Holey Perfect Mendelsohn Designs of Type 2^𝑛𝑢¹ with Block Size Four

Abstract

Let 4-HPMD denote a holey perfect Mendelsohn design with block size four. The existence of 4-HPMDs with 𝑛 holes of size 2 and one hole of size 3, that is, of type 2^𝑛3¹, was established by Bennett et al. in 1997. In this paper, we investigate the existence of 4-HPMDs of type 2^𝑛𝑢¹ for 1≤𝑢≤16: a 4-HPMD(2^𝑛𝑢¹) exists if and only if 𝑛≥max(4,𝑢+1), except possibly for (𝑛,𝑢)=(7,5), (7, 6), (11, 9), (11, 10). We also investigate the existence of 4-HPMD(2^𝑛𝑢¹) for general 𝑢 and prove that there exists a 4-HPMD(2^𝑛𝑢¹) for all 𝑛≥⌈5𝑢/4⌉+4. Moreover, if 𝑢≥35, then a 4-HPMD(2^𝑛𝑢¹) exists for all 𝑛≥⌈5𝑢/4⌉+1; if 𝑢≥95, then a 4-HPMD(2^𝑛𝑢¹) exists for all 𝑛≥⌈5𝑢/4⌉−2.

1. Introduction

Let 𝑣,𝑘 be positive integers, a (𝑣,𝑘,𝜆)-Mendelsohn design, briefly (𝑣,𝑘,𝜆)-MD, is a pair (𝑋,ℬ), where 𝑋 is a 𝑣-set (of points) and ℬ is a collection of cyclically ordered 𝑘-subset of 𝑋 (called blocks) such that every ordered pair of points of 𝑋 are consecutive in exactly 𝜆 blocks of ℬ. If for all 𝑡=1,2,…,𝑘−1, every ordered pair of points of 𝑋 is 𝑡-apart in exactly 𝜆 blocks of ℬ, then the (𝑣,𝑘,𝜆)-MD is called perfect and is denoted by (𝑣,𝑘,𝜆)-PMD.

The existence of a (𝑣,4,1)-PMD is equivalent to the existence of an idempotent quasigroup (𝑄,⋅) satisfying the identity, called Stein's third law, 𝑦⋅𝑥⋅𝑥⋅𝑦=𝑥 (1) for all 𝑥,𝑦∈𝑄. Let ℬ={(𝑥,𝑥⋅𝑦,𝑦,𝑦⋅𝑥)∶𝑥,𝑦∈𝑄,𝑥≠𝑦}. Then (𝑄,ℬ) is a (|𝑄|,4,1)-PMD.

For the existence of (𝑣,4,1)-PMD, we have the following theorem from [1–3].

Theorem 1. A (𝑣,4,1)-PMD exists if and only if 𝑣≡0,1(mod4) and 𝑣≠4,8.

In this paper, we are only interested in PMDs where 𝜆=1. A holey perfect Mendelsohn design (briefly HPMD) is a triple (𝑋,ℋ,ℬ) which satisfies the following properties. (1)ℋ is a partition of 𝑋 into subsets called holes. (2)ℬ is a family of cyclically ordered 𝑘-subset of 𝑋 (called blocks) such that a hole and a block contain at most one common point.(3)Every ordered pair of points from distinct holes are 𝑡-apart in exactly one block for 𝑡=1,2,…,𝑘−1. The type of the HPMD is the multiset {|𝐻|∶𝐻∈ℋ} and is described by an exponential notation. Throughout this paper, we shall use HPMD(𝑠^𝑛1₁𝑠^𝑛2₂⋯𝑠^𝑛𝑡_𝑡) to describe a PMD of the type in which 𝑠_𝑖 occurs 𝑛_𝑖 times, 1≤𝑖≤𝑡.

In graph theoretic terminology, an HPMD is a decomposition of a complete multipartite directed graph DK(𝑛₁,𝑛₂,…,𝑛_ℎ) into 𝑘-circuits such that for any two vertices 𝑥 and 𝑦 from different components, there is one circuit along which the directed distance from 𝑥 to 𝑦 is 𝑡, where 1≤𝑡≤𝑘−1.

Another class of designs related to HPMDs is group divisible design (GDD). A GDD is a 4-tuple (𝑋,𝒢,ℬ,𝜆) which satisfies the following properties. (1)𝒢 is a partition of 𝑋 into subsets called groups. (2)ℬ is a family of subsets of 𝑋 (called blocks) such that a group and a block contain at most one common point. (3)Every pair of points from distinct groups occurs in exactly 𝜆 blocks. The type of the GDD is the multiset {|𝐺|∶𝐺∈𝒢} and we will also use an “exponential” notation for the type of GDD. We also use the notation GDD(𝐾,𝑀;𝜆) to denote the GDD when its block sizes belong to 𝐾 and group sizes belong to 𝑀. In particular, a GDD({𝑘},{2,𝑢^∗},1), where there is only one group of size 𝑢, is denoted by 𝑘-GDD of type 2^𝑛𝑢¹.

Theorem 2 (see [4, 5]). There exists a 4-GDD of type 2^𝑛𝑢¹ for each 𝑛≥6, 𝑛≡0(mod3) and 𝑢≡2(mod3) with 2≤𝑢≤𝑛−1, except for (𝑛,𝑢)=(6,5) and possibly excepting (𝑛,𝑢)∈{(21,17),(33,23),(33,29),(39,35),(57,44)}.

If 𝑀={1}, then the GDD becomes a pairwise balanced design (PBD) [6]. If 𝐾={𝑘}, 𝑀={𝑛} and the type is 𝑛^𝑘, then the GDD becomes a transversal design, TD(𝑘,𝑛). The following results are well known (see [7, 8], e.g.).

Theorem 3. (a) There exists a TD(4,𝑚) for any positive integer 𝑚,𝑚∉{2,6}.
(b) There exists a TD(5,𝑚) for every positive integer 𝑚∉{2,3,6,10}.
(c) There exists a TD(6,𝑚) for 𝑚≥5 and 𝑚∉{6,10,14,18,22},
(d) There exists a TD(7,𝑚) for 𝑚≥7 and 𝑚∉{10,14,15,18,20,22,26,30,34,38,46,60}.

It is well known that the existence of a TD(𝑘,𝑛) is equivalent to the existence of 𝑘−2 mutually orthogonal Latin squares, denoted by (𝑘−2)-MOLS(𝑛). It is easy to see that if we ignore the direction of elements in the blocks, an HPMD becomes a GDD with 𝜆=3. But the converse may be not true. It is proved in [9] that a 4-GDD of type ℎ^𝑢 and index 𝜆=3 exists if and only if ℎ²𝑢(𝑢−1)≡0(mod4). However, for the existence of HPMDs with block size four (briefly 4-HPMDs), the following results are known ([8, 10–14]).

Theorem 4. (a) A 4-HPMD(ℎ^𝑢) exists if and only if ℎ²𝑢(𝑢−1)≡0(mod4) with the exception of types 1⁴, 2⁴, 1⁸ and ℎ⁴ for odd ℎ.
(b) For 𝑢≥2, a 4-HPMD(1^𝑛𝑢¹) exists if and only if 𝑛(𝑛−2𝑢−1)≡0(mod4) and 𝑛≥2𝑢+1, except (𝑛,𝑢)=(5,2) and except possibly (𝑛,𝑢)=(17,2).
(c) A 4-HPMD(2^𝑛3¹) exists if and only if 𝑛≥4.
(d) There exist HPMDs of type 2^𝑛𝑢¹ for 4≤𝑛≤8 and 0≤𝑢≤4, except (𝑛,𝑢)=(4,0),(4,4).
(e) For 0≤𝑢≤32, a 4-HPMD(4^𝑛𝑢¹) exists if and only if 𝑛≥max(4,⌈𝑢/2⌉+1), except possibly for (𝑛,𝑢)=(12,1), and 𝑛=19 and 𝑢∈{29,30,31,32}.

