPCW codes. A question mark means that nothing is known about the existence of optimal codes with this length.

Table 3. New optimal $(v,5,1)$ CPCW codes.

41	$\sout{\mathsf{42}}$	42–43	44	45	46–60	61	$\sout{62}$	63–64	65	66–80
81	82–87	88	89	90–100	101	$\sout{102}$	103–104	105	106–120	121
122–123	124	125	126–131	132	133–134	135	136–140	141	$\sout{142}$	?
145	?	147–160	161	?	164	165	166	?	168–180	181

Bold font for results in the present paper, normal font for theoretical results, italic for computer-aided, strikethrough for nonexistence and symbol ? when nothing is known.

and

Table 4. New optimal $(v,6,1)$ CPCW codes.

$\sout{61}$

$\sout{62}$

63

$\sout{\mathsf{64}}$

$\sout{\mathsf{65}}$

66–91

$\sout{\mathsf{92}}$

93–105

106–120

121

?

125

?

127–150

151

Bold font for results in the present paper, normal font for theoretical results, italic for computer-aided, strikethrough for nonexistence and symbol ? when nothing is known.

are organized in a similar way to Table 1 and Table 2, but in them the existence results from the present paper are implemented too, and bold font is used for them. The question marks mean that we could not find optimal codes for these values of v and the question of the existence of such codes remains open. In particular, we have not found examples of optimal $(v,5,1)$ CPCW codes for $v=143,144,146,162$, and 167, and of optimal $(v,6,1)$ CPCW codes for $v=122,123,124$, and 126. These are $\left(k\right(k-1)t+c,k,1)$ CPCW codes with relatively small c. The latter is also small for the optimal codes for which there are no theoretical nonexistence results, but they have been proved not to exist with the help of a computer [31], namely an optimal $(42,5,1)$ CPCW code and optimal $(v,6,1)$ CPCW codes for $v=61,64,65$, and 92. This motivated our study of the properties of optimal $\left(k\right(k-1)t+2,k,1)$ CPCW codes.

Table A1. Optimal (v,5,1) CPCW codes.

