Linear and circular single‐change covering designs revisited

A single‐change covering design (SCCD) is a v $v$ ‐set X $X$ and an ordered list ℒ ${\rm{ {\mathcal L} }}$ of b $b$ blocks of size k $k$ where every pair from X $X$ must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single‐change covering design, or more simply, a single‐change covering design. A single‐change covering design is circular when the first and last blocks also differ by one element. A single‐change covering design is minimum if no other smaller design can be constructed for a given v,k $v,k$ . In this paper, we use a new recursive construction to solve the existence of circular SCCD( v,4,b $v,4,b$ ) for all v $v$ and three residue classes of circular SCCD( v,5,b $v,5,b$ ) modulo 16. We solve the existence of three residue classes of SCCD (v,5,b) $(v,5,b)$ modulo 16. We prove the existence of circular SCCD (2c(k−1)+1,k,c2(2k−2)+c) $(2c(k-1)+1,k,{c}^{2}(2k-2)+c)$ , for all c≥1,k≥2 $c\ge 1,k\ge 2$ , using difference methods.


| INTRODUCTION
A single-change covering design (SCCD(v k b , , )) X ( , )  is a v-set X and an ordered list B B B = ( , , …, ) b 1 2  of blocks of size k where every pair from X must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. A single-change covering design (circular SCCD) is circular if the first and last blocks also differ by one element. These types of SCCD are considered strength two as every pair is present in at least one block. Instead of pairs we could require that each triple or larger t-sets occur in at least one block to build strength t designs.
The earliest discussion of SCCDs was in 1969 and focused on applications to efficient computing [5]. Consider the calculation of AA T for some matrix A; this requires processing all pairs of rows of A. In 1969 a limited number of pairs could be stored in core memory (RAM) at one time and updating RAM from a tape was slow. Therefore, visiting all pairs with the smallest number of accesses to the tape improves running time significantly. Similar constraints apply today with respect to CPU cache versus RAM access [1,3].
Gower and Preece continued the discussion in 1972 when they analyzed minimizing changes between successive blocks [2]. Work continued in the 1990s by various subsets of Preece, Constable, Zhang, Yucas, Wallis, McSorley, and Phillips in a series of 11 papers that substantially developed the theory. Most relevant to this work is the solution Preece et al. give about the existence of minimum SCCD with k = 4 using a recursive construction-based solution on "expansion sets" [7].
In 1998 McSorley explored circular single-change covering designs [4]. He completely solved minimum circular SCCD(v k b , , ), where k = 2, 3 and where v k 2 ≤ . Analyzing properties of elements in a design which are only introduced once, he explored substructures that must be present in circular designs to significantly reduce the work required in a computer search which lead to him constructing all the minimum circular SCCD (9,4,12) and circular SCCD(10, 4,15).
In this previous research, the powerful recursive construction used to successfully complete some families of linear SCCD(v k b , , ) had no analog for circular SCCD(v k b , , ). We introduce such a recursive construction and completely solve the existence of circular SCCD(v b , 4, ) and for three residue classes of SCCD(v, 5, b) when v (mod 16). We also solve the existence for linear SCCD(v b , 5, ) for three residue class of v (mod 16). Further, we construct an infinite family of circular SCCD(v k b , , ) for every fixed k using difference methods.

| BACKGROUND
In an SCCD, every block except the last differs from the next block by a single-change. That is If the SCCD is circular, one element is introduced in each block. If the SCCD is linear, every element in B 1 is introduced and one element is introduced in each subsequent block. A pair S is covered on block B i if S B i ⊆ and one element of S was introduced in B i . In this paper, we will only consider pairs, but the study of SCCD for higher strength sets is of interest. We say an An example SCCD(7, 3, 10) is given in Table 1. We emphasize the introduced element in each block with an asterisk, *. Note that elements 1, 2, and 3 are all introduced in the first block so the pairs {1, 2}, {1, 3}, {2, 3} are covered in B 1 . In B 2 only 4 is introduced, so only pairs involving 4 are covered, namely {1, 4}, {2, 4}.
In Table 1, we note that every pair is covered exactly once. However, in Table 2 we have a circular SCCD (6,3,8) where the pair {1, 4} is covered two times.
The set of elements that remain the same between B i and B i+1 of an SCCD is the ith unchanged subset between these blocks, [7]. If the SCCD is not circular, U 0 and U b can be any k ( − 1)-subset of B 1 or B b , respectively. In the SCCD(7, 3, 10) in Table 1  is an expansion set. If the SCCD is not circular and  contains U 0 , U b or both, then  is an outer expansion set, otherwise,  is an inner expansion set. We will denote expansion set locations in our tables with carets, . The carrot will be placed between the two blocks containing the unchanged subset. In Table 2, 6 8 is an expansion set.
We say a (circular) SCCD(v k b , , ) is economical if it has g v k g v k ( , , 2) ( ( , , 2) ) 1 2 is tight if it is economical, and g v k g v k ( , , 2)( ( , , 2)) 1 2 is an integer. Any economical SCCD is minimum. In a tight SCCD, every pair is covered exactly once. For some v k , combinations tight SCCD cannot exist, but economical designs can exist.