In this paper, we consider the existence of a 4-HPMD(2^𝑛𝑢¹) and extend the results in Theorems 4(c) and 4(d). It is fairly well known that the Latin square corresponding to a quasigroup satisfying Stein's third law is a self-orthogonal Latin square (briefly SOLS). So the existence of a 4-HPMD(2^𝑛𝑢¹) will imply the existence of a holey SOLS of the same type. Accordingly, the necessary condition for the existence of holey SOLS is also a necessary condition for the existence of 4-HPMD(2^𝑛𝑢¹), which can be stated as follows (see, e.g., [15, 16]).

Lemma 5. If a 4-HPMD(2^𝑛𝑢¹) exists, then 𝑛≥max(4,𝑢+1).

The main result of our investigation is the following theorem.

Theorem 6. (a) For 1≤𝑢≤16, a 4-HPMD(2^𝑛𝑢¹) exists if and only if 𝑛≥4, 𝑛≥𝑢+1, with the possible exception of (𝑛,𝑢)∈{(7,5),(7,6),(11,9),(11,10)}.
(b) For 𝑢≥1, a 4-HPMD(2^𝑛𝑢¹) exists if 𝑛≥⌈5𝑢/4⌉+4. Moreover, if 𝑢≥35, then a 4-HPMD(2^𝑛𝑢¹) exists for all 𝑛≥⌈5𝑢/4⌉+1; if 𝑢≥95, then a 4-HPMD(2^𝑛𝑢¹) exists for all 𝑛≥⌈5𝑢/4⌉−2.

For convenience in what follows, we shall simply refer to a 4-HPMD as an HPMD, where it is to be understood that that blocks are of size four.

2. Construction Tools

To construct HPMDs directly, sometimes we use starter blocks. Suppose the block set ℬ of an HPMD is closed under the action of some Abelian group 𝐺, then we are able to list only part of the blocks (starter or base blocks) which determines the structure of the HPMD. We can also attach some infinite points to an Abelian group 𝐺. When the group acts on the blocks, the infinite points remain fixed. Formally, let ℬ be the block set of an HPMD over the point set 𝑆=𝐺∪𝑋, where (𝐺,+) is a group, 𝑋 is a set of infinite points, 𝐺∩𝑋=∅. The addition (+) is extended over 𝑋 as follows: 𝑔+𝑥=𝑥+𝑔=𝑥 for any 𝑔∈𝐺 and 𝑥∈𝑋. A set 𝒜⊂ℬ is called starter blocks of ℬ if 𝒜 is a minimum subset of ℬ satisfying the following property: for any 𝐚∈𝒜 and any 𝑔∈𝐺, 𝐚+𝑔∈ℬ, and for any 𝐛∈ℬ, there exist 𝐚∈𝒜 and 𝑔∈𝐺 such that 𝐛=𝐚+𝑔, where 𝐚+𝑔=(𝑎₁+𝑔,𝑎₂+𝑔,𝑎₃+𝑔,𝑎₄+𝑔) when 𝐚=(𝑎₁,𝑎₂,𝑎₃,𝑎₄). In the following example 𝑥₁,𝑥₂,…, are infinite points.

Example 7. An HPMD(2⁹4¹) points: 𝑍₁₈∪{𝑥_𝑖∶1≤𝑖≤4} holes: {{𝑖,𝑖+9}∶0≤𝑖≤8}∪{{𝑥_𝑖∶1≤𝑖≤4}} starter blocks (+1mod18): (𝑥₁,0,14,16), (𝑥₂,0,17,7), (𝑥₃,0,3,1), (𝑥₄,0,5,12), (0,1,14,6), (0,4,10,7).

In this example, the entire set of 108 blocks are developed from the starter blocks by adding 𝑎∈𝑍₁₈ to each point of the starter blocks. To check the starter blocks, we need only to calculate whether the differences 𝑥_𝑖−𝑥_𝑖+1 from all pairs (𝑥_𝑖,𝑥_𝑖+1), 0≤𝑖≤3, in each of the starter blocks are precisely 𝐺⧵𝑆, where 𝑆 is the set of the differences of the holes. For the above example, the set of differences from the six blocks is exactly 𝑍₁₈⧵{0,8}. This is also true for the set of the differences 𝑥_𝑖−𝑥_𝑖+2.

The above idea of starter blocks can be generalized: instead of adding 𝑔∈𝐺 to each point of the starter blocks, we may have to add 𝑘𝑔, where 𝑘>1, to develop the block set; we refer to this as the +𝑘 method. In this case, for a set 𝒜 to be starter blocks, we require that for any 𝐚∈𝒜 and any 𝑔∈𝐺, 𝐚+𝑘𝑔∈ℬ. For quasigroups, we require that for all 𝑥,𝑦,𝑧∈𝑄,𝑥∗𝑦=𝑧 if and only if (𝑥+𝑘𝑔)∗(𝑦+𝑘𝑔)=(𝑧+𝑘𝑔) for any 𝑔∈𝐺 [17, 18].

Example 8. An HPMD(2⁶5¹) points: 𝑍₁₂∪{𝑥_𝑖∶1≤𝑖≤5} holes: {{𝑖,𝑖+6}∶0≤𝑖≤5}∪{{𝑥_𝑖∶1≤𝑖≤5}} starter blocks: (+4mod12): (𝑥₁,0,5,3), (𝑥₁,1,3,4), (𝑥₁,2,0,10), (𝑥₁,3,6,1), (𝑥₂,0,1,4), (𝑥₂,1,8,5), (𝑥₂,2,10,11), (𝑥₂,3,7,2), (𝑥₃,0,2,1), (𝑥₃,1,0,11), (𝑥₃,2,11,4), (𝑥₃,3,1,10), (𝑥₄,0,4,11), (𝑥₄,1,6,10), (𝑥₄,2,5,1), (𝑥₄,3,11,8), (𝑥₅,0,3,5), (𝑥₅,1,5,6), (𝑥₅,2,4,0), (𝑥₅,3,2,7).

By adding 4𝑔, where 𝑔∈𝑍₁₂, to the 20 starter blocks, we obtain a set of 60 blocks.

Next, we state several recursive constructions of HPMDs, which are commonly used in other block designs [8]. The following construction comes from the weighting construction of GDDs [6].

Construction 1 (Weighting). Suppose (𝑋,ℋ,ℬ) is a GDD with 𝜆=1 and let 𝑤∶𝑋↦𝑍⁺∪{0}. Suppose there exist HPMDs of type {𝑤(𝑥)∶𝑥∈𝐵} for every 𝐵∈ℬ. Then there exists an HPMD of type {∑_𝑥∈𝐻𝑤(𝑥)∶𝐻∈ℋ}.

Lemma 9. There exists an HPMD(2^𝑛𝑢¹) for each 𝑛≥6, 𝑛≡0(mod3) and 𝑢≡2(mod3) with 2≤𝑢≤𝑛−1, except for (𝑛,𝑢)=(6,5) and possibly excepting (𝑛,𝑢)∈{(21,17),(33,23),(33,29),(39,35),(57,44)}.