Length	Codewords
90	1 3 7 18	5 25 33 52	13 39 53 69	9 31 55 67
91	1 3 7 12	8 33 49 73	14 35 52 71	10 23 54 69
92	1 3 7 12	15 35 54 76	13 30 55 78	8 26 36 60
93	1 3 7 12	8 27 40 73	10 26 48 78	14 31 49 70
94	1 3 7 12	8 34 50 77	14 33 53 71	10 31 46 59
95	1 3 7 15	9 32 53 84	13 41 59 78	5 29 39 55
96	1 3 7 12	10 36 63 76	8 25 57 75	14 37 52 68
97	1 3 7 12	10 30 53 70	13 32 47 68	8 24 46 72
98	1 3 7 12	16 38 59 78	8 26 61 74	10 27 41 56
99	1 3 7 12	10 34 51 72	14 29 57 83	8 33 55 68
100	1 3 7 12	14 38 66 81	13 36 54 71	8 30 40 61
103	1 3 15 96	9 49 65 92	4 37 50 79	6 41 58 77	5 30 64 85
104	1 3 7 90	9 27 63 82	10 35 47 81	5 45 65 93	8 38 51 80
106	1 3 8 43	11 48 60 79	13 34 67 89	9 45 59 74	4 20 82 100
107	1 3 7 34	9 23 51 71	10 32 50 96	8 24 37 63	5 35 54 95
108	1 3 7 39	9 26 56 93	10 31 53 95	8 19 33 68	5 34 50 62
109	1 3 7 31	8 22 33 65	9 35 48 58	5 50 68 97	15 53 69 90
110	1 3 7 27	8 42 72 101	12 25 40 73	5 36 52 96	10 21 53 75
111	1 3 7 21	12 41 65 84	9 44 78 89	10 47 62 98	8 25 63 79
112	1 3 7 16	10 45 62 88	8 19 49 74	12 44 73 91	5 28 42 64
113	1 3 7 12	13 34 58 84	10 30 47 70	15 31 59 77	8 27 49 88
114	1 3 7 15	9 41 59 86	10 21 52 81	17 51 75 95	5 40 66 89
115	1 3 7 25	8 34 45 102	5 28 84 101	9 41 61 71	12 51 67 100
116	1 3 7 17	8 30 90 101	5 36 54 75	12 59 78 91	9 29 64 92
117	1 3 7 18	5 38 66 74	10 30 70 83	16 42 71 92	9 31 68 103
118	1 3 7 24	9 27 63 98	11 33 61 77	12 31 79 93	5 15 45 58
119	1 3 7 16	10 30 41 70	5 43 71 105	8 26 72 95	12 33 77 94
120	1 3 7 24	13 39 72 91	9 46 66 102	10 41 53 85	8 58 73 98
122	1 3 21 33	10 34 93 106	8 36 76 111	6 51 58 113	4 42 85 99	5 49 74 105
123	1 3 22 53	8 67 93 103	4 17 33 65	9 55 98 109	5 12 49 88	6 24 51 66
126	1 3 25 122	6 23 85 100	8 38 75 91	11 44 72 92	12 68 86 99	9 66 80 116
127	1 3 21 54	4 12 84 89	9 25 57 91	11 41 63 112	6 29 68 87	7 17 31 44
128	1 3 82 119	4 99 105 110	7 21 69 97	13 39 55 96	8 51 75 111	15 35 65 109
129	1 3 15 105	13 36 82 100	8 99 109 120	6 32 63 85	4 37 71 78	5 54 73 89
130	1 3 8 75	6 17 51 105	12 35 64 83	9 27 37 117	4 30 73 114	15 39 77 109
131	1 3 9 55	4 64 105 115	11 33 81 99	5 12 36 108	15 49 68 106	13 58 75 102
133	1 3 8 113	9 27 72 86	12 36 62 96	10 51 68 103	6 39 85 117	4 29 42 118
134	1 3 7 37	12 55 68 94	9 47 92 111	22 57 81 106	5 15 46 63	8 29 62 73
136	1 3 7 23	8 73 82 100	14 57 89 108	11 60 91 106	12 38 78 115	5 34 69 86
137	1 3 7 16	12 36 55 84	11 33 50 77	5 42 63 117	10 40 86 109	8 49 67 81
138	1 3 7 27	10 22 66 127	9 48 62 107	16 84 103 120	8 49 81 123	5 30 43 80
139	1 3 7 28	8 24 56 85	10 33 53 102	12 31 46 76	13 55 73 113	14 65 82 117
140	1 3 7 34	9 26 78 127	8 29 43 66	10 55 83 102	5 64 89 129	12 36 80 122
147	1 3 9 88	5 48 72 113	4 19 100 122	17 37 91 129	7 23 49 101	13 27 58 90
	12 40 76 126
148	1 3 44 58	6 108 119 141	4 27 81 86	15 45 83 132	8 34 84 131	12 32 85 124
	9 19 37 79
149	1 3 18 127	4 12 61 72	10 26 58 105	7 52 94 115	5 33 64 83	9 39 74 112
	13 27 56 80
150	1 3 36 110	5 39 58 138	9 81 104 130	6 56 63 71	11 32 123 137	10 52 83 113
	4 48 64 132
151	1 3 10 56	4 12 40 73	15 39 87 104	16 76 110 130	5 88 107 138	22 45 80 122
	11 25 92 124
152	1 3 10 95	11 33 110 133	8 54 91 125	6 31 72 93	5 101 114 140	4 28 44 107
	14 29 84 134
153	1 3 11 32	14 44 70 118	7 19 62 84	5 33 94 133	9 24 66 82	6 47 81 99
	4 17 40 67
154	1 3 10 100	11 50 71 129	8 49 89 135	6 35 67 126	5 20 53 137	18 56 80 103
	4 16 30 82
155	1 3 7 62	12 36 84 124	13 39 57 91	8 27 38 114	17 46 102 135	5 21 63 86
	22 47 82 127
156	1 3 7 87	5 15 35 123	9 55 73 105	8 29 57 71	16 39 61 98	19 66 100 131
	17 41 94 130
157	1 3 7 15	9 27 63 135	16 48 83 112	17 51 98 119	11 52 118 131	10 30 70 114
	19 42 75 99
158	1 3 7 33	9 27 63 139	8 43 58 141	10 41 57 94	21 45 87 135	12 77 97 136
	11 67 80 129
159	1 3 7 33	14 42 98 144	5 27 71 95	11 59 96 119	16 54 89 134	13 49 107 125
	10 53 72 92
160	1 3 7 23	13 39 91 139	8 41 79 107	9 27 63 74	12 29 43 87	10 25 93 128
	5 51 101 141
166	1 5 39 140	3 15 88 117	7 35 107 150	11 32 116 129	9 19 77 95	2 46 113 121
	17 79 109 142	6 42 112 152
168	1 3 16 125	4 38 117 149	11 37 104 129	10 41 81 110	5 52 85 146	8 53 73 159
	6 18 42 72	7 21 77 140
169	1 3 17 126	6 18 80 102	7 32 41 70	5 92 115 134	8 76 91 121	20 53 81 117
	11 24 122 143	4 69 79 118
170	1 3 16 116	7 21 112 132	5 27 80 164	12 47 72 141	10 61 92 128	8 74 97 114
	18 44 68 151	4 32 131 140
171	1 3 9 60	5 15 45 134	16 48 121 144	12 80 113 149	13 39 117 146	7 21 84 95
	20 44 99 130	18 53 124 143
172	1 3 10 130	11 31 114 137	6 57 151 159	15 33 62 100	5 59 84 133	14 30 91 131
	22 56 82 119	4 96 108 144
173	1 3 10 77	4 12 40 135	5 32 144 157	19 60 105 149	6 23 62 121	11 37 83 103
	18 65 100 122	25 79 110 143
174	1 3 7 127	5 15 35 113	13 36 82 150	11 56 84 133	12 44 86 153	9 34 119 157
	8 27 70 99	14 53 71 93
175	1 3 7 94	9 27 63 146	8 61 77 134	10 96 113 127	15 34 55 131	12 25 51 155
	11 33 75 140	5 85 128 152
176	1 3 8 34	9 27 72 130	14 94 129 153	11 49 71 147	6 56 97 140	4 93 106 123
	12 28 137 156	10 54 75 161
177	1 3 9 83	5 15 45 61	4 26 104 164	12 50 126 157	18 37 106 142	27 68 120 149
	7 21 54 65	23 85 110 134
178	1 3 7 133	9 27 63 129	14 70 92 153	12 38 148 161	8 90 113 154	15 34 74 94
	10 72 107 157	5 16 103 150
179	1 3 7 22	10 30 41 70	13 39 91 107	17 49 76 141	14 42 98 132	5 23 69 102
	24 67 104 129	12 57 108 143
180	1 3 7 146	11 33 77 166	19 42 88 137	13 39 67 117	12 70 127 163	15 55 116 164
	9 68 106 162	8 32 107 128