CHAFEE and STEVENS
| 407 Although not formally stated, Preece et al. proposed two recursive constructions for tight SCCD [7]. We state them explicitly and note that they apply to both linear and circular SCCD.
Proposition 2 (v + 1 construction [7]). If an SCCD v k b ( , , ) with an expansion set exists then a ( ) For example, consider  Table 3 by introducing a new block for each U i containing the new element, 13, with the respective unchanged subset in the expansion set from the SCCD(12, 4, 21). It is worth noting that this large SCCD(13, 4, 25) no longer contains an expansion set because the new element, 13, is in no unchanged subsets.
Theorem 3 (Building tight SCCD from two tight SCCDs [7]). If there exists a tight has an expansion set then the tight SCCD v k b ( *, , *) has an expansion set.
For example, Table 4 shows two tight SCCD ( The tight SCCD(10, 3, 22), X ( *, *)  , constructed by Theorem 3 is shown in Table 5 with the blocks from B i and B′ i shown in black and , respectively, while the inserted blocks are in . As of 2001, the following SCCD and circular SCCD were known to exist.
In this paper, we extend Theorem 3 to include economical SCCD and circular SCCD. We use this to almost complete the spectrum of circular SCCD with k = 4 and three residue classes of SCCD and circular SCCD with k = 5. There are still some small open cases when k = 4 and k = 5. Furthermore, we provide an infinite number of circular SCCD for each k using difference methods.

| RESULTS
We present two different kinds of constructions in this section: recursive and direct. The direct constructions use difference methods.

| Recursions
We first will generalize Theorem 3 to construct economical SCCD. We say that a block B i in a SCCD is tight if the pairs it covers are not covered in any other block of the SCCD. We say the excess, e, of a SCCD is the number of pairs covered repeatedly.
We say the excess, e i , of a block B i in an SCCD is the number of pairs B i covers that were covered in any block B j , j i < . We say the excess, e i , of a block B i with respect to an initial block B 0 in a circular SCCD is the number of pairs B i covers that were covered in any block B j , Lemma 5. The excess, e, of a (circular) SCCD is the sum of the excesses of blocks.
Proof. Consider the SCCD v k b ( , , ), X ( , )  . Each block B i will cover k − 1 pairs where e i of these have been previously covered. Summing through these blocks we have . The proof for the circular SCCD is similar with respect to an initial block. Proof. Suppose that we have an economical SCCD(v k b , , ). From the definition of e we find is tight if every pair is covered exactly once, therefore no pair can be previously covered, and e = 0.
The proof for circular SCCD is similar. □ Noting that the excess of a tight block is zero proves the following.
 has the same excess as X ( , )  .
We note that we can reverse the design Lemma 7 produced and generate a design where the tight blocks are at the start of the design.
We may now generalize Theorem 3. with excess e′ and an outer expansion set Since the reverse of a SCCD is a SCCD we may assume ′  contains U′ 0 . We relabel the elements of X To build X ( *, *)  we start by appending  with ′  .
Since X ( ′, ′)  has an outer expansion set and the first expansion location of X ∩ , we have that the remaining expansion locations partition X X ′\ . For all The blocks  and ′  cover the same pairs in *  as they did in  and ′  , with the exception of B′ 1   is a SCCD. , ) with excess e and an outer expansion set exists then an  , ) with an outer expansion set then there exists an economical SCCD v k b ( + 2, , ′).
For example, see Table 7. The black blocks form a tight SCCD (12,4,21). We insert the blocks to form the economical SCCD (14,4,30). Now we consider constructions of circular SCCD using two linear SCCDs. We say an expansion set is disjoint-capable if The tight SCCD(10, 3,22) in Table 8 of [7] contains a disjoint-capable expansion set: Theorem 11 (Build disjoint-capable). If there exists an SCCD v k b ( , , ) with excess e and an SCCD v k b ( ′, , ′) with excess e′ both with outer expansion sets that use both U U , b , and X X X * = ′ ∪ there exists an SCCD v k b ( *, , *) with excess e e e * = + ′ and a disjoint-capable outer expansion set.
Proof. Apply Theorem 3 to X ( , )  and X ( ′, ′)  to construct a tight SCCD v k b ( *, , *), X ( *, *)  , with an outer expansion set using U 0 and U′ b . Insert the last k − 1 blocks B x { } i ∪ so that the last k − 1 elements introduced are every x U 0 ∈ . Reverse the resulting SCCD. □ For example, consider the tight SCCD(21, 4, 69) in Table 9. The elements denote the unchanged subset and the elements highlight the elements in U′ b that are introduced in the first k − 1 blocks. The block numbers that are highlighted in black, , and are the blocks of the SCCD(12, 4, 21), the SCCD(12, 4, 21), and the inserted B″ i x , j blocks, respectively.
Proof. One of the tight SCCD(20, 5, 46) from [6] has an expansion set containing both U 0 and U b . Apply Theorem 11 to two copies. This SCCD(36, 5, 156) is shown in Table 10. The black, , and blocks are two copies of a tight SCCD(20, 5, 46) and the inserted B″ i x , j blocks, respectively.
We bring the previous lemmas and theorems together to construct circular SCCDs.
Theorem 13 (Constructing circular SCCD from two SCCD). If there exists an SCCD v k b ( , , ) with excess e and a disjoint-capable expansion set and there exists an − 1 a circular SCCD v k b ( *, , *) exists with excess e e e * = + ′ and X X X * = ′ ∪ . Furthermore, if the SCCD v k b ( ′, , ′) has an outer expansion set using both U′ 0 and U′ b′ then the circular SCCD v k b ( *, , *) has an expansion set.
the circular SCCD v v k k b ( + ′ − 2 + 2, , *), we delete the first k − 1 blocks from  and append the blocks from ′  . Note, so there is a single-change between the last block of the design and the first block. Therefore X ( *, *)  is a circular SCCD.
Then the circular SCCD, X ( *, *)  , will have an expansion set For example, consider the tight SCCD(10, 3, 22) in Table 8 which contains a disjoint-capable expansion set and the SCCD(6, 3, 7) in Table 11. We combine these two SCCDs as prescribed in Theorem 13 to construct a tight circular SCCD(12, 3, 33) as seen in Table 12.