Proof. From Theorem 2, there exist 4-GDDs of the same type. So we can give all points of this GDD weight one to get the desired HPMD(2^𝑛𝑢¹) by Construction 1.

Using Theorem 3(a), if we give every point of an HPMD weight 𝑚 and input TD(4,𝑚) to each block of the HPMD, we can obtain the following construction.

Construction 2. Suppose there exists an HPMD of type (ℎ^𝑛1₁ℎ^𝑛2₂⋯ℎ^𝑛𝑘_𝑘), then there exists an HPMD((𝑙ℎ₁)^𝑛1(𝑙ℎ₂)^𝑛2⋯(𝑙ℎ_𝑘)^𝑛𝑘), where 𝑙≠2,6.

The next construction may be called “filling in holes,” which is used commonly in constructing designs.

Construction 3. Suppose there exist an HPMD of type {𝑠_𝑖∶1≤𝑖≤𝑘} and HPMDs of type {ℎ_𝑖𝑗∶1≤𝑗≤𝑛_𝑖}∪{𝑎}, where ∑^𝑛𝑖_𝑗=1ℎ_𝑖𝑗=𝑠_𝑖 and 1≤𝑖≤𝑘−1, then there exists an HPMD of type {ℎ_𝑖𝑗∶1≤𝑗≤𝑛_𝑖,1≤𝑖≤𝑘−1}∪{𝑠_𝑘+𝑎}.

Lemma 10. If there exists an HPMD(2^𝑚𝑘¹), there exists an HPMD(2^3𝑚(2𝑚+𝑘)¹).

Proof. By Theorem 4(a), there exists an HPMD((2𝑚)⁴) for any 𝑚≥4. We adjoin 𝑘 points to this HPMD((2𝑚)⁴) and fill three holes of size 2𝑚 with an HPMD(2^𝑚𝑘¹), leaving one hole of size 2𝑚+𝑘. The result is the desired HPMD(2^3𝑚(2𝑚+𝑘)¹).

Lemma 11. If there exists HPMD(2^𝑚𝑠¹), then there exists an HPMD(2^4𝑚𝑡¹) for 4𝑠+1≤𝑡≤4𝑠+3.

Proof. Applying Construction 2 with 𝑙=4 to the HPMD(2^𝑚𝑠¹), we have an HPMD of type 8^𝑚(4𝑠)¹. Add 𝑘 points, 1≤𝑘≤3, to this HPMD and fill an HPMD(2⁴𝑘¹), which exists by Theorem 4(d), into the holes of size 8, we have an HPMD(2^4𝑚(4𝑠+𝑘)¹), where 1≤𝑘≤3.

Lemma 12. If an HPMD(2^𝑚𝑠¹) exists, then an HPMD(2^5𝑚𝑡¹) exists for 5𝑠≤𝑡≤5𝑠+4.

Proof. Applying Construction 2 with 𝑙=5 to the HPMD(2^𝑚𝑠¹), we obtain an HPMD of type 10^𝑚(5𝑠)¹. To this HPMD we adjoin 𝑡 points, where 0≤𝑡≤4, and fill the holes of size 10 with an HPMD(2⁵𝑡¹), which exists by Theorem 4(d), we obtain an HPMD(2^5𝑚(5𝑠+𝑡)¹).

Lemma 13. If there exists a TD(5,𝑚), then there exists an HPMD((2𝑚)⁴𝑠¹), where 𝑚≤𝑠≤3𝑚.

Proof. Give weight 2 to each point of first four groups of a TD(5,𝑚). Give weight 1, 2 or 3 to the 𝑚 points of the fifth group so that the total is 𝑠. Since there exist HPMDs of type 2⁴𝑡¹, 𝑡=1,2,3, from Theorem 4(d), we obtain the desired HPMD by Construction 1.

Lemma 14. If there exists a TD(6,𝑚), then there exist(a)HPMD((2𝑚)⁴(2𝑘)¹𝑠¹), where 0≤𝑘≤𝑚 and 𝑚≤𝑠≤3𝑚,(b)HPMD(2^4𝑚+𝑘𝑠¹) for 𝑘=0,1,5,6,…,𝑚 and 𝑚≤𝑠≤3𝑚.

Proof. (a) We will use Construction 1: give weight 2 to each point of first four groups of a TD(6,𝑚); give weight 2 to the 𝑘 points of the fifth group and give weight 1, 2, or 3 to the 𝑚 points of the sixth group so that the total is 𝑠. Since there exist HPMDs of type 2^𝑛𝑡¹ for 𝑛=4,5 and 𝑡=1,2,3 from Theorem 4(d), we obtain the HPMD of type (2𝑚)⁴(2𝑘)¹𝑠¹.
(b) Take the HPMD((2𝑚)⁴(2𝑘)¹𝑠¹) from (a), where 𝑚≤𝑠≤3𝑚, we fill the holes of sizes 2𝑚 and 2𝑘 with an HPMD(2^𝑚) and an HPMD(2^𝑘), to obtain an HPMD(2^4𝑚+𝑘𝑠¹), where 𝑚≤𝑠≤3𝑚.

Lemma 15. If there exist a TD(6,𝑚), an HPMD(2^𝑚𝑡¹), and an HPMD(2^𝑘𝑡¹), where 4≤𝑘≤𝑚, then there exists an HPMD(2^4𝑚+𝑘𝑠¹) for 𝑚+𝑡≤𝑠≤3𝑚+𝑡.

Proof. Take the HPMD((2𝑚)⁴(2𝑘)¹𝑢¹) from Lemma 14(a), where 0≤𝑘≤𝑚 and 𝑚≤𝑢≤3𝑚, we first adjoin 𝑡 points to the HPMD((2𝑚)⁴(2𝑘)¹𝑢¹) and then fill the holes of sizes 2𝑚 and 2𝑘 with an HPMD(2^𝑚𝑡¹) and an HPMD(2^𝑘𝑡¹), we obtain an HPMD(2^4𝑚+𝑘(𝑢+𝑡)¹) where 𝑚+𝑡≤𝑢+𝑡≤3𝑚+𝑡.

Lemma 16 (see [8, Lemma 6.3]). If there exists a TD(7,𝑚), then there exist(a)HPMD((2𝑚)⁵(2𝑘)¹𝑠¹) for 0≤𝑘≤𝑚 and 0≤𝑠≤4𝑚,(b)HPMD(2^5𝑚+𝑘𝑠¹) for 𝑘=0,1,5,6,…,𝑚 and 0≤𝑠≤4𝑚.

Proof. (a) Give weight 2 to each points of five groups of the TD(7,𝑚), give weight 2 to 𝑘 points of the sixth group and weight 0 to the remainder points of this group, and give weight 𝑡, where 0≤𝑡≤4, to the points of the seventh group such that the sum of these weights is equal to 𝑢. As there exists an HPMD(2^𝑛𝑡¹) for 𝑛=5,6 and 0≤𝑡≤4, we obtain an HPMD((2𝑚)⁵(2𝑘)¹𝑢¹).
(b) Take the HPMD((2𝑚)⁵(2𝑘)¹𝑢¹) from (a), we obtain an HPMD(2^5𝑚+𝑘𝑢¹) by filling HPMDs of types 2^𝑚 and 2^𝑘 into the holes of sizes 2𝑚 and 2𝑘.