and

Table A2. Optimal (v,6,1) CPCW codes.

Length	Codewords
106	1 3 7 15 36	5 45 56 65 84	10 26 44 57 74
107	1 3 7 12 89	8 34 47 63 83	10 27 41 69 84
108	1 3 7 12 37	8 29 51 69 93	10 45 58 77 91
109	1 3 7 12 39	8 28 41 58 83	10 24 53 72 88
110	1 3 7 12 33	8 36 54 76 95	10 27 47 71 85
111	1 3 7 12 29	8 24 56 66 96	13 33 47 74 93
112	1 3 7 12 28	8 31 57 77 90	10 47 64 83 98
113	1 3 7 12 25	8 27 43 60 81	10 36 51 65 93
114	1 3 7 12 25	10 30 44 71 86	8 31 48 67 93
115	1 3 7 12 25	8 37 54 81 95	10 26 45 77 92
116	1 3 7 12 26	8 43 60 70 100	13 28 49 78 96
117	1 3 7 12 22	13 33 50 74 92	8 31 47 77 91
118	1 3 7 12 22	13 39 57 77 91	8 24 53 70 95
119	1 3 7 12 25	8 28 43 62 89	10 42 59 82 98
120	1 3 7 12 22	8 24 51 74 92	13 33 47 78 95
125	1 3 19 31 54	5 47 64 93 104	7 41 50 77 110	4 67 80 105 119
127	1 3 8 51 113	6 38 61 96 108	4 20 33 60 101	10 34 52 88 116
128	1 3 10 30 53	6 21 45 87 100	4 18 35 40 72	8 33 59 71 117
129	1 3 9 25 97	5 15 60 80 98	12 52 73 102 115	4 23 71 82 99
130	1 3 7 15 107	9 41 57 75 119	5 33 50 72 93	10 35 71 84 111
131	1 3 23 36 105	6 65 73 97 116	9 37 54 79 93	4 11 16 57 87
132	1 3 10 26 87	5 57 74 95 108	11 33 73 100 112	6 14 50 78 97
133	1 3 7 25 95	5 48 78 105 125	9 26 49 68 80	11 46 61 75 112
134	1 3 7 92 107	8 48 66 101 113	9 46 63 82 120	5 25 84 95 108
135	1 3 7 38 61	8 24 34 56 107	5 17 30 50 93	9 27 66 80 95
136	1 3 7 31 47	8 49 74 88 124	10 52 65 103 125	5 64 83 109 118
137	1 3 7 20 56	8 46 58 83 122	9 44 72 104 115	10 26 78 96 123
138	1 3 7 15 48	9 43 64 96 116	5 24 62 80 108	10 23 49 60 121
139	1 3 7 16 57	11 42 60 74 111	10 48 68 103 115	5 22 66 99 118
140	1 3 7 15 59	11 35 48 86 112	5 30 46 79 113	9 62 80 109 130
141	1 3 7 15 102	9 19 35 53 128	5 33 60 89 110	11 41 58 78 103
142	1 3 7 15 46	9 29 61 87 114	10 35 75 86 108	5 23 93 106 123
143	1 3 7 15 47	5 27 43 62 112	9 29 77 101 118	10 55 83 94 120
144	1 3 7 15 93	5 39 81 118 128	17 46 73 91 121	9 22 50 109 133
145	1 3 7 12 33	8 57 74 111 125	13 44 73 97 120	16 59 81 99 126
146	1 3 7 15 105	16 34 54 86 119	5 40 62 87 115	10 29 55 68 79
147	1 3 7 15 44	11 39 56 69 90	5 24 40 72 105	9 31 83 93 129
148	1 3 7 12 42	10 56 88 115 135	15 34 51 72 96	18 44 73 98 126
149	1 3 7 12 102	8 34 71 122 139	13 32 55 75 96	14 30 45 103 127
150	1 3 7 12 45	8 39 87 124 134	13 54 69 86 136	19 49 72 107 129