| Difference methods
We can also construct circular SCCD using difference methods.

Proof. Let
0, and  be the concatenation of the L j in order j v 0 ≤ ≤ .  is a circular single-change list of blocks where the element introduced in block , thus B 0,0 covers the differences k ±{1, 2, …, − 1}. This is all the differences in v , so as  is formed by the development of L i all pairs are covered. The circular SCCD is tight as b g = 2 and is an integer. apply Theorem 9 to each member of 5  . To each of these, we apply Theorem 13 with the previously constructed tight SCCD(21, 4, 69) with a disjoint-capable expansion set.
Apply Proposition 2 to give v 7 (mod 9) ≡ . For v 8 (mod 9) ≡ we create a family 8  , of tight SCCD i b (9 , 4, ) applying Theorem 8 recursively to the tight SCCD(18, 4, 50) using the tight SCCD(12, 4, 21). We then apply Theorem 9 to each member of 8  . To each of these, in turn, we apply Theorem 13 with the previously constructed tight SCCD(21, 4, 69) with a disjoint-capable expansion set. Thus completing the proof of (1). To prove (2), we use the tight SCCD(20, 5, 46) from the black blocks of Table 10, which has an outer expansion set. Use Theorem 8 recursively to construct tight SCCD i b (16 + 4, 5, ) for all i 1 ≤ . Apply Proposition 2 and Theorem 9 to add one and two points to each, respectively.
To prove (3), we use the tight circular SCCD(20, 5, 46) from the black blocks of Table 10 and the tight SCCD(36, 5, 156) from Table 10 in conjunction with (2), Proposition 2, Theorem 9, and Theorem 13 following the same construction structure as proving (1) given in detail above. Use Theorem 14 to construct v 9 (mod 16) ≡ . To prove (4), we use the construction given in Theorem 14. □ We note that the flexibility of our theorems means that there are other construction pathways to many of these designs.
The missing cases for economical SCCD v b ( , 4, ) are v 12 2 6 ≤ ≤ . These are too small to produce with our recursive constructions and thus we need to search for them directly, perhaps by computer. For k = 5, the existence of a tight SCCD(28, 5, 94) with an expansion set would allow the use of Proposition 2, Theorem 3, and Theorem 10 to construct economical SCCD (v b , 5, ) exists for v 4, 5, 6 (mod 8) ≡ , v 20 ≥ . These are tight if v 4, 5 (mod 8) ≡ . Using these designs in conjunction with Theorem 13 and the SCCD(36, 5, 156) with a disjoint-capable expansion set given in Table 10 would allow us to construct circular SCCD(v b , 5, ) for T A B L E 16 A tight SCCD(18,4,50) with outer expansion set [7].