Lemma 17. If there exist a TD(7,𝑚), an HPMD(2^𝑚𝑡¹), and an HPMD(2^𝑘𝑡¹), where 4≤𝑘≤𝑚, then there exists HPMD(2^5𝑚+𝑘𝑠¹) for 𝑡≤𝑠≤4𝑚+𝑡.

Proof. We adjoin 𝑡 points to the HPMD((2𝑚)⁵(2𝑘)¹𝑢¹) from Lemma 16(a), where 0≤𝑢≤4𝑚 and fill the holes of sizes 2𝑚 and 2𝑘 with an HPMD(2^𝑚𝑡¹) and an HPMD(2^𝑘𝑡¹), we obtain an HPMD(2^5𝑚+𝑘(𝑢+𝑡)¹), where 𝑡≤𝑢+𝑡≤4𝑚+𝑡.

Lemma 18. An HPMD(12⁴𝑡¹) exists for 4≤𝑡≤16.

Proof. We start with a TD(5,4). In the first four groups of the TD, we give all the points weight 3. In the last group we give the points a weight of 1, 2, 3, or 4 for a total weight of 𝑡. We need HPMDs of type (3⁴𝑘¹), where 1≤𝑘≤4. For 𝑘=3, the design is given in Theorem 4(a); for 𝑘=1,2,4, they are given in Appendix J. The resulting design is an HPMD(12⁴𝑡¹) by Construction 1.

3. HPMD(2^𝑛𝑢¹) for Some Specific 𝑛

From the results in the previous section, we can obtain the following results.

Lemma 19. There exists an HPMD(2^𝑛1¹) for any 𝑛≥4.

Proof. For 4≤𝑛≤8, we have Theorem 4(d). For 9≤𝑛≤15 except 𝑛=14, please see Appendix A. For 𝑛=14,17,18,19, we can obtain the design from an HPMD(2^𝑛−49¹) given in Appendix G, by filling the hole of size 9 with an HPMD(2⁴1¹). For 𝑛=16, we take an HPMD(8⁴) from Theorem 4(a), adjoin one point to this HPMD, and fill each hole of size 8 with an HPMD(2⁴1¹).
For 𝑛≥20, let 𝑛=4𝑚+𝑡, where 𝑚≥5 and 0≤𝑡≤3. For 𝑡=0,1, we apply Construction 2 with 𝑙=4 on an HPMD(2^𝑚) (Theorem 4(a)) to get an HPMD(8^𝑚). Adjoin one point to this HPMD and fill the holes of size 8 with an HPMD(2⁴1¹), we obtain an HPMD(2^4𝑚1¹). If we adjoin 3 points to the HPMD(8^𝑚) and fill 𝑚−1 holes of size 8 with an HPMD(2⁴3¹) and the remaining hole with an HPMD(2⁵1¹), we obtain an HPMD(2^4𝑚+11¹).
For 𝑡=2,3, because an HPMD(2^𝑚−13¹) exists for any 𝑚≥5 by Theorem 4(c), we have an HPMD(8^𝑚−112¹) by Construction 2 with 𝑙=4. Adjoin 𝑘 points, where 𝑘=1,3, to this HPMD, and fill the holes of size 8 with an HPMD(2⁴𝑘¹), the hole of size 12 with an HPMD(2⁶1¹) if 𝑘=1 and an HPMD(2⁷1¹) if 𝑘=3, we obtain an HPMD(2^4𝑚+𝑡1¹), where 𝑡=2,3.

Combining Theorems 4(a) (with ℎ=2) and 4(c), and the above lemma, we have the following.

Lemma 20. An HPMD(2^𝑛𝑢¹) exists for all 𝑛≥4 and 1≤𝑢≤3.

Now we show the existence of some HPMD(2^𝑛𝑢¹) for 𝑛∈{17,18,19,21,22,23,26,27,29,31,37}.

Lemma 21. An HPMD(2¹⁷𝑢¹) exists for 1≤𝑢≤16.

Proof. For 1≤𝑢≤3, we apply Lemma 20. For 4≤𝑢≤8 please see Appendices B–F. For 9≤𝑢≤11, we first get an HPMD(8⁴10¹) by applying Lemma 13 with 𝑚=4 and 𝑠=10. Adjoin 𝑘 points, where 1≤𝑘≤3, to this HPMD, and fill three holes of size 8 with an HPMD(2⁴𝑘¹) and the hole of size 10 with an HPMD(2⁵𝑘¹), we obtain an HPMD(2¹²⁺⁵(8+𝑘)¹) for 1≤𝑘≤3. For 12≤𝑢≤16, please see Appendix I, except 𝑢=14, which is given in [8].

Lemma 22. An HPMD(2¹⁸𝑢¹) exists for 1≤𝑢≤17.

Proof. For 1≤𝑢≤3, we apply Lemma 20. For 𝑢=4,6,7, please see Appendices B, D, and E. For 𝑢=5,8, we apply Lemma 9.
For 9≤𝑢≤11, we first get an HPMD(8⁴12¹) by applying Lemma 13 with 𝑚=4,𝑠=12. Adjoin 𝑘 points, where 1≤𝑘≤3, to this HPMD, and fill three holes of size 8 with an HPMD(2⁴𝑘¹) and the hole of size 12 with an HPMD(2⁶𝑘¹), we obtain an HPMD(2¹²⁺⁶(8+𝑘)¹) for 1≤𝑘≤3.
For 12≤𝑢≤17, we obtain an HPMD(2^𝑛𝑢¹) by Lemma 10 with 𝑚=6, because an HPMD(2⁶𝑘¹) exists for 0≤𝑘≤4 by Theorem 4(d) and an HPMD(2⁶5¹) is given in Example 8.

Lemma 23. An HPMD(2¹⁹𝑢¹) exists for 1≤𝑢≤16.

Proof. For 1≤𝑢≤3, we apply Lemma 20. For 4≤𝑢≤9, please see Appendices B–G. For 11≤𝑢≤13, we first get an HPMD(10⁴8¹) by applying Lemma 13 with 𝑚=5, 𝑠=8. Adjoin 𝑘 points, where 1≤𝑘≤3, to the HPMD, and fill three holes of size 10 with an HPMD(2⁵𝑘¹) and the hole of size 8 with an HPMD(2⁴𝑘¹), we obtain an HPMD(2¹⁹(10+𝑘)¹) for 1≤𝑘≤3. For 𝑢=10 or 14≤𝑢≤16, please see Appendix I.

Lemma 24. An HPMD(2²¹𝑢¹) exists for 1≤𝑢≤20.

Proof. For 1≤𝑢≤3, we apply Lemma 20. For 𝑢=4, we first obtain an HPMD(8⁴12¹) from HPMD(2⁴3¹) by Construction 2 with 𝑙=4. Add two points to this HPMD and fill the holes of size 8 with an HPMD(2⁴2¹) and the hole of 12 with an HPMD(2⁵4¹), we obtain an HPMD(2²¹4¹).
For 5≤𝑢≤15, from Lemma 14(b) with 𝑚=5, 𝑘=1, we have HPMD(2^𝑛𝑢¹). For 16≤𝑢≤18, we apply Lemma 10 with 𝑚=7 and 𝑘=2,3,4. For 𝑢=19, please see Appendix I. For 𝑢=20, we apply Lemma 9.

Lemma 25. An HPMD(2²²𝑢¹) exists for 1≤𝑢≤16.