in Appendix A contain all the examples of new optimal $(v,5,1)$ and $(v,6,1)$ CPCW codes. Each codeword contains the element $0\in {\mathbb{Z}}_v$, and therefore this element is not presented.

3.2. Properties of Optimal $\left(k\right(k-1)t+2,k,1)$ CPCW Codes

The optimal $(v,k,1)$ CPCW codes of length $k(k-1)t+1$ correspond to 2-$\left(k\right(k-1)t+1,k,1)$ designs. They are perfect because their codewords cover all $v-1$ differences. Optimal CPCW codes of length $k(k-1)t+2$, however, have remarkable combinatorial properties too because they do not cover only one difference. We derive here some necessary conditions for the existence of such codes.

Lemma 1.

The element ${\displaystyle \frac{v}{2}}\in {\mathbb{Z}}_v$ cannot be covered by the differences of the codewords of a $(v,k,1)$ CPCW code of even length. For even v, the element ${\displaystyle \frac{v}{2}}\in {\mathbb{Z}}_v$ can be obtained as the difference $a-b$ of ${\displaystyle \frac{v}{2}}$ different pairs $\{a,b\}$.

Proof. 

Let C be a codeword of a $(v,k,1)$ CPCW code $\mathcal{C}$ of even length. Let $a,b\in C$. Then, $a-b$ and $b-a$ must be two different differences from $▵C$. Suppose $a-b={\displaystyle \frac{v}{2}}$, and then $b-a=v-{\displaystyle \frac{v}{2}}={\displaystyle \frac{v}{2}}$, and ${\displaystyle \frac{v}{2}}$ appears twice in $▵C$, but this is impossible for a $(v,k,1)$ CPCW code. Since $a-b=b-a={\displaystyle \frac{v}{2}}$, the element ${\displaystyle \frac{v}{2}}\in {\mathbb{Z}}_v$ can be obtained as the difference $a-b$ of the ${\displaystyle \frac{v}{2}}$ different pairs $\{a,a+{\displaystyle \frac{v}{2}}\}$, $a\le {\displaystyle \frac{v}{2}}$. □

Lemma 2.

The codewords of an optimal $\left(k\right(k-1)t+2,k,1)$ CPCW code leave uncovered only the difference ${\displaystyle \frac{v}{2}}$.

Proof. 

Since $k(k-1)t+2$ is even, by Lemma 1, the possible differences are $v-2=k(k-1)t$. The size of such a code is

$$⌊{\displaystyle \frac{v-1}{k(k-1)}}⌋=⌊{\displaystyle \frac{k(k-1)t+1}{k(k-1)}}⌋=t,$$

and each codeword covers $k(k-1)$ differences. Then, all the codewords cover $k(k-1)$$t=v-2$ differences; that is, only one difference remains uncovered. □

An optimal $\left(k\right(k-1)t+2,k,1)$ CPCW code corresponds to an optimal partial cyclic 2-$\left(k\right(k-1)t+2,k,1)$ design. The translates of each codeword correspond to v blocks of this partial design and to a circulant submatrix of the incidence matrix of the partial design. We next consider some properties of this partial design.