Proof. For 1≤𝑢≤3, we apply Lemma 20. For 𝑢=4, we take the HPMD(2¹⁷14¹) from Lemma 21 and fill the hole of size 17 with an HPMD(2⁵4¹). For 5≤𝑢≤8, please see Appendices C–F.
For 𝑢=9, we generate an HPMD(8⁵12¹) from an HPMD(2⁵3¹) (Theorem 4(c)) by Construction 2 with 𝑙=4. To this HPMD, we adjoin one point, fill four holes of size 8 by an HPMD(2⁴1¹) and one hole of size 12 by an HPMD(2⁶1¹), we obtain an HPMD(2²²(8+1)¹).
For 10≤𝑢≤14, we first get an HPMD(10⁴14¹) by applying Lemma 13 with 𝑚=5,𝑠=14. Adjoin 𝑘 points, where 0≤𝑘≤4, to this HPMD, and fill three holes of size 10 with an HPMD(2⁵𝑘¹) and the hole of size 14 with an HPMD(2⁷𝑘¹), we obtain an HPMD(2²²(10+𝑘)¹) for 0≤𝑘≤4.
For 𝑢=15, we take an HPMD(12⁴8¹) from Lemma 18. Adjoin three points to this HPMD and fill three holes of size 12 with an HPMD(2⁶3¹) and one hole of size 8 with an HPMD(2⁴3¹), we obtain an HPMD(2²²(12+3)¹).
Finally for 𝑢=16, please see Appendix I.

Lemma 26. An HPMD(2²³𝑢¹) exists for 1≤𝑢≤17.

Proof. For 1≤𝑢≤3, we apply Lemma 20. For 𝑢=4, we add 2 points to an HPMD(12⁴) and fill in three holes with an HPMD(2⁶2¹) and one hole with an HPMD(2⁵4¹), to obtain an HPMD(2²³4¹). For 5≤𝑢≤8, please see Appendices C–F.
For 9≤𝑢≤11, we take the HPMD(8⁵14¹) from Appendix J, adjoin 𝑘 points, 1≤𝑘≤3, to the HPMD, and fill four holes of size 8 with an HPMD(2⁴𝑘¹) and one hole of size 14 with an HPMD(2⁷𝑘¹), we obtain an HPMD(2²³(8+𝑘)¹).
For 12≤𝑢≤16, we take an HPMD(12⁴10¹) from Lemma 18. Adjoin 𝑘 points, where 0≤𝑘≤4, to this HPMD, and fill three holes of size 12 with an HPMD(2⁶𝑘¹) and the hole of size 10 with an HPMD(2⁵𝑘¹) (Theorem 4(d)), we obtain an HPMD(2²³(12+𝑘)¹) for 0≤𝑘≤4.
For 𝑢=17, please see Appendix I.

Lemma 27. An HPMD(2²⁶𝑢¹) exists for 0≤𝑢≤18.

Proof. For 0≤𝑢≤14, we first apply Lemma 14(a) with 𝑚=5 and 𝑠=12 to get an HPMD(10⁴(2𝑘)¹12¹), where 0≤𝑘≤5. Adjoin 𝑡 points, where 0≤𝑡≤4, to this HPMD, and fill the holes of sizes 10 and 12 with an HPMD(2⁵𝑡¹) and an HPMD(2⁶𝑡¹), respectively. The result is an HPMD(2²⁶(2𝑘+𝑡)¹), where 0≤2𝑘+𝑡≤14.
For 14≤𝑢≤18, we first get an HPMD(14⁴10¹) by applying Lemma 13 with 𝑚=7, 𝑠=10. Adjoin 𝑘 points, where 0≤𝑘≤4, to this HPMD, and fill three holes of size 14 with an HPMD(2⁷𝑘¹) and the hole of size 10 with an HPMD(2⁵𝑘¹), we obtain an HPMD(2²⁶(14+𝑘)¹) for 0≤𝑘≤4.

Lemma 28. An HPMD(2²⁷𝑢¹) exists for 0≤𝑢≤26.

Proof. For 0≤𝑢≤14, we first apply Lemma 14(a) with 𝑚=5 and 𝑠=14 to get an HPMD(10⁴(2𝑘)¹14¹), where 0≤𝑘≤5, then adjoin 𝑡 points, where 0≤𝑡≤4, to this HPMD, and fill the holes of sizes 10 and 14 with an HPMD(2⁵𝑡¹) and an HPMD(2⁷𝑡¹), respectively. The result is an HPMD(2²⁷(2𝑘+𝑡)¹), where 0≤2𝑘+𝑡≤14.
For 14≤𝑢≤18, we first get an HPMD(14⁴12¹) by applying Lemma 13 with 𝑚=7, 𝑠=12. Adjoin 𝑘 points, where 0≤𝑘≤4, to this HPMD, and fill three holes of size 14 with an HPMD(2⁷𝑘¹) and the hole of size 12 with an HPMD(2⁶𝑘¹), we obtain an HPMD(2²⁷(14+𝑘)¹) for 0≤𝑘≤4.
For 18≤𝑢≤26, we start with an HPMD(18⁴), adjoin 𝑘 points to this HPMD, and fill in the first three holes of size 18, where 0≤𝑘≤8, with an HPMD(2⁹𝑘¹). The resulting design is an HPMD of type 2²⁷(18+𝑘)¹, where 18≤18+𝑘≤26, and this completes the proof of the lemma.

Lemma 29. An HPMD(2²⁹𝑢¹) exists for 1≤𝑢≤21.

Proof. For 𝑢=1,2,3, we apply Lemma 20. For 𝑢=4, we take an HPMD(12⁴14¹) from Lemma 18 and fill the holes of sizes 12 and 14 with an HPMD(2⁶) and an HPMD(2⁵4¹), respectively. For 𝑢=5, we take an HPMD(2²³17¹) from Lemma 26 and fill the hole of size 17 with an HPMD(2⁶5¹). For 𝑢=6, please see Appendix D. For 7≤𝑢≤21, we can get the designs from Lemma 14(b) with 𝑚=7 and 𝑘=1.

Lemma 30. An HPMD(2³¹𝑢¹) exists for 1≤𝑢≤21.

Proof. For 1≤𝑢≤5, we can get an HPMD(2²⁵(12+𝑢)¹) from Lemma 12 with 𝑛=5, 𝑠=2 and 𝑘=2,3,4, and 𝑠=3 and 𝑘=1,2. Fill the hole of size (12+𝑢) with an HPMD(2⁶𝑢¹), we have an HPMD(2³¹𝑢¹).
For 6≤𝑢≤18, we form a {6,7}-GDD of type 6⁷ by deleting one block from a TD(7,7). In the first five groups of this GDD, we give all of the points weight 2. In the fifth group, we give one point of weight 2 and the other points weight 0. In the last group, we give the points a weight of 1, 2, or 3 for a total weight of 𝑢 where 6≤𝑢≤18. Since there are HPMDs of types 2^𝑛 for 𝑛=5,6,7 and 2^𝑛𝑘¹ for 𝑛=4,5,6 and 𝑘=1,2,3 by Theorem 4(c), we get an HPMD of type 12⁵2¹𝑢¹ for 6≤𝑢≤18. To this HPMD, we fill the holes of size 12 with an HPMD(2⁶) and obtain an HPMD(2³¹𝑢¹) for 6≤𝑢≤18.
For 7≤𝑢≤21, we use a TD(8,7): in the first four groups of the TD(8,7), we give all of the points a weight of two. In the fifth, sixth, and seventh groups, we give one point weight two and the other points weight zero. In the last group, we give the points a weight of 1, 2, or 3, for a total weight of 𝑢. Since we have HPMDs of types 2^𝑛𝑘¹ for 𝑛=4,5,6,7 and 𝑘=1,2,3, we get an HPMD of type 14⁴2³𝑢¹ for 7≤𝑢≤21. By filling in the holes of size 14 with an HPMD(2⁷), the resulting design is an HPMD(2³¹𝑢¹) for 7≤𝑢≤21.