Lemma 3.

The ${\displaystyle \frac{v}{2}}$ pairs of points $\{a,a+{\displaystyle \frac{v}{2}}\}$ are not contained in the blocks of an optimal partial cyclic 2-$\left(k\right(k-1)t+2,k,1)$ design corresponding to an optimal $\left(k\right(k-1)t+2,k,1)$ CPCW code. Each of the remaining ${\displaystyle \frac{v(v-2)}{2}}$ pairs of points is in exactly one block of this partial design.

Proof. 

The ${\displaystyle \frac{v}{2}}$ pairs of points that are not contained in the blocks of the design correspond to the ${\displaystyle \frac{v}{2}}$ different pairs $\{a,a+{\displaystyle \frac{v}{2}}\}$ from Lemma 1. □

Lemma 4.

Denote the rows/columns of a circulant M of an even order v by the numbers $0,1,2,\dots v-1$. Let φ be a permutation that rearranges the rows and the columns of M in such a way that they follow each other as $0,2,4\dots v-2,1,3,5,\dots v-1$. Then, the resulting matrix $\phi M$ is composed of 4 circulants A, B, C, and D of order ${\displaystyle \frac{v}{2}}$.

$$\phi M=\left(\begin{array}{cc}\hfill A& B\\ \hfill C& D\end{array}\right).$$

Denote by $u_A$, $u_B$, $u_C$, and $u_D$ the number of ones in a row/column of the matrices A, B, C, and D, respectively. Then, $u_A=u_D$ and $u_B$ = $u_C$.

Proof. 

Cutting off the even rows and even columns of M leads to a matrix A of dimension ${\displaystyle \frac{v}{2}}\times {\displaystyle \frac{v}{2}}$. Let $m_{i,j}$ be the element in row i and column j of M and $a_{i,j}$ be the element in row i and column j of A. Then, $a_{i,j}=m_{2i,2j}$.

We want to show that A is a circulant matrix. Addition of the indices of A is modulo ${\displaystyle \frac{v}{2}}$, and of M modulo v. Since $2i (\mathrm{mod} v)=2(i (\mathrm{mod} {\displaystyle \frac{v}{2}}))$, the addition of the even indices of M can be considered modulo ${\displaystyle \frac{v}{2}}$ too. Then, $a_{i+1,j+1}=m_{2i+2,2j+2}$. M is a circulant, and therefore $m_{2i+2,2j+2}=m_{2i,2j}=a_{i,j}$. It follows that $a_{i+1,j+1}=a_{i,j}$; that is, A is a circulant matrix.

Similarly, the circulants B, C, and D are obtained by cutting off the even rows and odd columns, the odd rows and even columns, and the odd rows and odd columns, respectively.

The number of ones in each row and column of $\phi M$ is the same, and therefore $u_A=u_D$ and $u_B$ = $u_C$. □

Example 1.

We illustrate the action of φ on two arbitrary circulants of even orders 10 and 8.

0 1 2 3 4 5 6 7 8 9 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 1 0 1 0 2 0 0 0 1 0 0 0 1 0 1 3 1 0 0 0 1 0 0 0 1 0 4 0 1 0 0 0 1 0 0 0 1 5 1 0 1 0 0 0 1 0 0 0 6 0 1 0 1 0 0 0 1 0 0 7 0 0 1 0 1 0 0 0 1 0 8 0 0 0 1 0 1 0 0 0 1 9 1 0 0 0 1 0 1 0 0 0	0 1 2 3 4 5 6 7 8 9 0 0 1 0 0 0 1 0 1 0 0 2 0 0 0 1 0 0 0 1 0 1 4 0 1 0 0 0 1 0 0 0 1 6 0 1 0 1 0 0 0 1 0 0 8 0 0 0 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 1 0 3 1 0 0 0 1 0 0 0 1 0 5 1 0 1 0 0 0 1 0 0 0 7 0 0 1 0 1 0 0 0 1 0 9 1 0 0 0 1 0 1 0 0 0	0 2 4 6 8 1 3 5 7 9 0 0 0 0 0 0 1 0 1 1 0 2 0 0 0 0 0 0 1 0 1 1 4 0 0 0 0 0 1 0 1 0 1 6 0 0 0 0 0 1 1 0 1 0 8 0 0 0 0 0 0 1 1 0 1 1 0 1 0 1 1 0 0 0 0 0 3 1 0 1 0 1 0 0 0 0 0 5 1 1 0 1 0 0 0 0 0 0 7 0 1 1 0 1 0 0 0 0 0 9 1 0 1 1 0 0 0 0 0 0
0 1 2 3 4 5 6 7 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 2 1 0 0 1 0 1 0 0 3 0 1 0 0 1 0 1 0 4 0 0 1 0 0 1 0 1 5 1 0 0 1 0 0 1 0 6 0 1 0 0 1 0 0 1 7 1 0 1 0 0 1 0 0	0 1 2 3 4 5 6 7 0 0 1 0 1 0 0 1 0 2 1 0 0 1 0 1 0 0 4 0 0 1 0 0 1 0 1 6 0 1 0 0 1 0 0 1 1 0 0 1 0 1 0 0 1 3 0 1 0 0 1 0 1 0 5 1 0 0 1 0 0 1 0 7 1 0 1 0 0 1 0 0	0 2 4 6 1 3 5 7 0 0 0 0 1 1 1 0 0 2 1 0 0 0 0 1 1 0 4 0 1 0 0 0 0 1 1 6 0 0 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 3 0 0 1 1 1 0 0 0 5 1 0 0 1 0 1 0 0 7 1 1 0 0 0 0 1 0