Lemma 31. An HPMD(2³⁷𝑢¹) exists for 1≤𝑢≤28.

Proof. We will use a TD(8,7): in the first five groups of the TD(8,7), we give all of the points a weight of two. In the sixth and seventh groups, we give one point weight two and the other points weight zero. In the last group, we give the points a weight of 0, 1, 2, 3, or 4, for a total weight of 𝑢. Since we have HPMDs of types 2^𝑛𝑘¹ for 𝑛=5,6,7 and 0≤𝑘≤4, we get an HPMD of type 14⁵2²𝑢¹ for 0≤𝑢≤28. By filling in the holes of size 14 with an HPMD(2⁷), the resulting design is an HPMD(2³⁷𝑢¹) for 1≤𝑢≤28.

4. HPMD(2^𝑛𝑢¹) for 1≤𝑢≤16

Lemma 32. An HPMD(2^𝑛𝑢¹) exists for 1≤𝑢≤16 and 𝑢+1≤𝑛≤25, except possibly (𝑛,𝑢)∈{(7,5),(7,6),(11,9),(11,10)}.

Proof. For 1≤𝑢≤3, we apply Lemma 20.
For 4≤𝑢≤16, let us consider 𝑢≤𝑛≤15 first. For 5≤𝑛≤8 and 𝑢=4, an HPMD(2^𝑛𝑢¹) exists by Theorem 4(d). For 𝑛=9,12,15 and 𝑢=5,8,11,14, we apply Lemma 9 when 𝑢<𝑛. For 𝑛=12 and 9≤𝑢≤11, we obtain an HPMD(2^𝑛𝑢¹) by Lemma 10 with 𝑚=4, because an HPMD(2⁴𝑘¹) exists for 𝑘=1,2,3. For 𝑛=15 and 10≤𝑢≤14, we obtain an HPMD(2^𝑛𝑢¹) by Lemma 10 with 𝑚=5, because an HPMD(2⁵𝑘¹) exists for 𝑘=0,1,2,3,4. Besides an HPMD(2⁶5¹) in Example 8 and an HPMD(2⁹4¹) in Example 7, an HPMD(2^𝑛11¹) for 𝑛=13,14 can be found in [8]. The other designs for 𝑢≤𝑛≤15 are given in Appendices B–H, except (𝑛,𝑢)∈{(7,5),(7,6),(11,9),(11,10)}.
Now let us consider 4≤𝑢≤16 and 16≤𝑛≤25. For 𝑛=16,24, 4≤𝑢≤15, and 𝑢≠4,8,12, by Lemma 11, we have an HPMD(2^𝑛𝑢¹). For 𝑛=16 and 𝑢=4,8,12, let 𝑡=𝑢−2, we obtain first an HPMD(8⁴𝑡¹) by Lemma 13 with 𝑚=4 and 𝑠=𝑡 when 𝑡≥4; for 𝑡=2, we have an HPMD(8⁴𝑡¹) in Appendix J. To the HPMD(8⁴𝑡¹), we adjoin 2 points and fill the holes of size 8 with an HPMD(2⁴2¹), to obtain an HPMD(2¹⁶𝑢¹) for 𝑢=4,8,12. For 𝑛=24 and 𝑢=4, we adjoin 4 points to an HPMD(12⁴) and fill the holes of size 12 with an HPMD(2⁶4¹). For 𝑛=24 and 𝑢=8,12,16, we get them from Lemma 15 with 𝑚=5,𝑘=4 and 1≤𝑡≤3. The cases of 𝑛=17,18,19,21,22,23 are covered by Lemmas 21, 22, 23, 24, 25, and 26, respectively. For 𝑛=20,25 and 4≤𝑢≤19, we can get an HPMD(2^𝑛𝑢¹) from Lemma 12 with 𝑚=4,5, 1≤𝑠≤3 and 𝑡=𝑢.

Lemma 33. An HPMD(2^𝑛𝑢¹) exists for 1≤𝑢≤18 and 26≤𝑛≤32.

Proof. For 𝑛=26,27,29, and 31, we have Lemmas 27, 28, 29, and 30 to cover these cases, respectively.
For 𝑛=28,32 and 𝑢≠8,12,16, the designs are provided by Lemma 11 with 𝑚=7,8. For 𝑛=28,32 and 𝑢=8,12,16, we get these designs from Lemma 14(b) with 𝑚=7,8 and 𝑘=0.
For 𝑛=30, we apply Lemma 12 with 𝑚=6, 1≤𝑠≤4, and 𝑡=𝑢.

Lemma 34. An HPMD(2^𝑛𝑢¹) exists for 1≤𝑢≤24 and 33≤𝑛≤40.

Proof. For 𝑛=33,34 and 1≤𝑢≤6, we obtain at first an HPMD(2²⁵𝑡¹) by Lemma 12 with 𝑚=5, 𝑠=4, and 𝑡=𝑢+16, 𝑢+18. Fill an HPMD(2⁸𝑢¹) and an HPMD(2⁹𝑢¹) into the holes of sizes 𝑢+16 and 𝑢+18, we obtain an HPMD(2³³𝑢¹) and an HPMD(2³⁴𝑢¹), respectively. For 𝑛=33,34 and 7≤𝑢≤24, an HPMD(2^𝑛𝑢¹) exists by Lemma 15 with 𝑚=7, k=5,6, 1≤𝑡≤3.
For 𝑛=35,36,40, the designs come from Lemma 16(b) with 𝑚=7,𝑘=0,1,5. For 𝑛=37, the designs are provided by Lemma 31.
For 𝑛=38 and 1≤𝑢≤7, we first apply Lemma 12 with 𝑚=6, 𝑠=4, and 𝑡=𝑢+16, to get an HPMD(2³⁰(𝑢+16)¹). Fill the hole of size 𝑢+16 by an HPMD(2⁸𝑢¹), we obtain an HPMD(2³⁸𝑢¹). For 𝑛=38 and 8≤𝑢≤24, we have an HPMD(2^𝑛𝑢¹) by Lemma 14(b) with 𝑚=8 and 𝑘=6. Finally for 𝑛=39, we apply Lemma 17 with 𝑚=7, 𝑘=4, 1≤𝑡≤3.

Lemma 35. An HPMD(2^𝑛𝑢¹) exists for 1≤𝑢≤30 and 41≤𝑛≤49.

Proof. For 𝑛=41,42,44 and 1≤𝑢≤31, an HPMD(2^𝑛𝑢¹) exists by Lemma 17 with 𝑚=7 and 𝑘=6,7, and 𝑚=8 and 𝑘=4, and 1≤𝑡≤3.
For 𝑛=43 and 1≤𝑢≤7, there exists an HPMD(2³⁵(𝑢+16)¹) by Lemma 12 with 𝑚=7, 0≤𝑠≤6, and 𝑡=𝑢+16. Fill an HPMD(2⁸𝑢¹) into the hole of size 𝑢+16, we obtain an HPMD(2⁴³𝑢¹). For 𝑛=43 and 𝑢=8, because we have an HPMD(2⁸7¹), by Construction 2 with 𝑙=4, we have an HPMD(8⁸28¹). Add two points to this design and fill an HPMD(2⁴2¹) into the holes of size 8 and an HPMD(2¹¹8¹) into the hole of size 28, we obtain an HPMD(2⁴³8¹). For 𝑛=43 and 𝑢=9, an HPMD(2^𝑛𝑢¹) exists by Lemma 14(b) with 𝑚=9 and 𝑘=7. For 𝑛=43 and 10≤𝑢≤30, an HPMD(2^𝑛𝑢¹) exists by Lemma 15 with 𝑚=9, 𝑘=7, and 1≤𝑘≤3.
For 𝑛=45,46,47,48 and 1≤𝑢≤35, the designs come from Lemma 17 with 𝑚=8, 𝑘=5,6,7,8 and 1≤𝑡≤3.
For 𝑛=49 and 0≤𝑢≤32, because an HPMD(2⁷𝑡¹) exists for 0≤𝑡≤4, we get an HPMD(14⁷(7𝑡)¹) from Construction 2 with 𝑙=7. Add 𝑘 points to this design, 0≤𝑘≤4, and fill an HPMD(2⁷𝑘¹) into the holes of size 14, we obtain an HPMD(2⁴⁹(7𝑡+𝑘)¹) for 0≤7𝑡+𝑘≤32.

Lemma 36. An HPMD(2^𝑛𝑢¹) exists for 1≤𝑢≤39 and 50≤𝑛≤66.

Proof. For 50≤𝑛≤54 and 1≤𝑢≤39, we apply Lemma 17 with 𝑚=9, 5≤𝑘≤9 and 1≤𝑡≤3. For 𝑛=55,56 and 1≤𝑢≤44, we apply Lemma 16(b) with 𝑚=11 and 𝑘=0,1.
For 𝑛=57,58 and 1≤𝑢≤11, by Lemma 16(b) with 𝑚=9 and 𝑘=0,1, there exist HPMD(2^𝑠(𝑢+24)¹) for 𝑠=45,46 and 1≤𝑢≤11. Fill in the hole of size 𝑢+24 with an HPMD(2¹²𝑢¹), we obtain an HPMD(2^𝑠+12𝑢¹) for 𝑠=45,46. For 𝑛=57,58 and 12≤𝑢≤39, an HPMD(2^𝑛𝑢¹) exists from Lemma 15 with 𝑚=12 and 𝑘=9,10.
For 59≤𝑛≤66 and 1≤𝑢≤47, an HPMD(2^𝑛𝑢¹) exists by Lemma 17 with 𝑚=11, 4≤𝑘≤11, and 1≤𝑡≤3.

Lemma 37. An HPMD(2^𝑛𝑢¹) exists for 1≤𝑢≤20 and 𝑛∈{21,25,29,33,37,45,59}.

Proof. The cases of 𝑛=21,29,37 are covered by Lemmas 24, 29, and 31, respectively. The cases of 𝑛=33,45,59 are covered by Lemmas 34, 35, and 36, respectively. For 𝑛=25, we apply Lemma 12 with 𝑚=5 and 1≤𝑠≤4 because HPMDs of type 2⁵𝑠¹ exist for all 1≤𝑠≤4.

Lemma 38. An HPMD(2^𝑛𝑢¹) exists for 1≤𝑢≤51 and 67≤𝑛≤87.

Proof. For 67≤𝑛≤78 and 1≤𝑢≤51, an HPMD(2^𝑛𝑢¹) exists by by Lemma 17 with 𝑚=12 and 7≤𝑘≤12, and 𝑚=13 and 8≤𝑘≤13, and 1≤𝑡≤3.
For 𝑛∈{79,82,83} and 1≤𝑢≤15, by Lemma 17 with 𝑚=11 and 𝑘=8, and 𝑚=12 and 𝑘=6,7, and 1≤𝑡≤3. there exist HPMD(2^𝑠(𝑢+32)¹) for 𝑠=63,66,67. Fill in the hole of size 𝑢+32 with HPMD(2¹⁶𝑢¹) in HPMD(2^𝑠(𝑢+32)¹), we obtain an HPMD(2^𝑛𝑢¹) for 𝑛∈{79,82,83}. For 𝑛=79 and 16≤𝑢≤67, we apply Lemma 15 with 𝑚=16 and 𝑘=15, and 1≤𝑘≤3. For 𝑛=82,83 and 𝑢=16, we apply Lemma 14(a) with 𝑚=18, 𝑘=8 and 𝑠=2𝑡, where 𝑡=10,11, to obtain an HPMD(36⁴16¹(2t)¹). Fill the holes of sizes 36 and 2𝑡 with an HPMD of types 2¹⁸, 2^𝑡, respectively, we obtain an HPMD(2^72+𝑡16¹) for 𝑡=10,11. For 𝑛=82,83 and 17≤𝑢≤54, we apply Lemma 15 with 𝑚=17, 𝑘=14,15, 1≤𝑡≤3.
For 80≤𝑛≤81 and 1≤𝑢≤64, we apply Lemma 16(b) with 𝑚=16 and 𝑘=0,1. Finally for 84≤𝑛≤87 and 1≤𝑢≤67, we apply Lemma 17 with 𝑚=16, 4≤𝑘≤7, and 1≤𝑡≤3.

The required HPMDs of type 2^𝑘𝑡¹ for the application of Lemmas 15 and 17 in the above proof as well as in the proofs of the next two lemmas come from either Lemma 32 or Lemma 37.

Lemma 39. An HPMD(2^𝑛𝑢¹) exists for 88≤𝑛≤116 and 1≤𝑢≤⌊4𝑛/5⌋.

Proof. Let 𝜇(𝑛)=⌊4𝑛/5⌋.
For 88≤𝑛≤92, 𝜇(92)=73, let 𝑛=5𝑠+𝑘, where 𝑠=16 and 8≤𝑘≤12. Because an TD(7,16) exists, using Lemma 17 with 𝑚=16, 8≤𝑘≤12, 1≤𝑡≤𝑘−3, we obtain an HPMD(2^𝑛𝑢¹) for 1≤𝑢≤𝑘+61 and 8≤𝑘≤12.
For 93≤𝑛≤102, 𝜇(102)=81, let 𝑛=5𝑠+𝑘, where 𝑠=17 and 8≤𝑘≤17. using Lemma 17 with 𝑚=17, 8≤𝑘≤17, 1≤𝑡≤𝑘−3, we obtain an HPMD(2^𝑛𝑢¹) for 1≤𝑢≤𝑘+65 and 8≤𝑘≤17.
For 103≤𝑛≤114, 𝜇(114)=91, let 𝑛=5𝑠+𝑘, where 𝑠=19 and 8≤𝑘≤19. using Lemma 17 with 𝑚=19, 8≤𝑘≤19, 1≤𝑡≤𝑘−3, we obtain an HPMD(2^𝑛𝑢¹) for 1≤𝑢≤𝑘+73 and 8≤𝑘≤19.
For 115≤𝑛≤116, 𝜇(116)=92, we apply Lemma 16(b) with 𝑚=23 and 𝑘=0,1 to obtain an HPMD(2^𝑛𝑢¹) for 1≤𝑢≤92.

Lemma 40. An HPMD(2^𝑛𝑢¹) exists for 𝑛≥117 and 1≤𝑢≤⌊4𝑛/5⌋+2.