Lemma 5.

Let M be the incidence matrix of an optimal partial cyclic 2-$\left(k\right(k-1)t+2,k,1)$ design D corresponding to an optimal $\left(k\right(k-1)t+2,k,1)$ CPCW code. If the permutation φ from Lemma 4 is applied to the rows of M and to the columns of each of its t circulant submatrices, a matrix with $4t$ circulant submatrices of order ${\displaystyle \frac{v}{2}}$ is obtained. If $v\equiv 2 (\mathrm{mod} 4)$, the columns of each of the two rows of submatrices cover all possible ${\displaystyle \frac{v(v-2)}{8}}$ pairs of points, and the missing ${\displaystyle \frac{v}{2}}$ pairs of points in the partial design have one point from the first and one from the second row of submatrices. If $v\equiv 0 (\mathrm{mod} 4)$, there are ${\displaystyle \frac{v}{4}}$ pairs of points that are not covered by the columns of each row of submatrices, and all possible ${\displaystyle \frac{v^2}{4}}$ pairs of points having one point from the first and one from the second row of submatrices are covered.

Proof. 

By Lemma 4, the permutation $\phi$ transforms each of the t circulants of order v to 4 circulants of order ${\displaystyle \frac{v}{2}}$. That is why the incidence matrix $\phi M$ of the partial design has $4t$ circulant submatrices of order ${\displaystyle \frac{v}{2}}$. If $v\equiv 2 (\mathrm{mod} 4)$, then ${\displaystyle \frac{v}{2}}$ is odd. That is why, if a is odd, $a+{\displaystyle \frac{v}{2}}$ is even and vice versa. Since the first row of submatrices of $\phi M$ contains the even rows of M and the second the odd rows, the missing pairs of points $\{a,a+{\displaystyle \frac{v}{2}}\}$ (Lemma 3) have one point from the first row of submatrices and one from the second. If $v\equiv 0 (\mathrm{mod} 4)$, then ${\displaystyle \frac{v}{2}}$ is even, a and $a+{\displaystyle \frac{v}{2}}$ are both even or both odd, and the missing pairs of points consist of two points from one and the same row of submatrices. □

With respect to the number of ones in the four submatrices, $\phi$ can transform a circulant of order v to $⌊{\displaystyle \frac{k}{2}}⌋+1$ types of matrices $A_0,A_1,\dots A_{⌊\frac{k}{2}⌋}$ such that $A_i$ has two circulant submatrices with i ones in each row/column and two with $k-i$ ones in each row/column.

Example 2.

Types of matrices for $k=5$ and 6

For $k=5$, the types of matrices are

$A_0$	$A_1$	$A_2$
0 5	1 4	2 3
5 0	4 1	3 2

For $k=6$, the types of matrices are

$A_0$	$A_1$	$A_2$	$A_3$
0 6	1 5	2 4	3 3
6 0	5 1	4 2	3 3

Theorem 1.

For an optimal $(v,k,1)$ CPCW code with $v=k(k-1)t+2$ and $v\equiv 2 (\mathrm{mod} 4)$, the following equations hold:

$$\begin{array}{c}\hfill \sum _{i=0}^{{\displaystyle \frac{v}{2}}}{a}_i={\displaystyle \frac{v-2}{k(k-1)}}\end{array}$$

(3)

$$\begin{array}{c}\hfill \sum _{i=0}^{{\displaystyle \frac{v}{2}}}\left({\displaystyle \frac{i(i-1)}{2}}+{\displaystyle \frac{(k-i)(k-i-1)}{2}}\right)a_i={\displaystyle \frac{v-2}{4}}\end{array}$$

(4)

$$\begin{array}{c}\hfill \sum _{i=1}^{{\displaystyle \frac{v}{2}}}i(k-i)a_i={\displaystyle \frac{v-2}{4}}\end{array}$$

(5)

Proof. 

We obtain the equations by counting in different ways the pairs of points in $\phi M$. Denote by $a_i$ the number of $v\times v$ matrices of type $A_i$ that are contained in $\phi M$. Their number is $t={\displaystyle \frac{v-2}{k(k-1)}}$, and this is Equation (3). Equation (4) counts the pairs of points from one and the same row of submatrices. By Lemma 5, their number is ${\displaystyle \frac{v(v-2)}{8}}$. On the other hand, each ${\displaystyle \frac{v}{2}}\times {\displaystyle \frac{v}{2}}$ submatrix of a matrix of type $A_i$ covers ${\displaystyle \frac{v}{2}}\left({\displaystyle \frac{i(i-1)}{2}}+{\displaystyle \frac{(k-i)(k-i-1)}{2}}\right)$ pairs of points. Equation (5) counts the pairs in which the two points are from different rows of submatrices. By Lemma 5, all but the missing ${\displaystyle \frac{v}{2}}$ pairs are covered; that is, all such pairs are ${\displaystyle \frac{v^2}{4}}-{\displaystyle \frac{v}{2}}={\displaystyle \frac{v(v-2)}{4}}$. On the other hand, each $v\times v$ submatrix of type $A_i$ covers $i(k-i)v$ such pairs of points. □

Theorem 2.

For an optimal $(v,k,1)$ CPCW code with $v=k(k-1)t+2$ and $v\equiv 0 (\mathrm{mod} 4)$, the following equations hold:

$$\begin{array}{c}\hfill \sum _{i=0}^{{\displaystyle \frac{v}{2}}}{a}_i={\displaystyle \frac{v-2}{k(k-1)}}\end{array}$$

(6)

$$\begin{array}{c}\hfill \sum _{i=0}^{{\displaystyle \frac{v}{2}}}\left({\displaystyle \frac{i(i-1)}{2}}+{\displaystyle \frac{(k-i)(k-i-1)}{2}}\right)a_i={\displaystyle \frac{v-4}{4}}\end{array}$$

(7)

$$\begin{array}{c}\hfill \sum _{i=1}^{{\displaystyle \frac{v}{2}}}i(k-i)a_i={\displaystyle \frac{v}{4}}\end{array}$$

(8)

Proof. 

The proof is similar to that of Theorem 1, but, by Lemma 5, the pairs of points from one and the same row of submatrices are ${\displaystyle \frac{v(v-2)}{8}}-{\displaystyle \frac{v}{4}}={\displaystyle \frac{v(v-4)}{8}}$. All the pairs in which the two points are from different rows of submatrices are covered. That is why their number is ${\displaystyle \frac{v^2}{4}}$. □

Corollary 1.

An optimal $(20t+2,5,1)$ CPCW code can only exist if t is even.

Proof. 

In this case, $v\equiv 2 (\mathrm{mod} 4)$, and Equation (5) gives

$$4a_1+6a_2={\displaystyle \frac{v-2}{4}}.$$

That is why $v-2$ should be divisible by 8. □

Corollary 1 is not a new result. It is part of the nonexistence results presented in [7], Theorem 4.6. The next example shows the application of Theorem 1 on an optimal $(20t+2,5,1)$ CPCW code with even t.

Example 3.

For an optimal $(82,5,1)$ CPCW code by Theorem 1:

$$\begin{array}{c}\hfill a_0+a_1+a_2+a_3=4\end{array}$$

(9)

$$\begin{array}{c}\hfill 5a_0+3a_1+2a_2=10\end{array}$$

(10)

$$\begin{array}{c}\hfill 2a_1+3a_2=10\end{array}$$

(11)

There exists a unique integer solution: $a_0=0,a_1=2,a_2=2$, and the optimal $(82,5,1)$ CPCW codes comply with it.

3.3. Nonexistence of an Optimal $(92,6,1)$ CPCW Code

Multiple nonexistence results on optimal $(v,k,1)$ CPCW codes are obtained in [7]. They include the nonexistence of an optimal $(62,6,1)$ code. We do not know other theoretical nonexistence results for optimal $(30t+2,6,1)$ CPCW codes. Our computer-aided classification in [31] shows that an optimal $(92,6,1)$ CPCW code does not exist. Using Theorem 2, we shall now prove this without a computer.

Proposition 1.

An optimal $(92,6,1)$ CPCW code does not exist.

Proof. 

By Theorem 2:

$$\begin{array}{c}\hfill a_0+a_1+a_2+a_3=3\end{array}$$

(12)

$$\begin{array}{c}\hfill 15a_0+10a_1+7a_2+6a_3=22\end{array}$$

(13)

$$\begin{array}{c}\hfill 5a_1+8a_2+9a_3=23\end{array}$$

(14)

There exists a unique integer solution:

$$\begin{array}{c}\hfill a_0=0,a_1=1,a_2=0,a_3=2\end{array}$$

(15)

It follows that the number of ones in $\phi M$ is distributed as

$$\begin{array}{ccc}5 1& 3 3& 3 3\\ 1 5& 3 3& 3 3\end{array}$$

We can see from (15) that the submatrices of dimension $46\times 46$ have $1,3$, or 5 ones in each row/column. Since 46 is even and $v/4=23$ is odd, we can use the same ideology as in Lemma 5 and Theorem 1 on the variable-length code corresponding to the first 46 rows of $\phi M$; that is, we apply to the first 46 rows of $\phi M$ a permutation ${\phi }^{\prime }$ such that after its application these rows follow each other as $0,2,4\dots 44,1,3,5,\dots 45$, and we apply the same permutation to the columns of the submatrices of dimension 46.

A $46\times 46$ circulant with 1, 3, or 5 ones in each row/column can be transformed by ${\phi }^{\prime }$ to one, two, or three types of matrices, respectively. We denote them $A_{1,0}$,$A_{3,0}$, $A_{3,1}$, $A_{5,0}$, $A_{5,1}$, and $A_{5,2}$. The matrix $A_{i,j}$ is of dimension $46\times 46$ and has four submatrices of dimension $23\times 23$, two of them with j and two with $i-j$ ones in each row/column.

$$\begin{array}{cccccc}\hfill A_{1,0}& A_{3,0}& A_{3,1}\hfill & A_{5,0}& A_{5,1}& A_{5,2}\\ \hfill 0 1& 0 3& 1 2\hfill & 0 5& 1 4& 2 3\\ \hfill 1 0& 3 0& 2 1\hfill & 5 0& 4 1& 3 2\end{array}$$

Denote by $a_{i,j}$ the number of $23\times 23$ matrices of type $A_{i,j}$ that are contained in the first 23 rows of ${\phi }^{\prime }\phi M$. Since 23 is odd, we can apply Theorem 1 and write an analogue of Equation (5) for $a_{i,j}$; thus, we obtain

$$\begin{array}{c}\hfill \sum _{(i,j)\in S}j(i-j)a_{i,j}={\displaystyle \frac{46-2}{4}}=11\end{array}$$

(16)

where $S=\left\{\right(1,0),(3,0),(3,1),(5,0),(5,1),(5,2\left)\right\}$. It follows that

$$\begin{array}{c}\hfill 2a_{3,1}+4a_{5,1}+6a_{5,2}=11.\end{array}$$

(17)

The same result can be obtained by counting the pairs of points such that one is from the first 23 rows of ${\phi }^{\prime }\phi M$ and the other from the second 23 rows. The missing 23 pairs of points are among them, and therefore

$$46(2a_{3,1}+4a_{5,1}+6a_{5,2})=23·23-23=23·22=46·11.$$

This results in Equation (17) again. Equation (17), however, has no integer solutions because its left side is even and the right one is odd. That is why an optimal $(92,6,1)$ CPCW code cannot exist. □

4. Discussion

The presented examples of optimal $(v,5,1)$ and $(v,6,1)$ CPCW codes might be useful in relevant applications, as well as in future research on the topic. The results for small parameters show that nonexistence of some optimal $\left(k\right(k-1)t+c,k,1)$ CPCW codes with small c can be expected. The solutions of the equations from Theorems 1 and 2 might be useful in further theoretical or computer-aided research on optimal $\left(k\right(k-1)t+2,k,1)$ CPCW codes whose existence remains undecided.