Proof. Let 𝑛=5𝑠+𝑗, where 12≤𝑗≤16. Since 𝑛≥117, we have 𝑠≥21. If there exists a TD(7,𝑠), then using Lemma 17 with 𝑚=𝑠, 𝑘=𝑗, 1≤𝑡≤𝑘−1, we obtain an HPMD(2^𝑛𝑢¹) for 1≤𝑢≤4𝑠+𝑘−1. Because 𝑘=𝑗=𝑛−5𝑠 and 𝑠≥(𝑛−16)/5, we have 𝑢≤⌊4𝑛/5⌋+2.
For 𝑠∈𝑀₇={22,26,30,34,38,46,60}, we do not have a TD(7,𝑠). However, since a TD(7,𝑠−1) exists, we may use Lemma 17 with 𝑚=𝑠−1,17≤k≤21, 1≤𝑡≤𝑘−1, where 𝑘=𝑗+5, to obtain an HPMD(2^𝑛𝑢¹) for 1≤𝑢≤4(𝑠−1)+𝑘−1=4𝑠+𝑗. Because 𝑗=𝑛−5𝑠 and 𝑠≥(𝑛−21)/5, we have 𝑢≤⌊4𝑛/5⌋+3. The required HPMDs of type 2^𝑘𝑡¹ in the above proof come from Lemma 37.

As a summary of Lemma 5, Theorem 4(c), and Lemmas 19, 32–40, we have the following result.

Theorem 41. For 1≤𝑢≤16, an HPMD(2^𝑛𝑢¹) exists if and only if 𝑛≥4 and 𝑛≥𝑢+1, except possibly (𝑛,𝑢)∈{(7,5),(7,6),(11,9),(11,10)}.

5. HPMD(2^𝑛𝑢¹) for General 𝑢

In this section, we will provide a general result regarding the existence of HPMD(2^𝑛𝑢¹). That is, we will show the existence of HPMD(2^𝑛𝑢¹) for any 𝑢 and all 𝑛≥⌈5𝑢/4⌉+4. Let us start with a lemma.

Lemma 42. An HPMD(2⁷⁹𝑢¹) exists for 1≤𝑢≤62.

Proof. For 𝑢≤16, the designs are provided by Theorem 41. For 17≤𝑢≤62, we obtain first an HPMD(32⁴30¹𝑡¹) from Lemma 14(a) with 𝑚=16, 𝑘=15, and 𝑠=𝑡, where 17≤𝑡≤48. Adjoin 𝑘 points to this HPMD, where 0≤𝑘≤14, and fill the holes of sizes 32 and 30 with HPMDs of types 2¹⁶𝑘¹, 2¹⁵𝑘¹, respectively, we obtain an HPMD(2⁷⁹(𝑡+𝑘)¹) for 16≤𝑡+𝑘≤62.

In the proofs of the remaining lemmas in this section, the required HPMDs of type 2^𝑘𝑡¹ for the application of Lemmas 15 and 17 come from Theorem 41.

Lemma 43. There exists an HPMD(2^𝑛𝑢¹) for all 𝑢≥1 and ⌈5𝑢/4⌉+4≤𝑛≤44.

Proof. For 1≤𝑢≤16, Theorem 41 applies. Now we assume 𝑢≥17, it must be the case that 𝑛≥26 because 𝑛≥⌈5𝑢/4⌉+4. For the same reason, we have 𝑢≤⌊4𝑛/5⌋−3. Table 1 shows the existence of an HPMD(2^𝑛𝑢¹) for 26≤𝑛≤44 and 17≤𝑢≤⌊4𝑛/5⌋−3.

Table 1 

Existence proof of HPMD (2^𝑛𝑢¹) for Lemma 43.

Lemma 44. An HPMD(2^𝑛𝑢¹) exists for 45≤𝑛≤87 and 1≤𝑢≤⌊4𝑛/5⌋−1.

Proof. For 1≤𝑢≤16, Theorem 41 applies. Let 𝜇(𝑛)=⌊4𝑛/5⌋−1.
For 45≤𝑛≤48, 𝜇(48)=37, we apply Lemma 17 with 𝑚=8, 5≤𝑘≤8, and 1≤𝑡≤𝑘−3 to obtain an HPMD(2^𝑘+40𝑢¹) where 1≤𝑢≤𝑘+29 and 5≤𝑘≤8.
For 49≤𝑛≤54, 𝜇(54)=42 and we use Lemma 17 with 𝑚=9, 4≤𝑘≤9, and 1≤𝑡≤𝑘−3 to obtain an HPMD(2^𝑘+45𝑢¹) where 1≤𝑢≤𝑘+33 and 4≤𝑘≤9.
For 55≤𝑛≤56, 𝜇(56)=43 and we apply Lemma 16(b) with 𝑚=11 and 𝑘=0,1 to obtain an HPMD(2^𝑛𝑢¹) where 0≤𝑢≤44.
For 𝑛=57,58, 𝜇(58)=45 and we apply Lemma 15 with 𝑚=12, 𝑘=9,10, and 0≤𝑡≤𝑘−1, to obtain an HPMD(2^48+𝑘𝑢¹), where 𝑘=9,10 and 12≤𝑢≤35+𝑘.
For 59≤𝑛≤66, 𝜇(66)=51 and an HPMD(2^𝑛𝑢¹) exists by Lemma 17 with 𝑚=11, 4≤𝑘≤11 and 1≤𝑡≤𝑘−3. to obtain an HPMD(2^𝑘+55𝑢¹) where 4≤𝑘≤11 and 1≤𝑢≤𝑘+41.
For 67≤𝑛≤72, 𝜇(72)=56 and an HPMD(2^𝑛𝑢¹) exists by Lemma 17 with 𝑚=12, 7≤𝑘≤12 and 1≤𝑡≤𝑘−3, to obtain an HPMD(2^𝑘+60𝑢¹) where 7≤𝑘≤12 and 1≤𝑢≤𝑘+45.
For 73≤𝑛≤78, 𝜇(78)=61 and an HPMD(2^𝑛𝑢¹) exists by Lemma 17 with 𝑚=13, 8≤𝑘≤13 and 1≤𝑡≤𝑘−3, to obtain an HPMD(2^𝑘+65𝑢¹) where 8≤𝑘≤13 and 1≤𝑢≤𝑘+49.
For 𝑛=79, the case is covered by Lemma 42. For 𝑛=80,81, 𝜇(81)=63, we apply Lemma 16(b) with 𝑚=16 and 𝑘=0,1, to obtain an HPMD(2^𝑛𝑢¹) where 0≤𝑢≤64.
For 𝑛=82,83, 𝜇(83)=65 and we apply Lemma 15 with 𝑚=17, 𝑘=14,15, 1≤𝑡≤𝑘−1, to obtain an HPMD(2^68+𝑘𝑢¹) where 𝑘=14,15 and 17≤𝑢≤𝑘+50.
For 84≤𝑛≤87, 𝜇(87)=68 and we apply Lemma 17 with 𝑚=16, 4≤𝑘≤7, 1≤𝑡≤𝑘−3, to obtain an HPMD(2^𝑘+80𝑢¹) where 4≤𝑘≤7, 1≤𝑢≤𝑘+61.

Theorem 45. There exists an HPMD(2^𝑛𝑢¹) for 𝑢≥1 and 𝑛≥⌈5𝑢/4⌉+4.

Proof. For 1≤𝑢≤16, Theorem 41 applies. Now assume that 𝑢≥17, it must be the case that 𝑛≥26 because 𝑛≥⌈5𝑢/4⌉+4. For 26≤𝑛≤44, the theorem holds by Lemma 43. For the case of 𝑛≥45, the theorem holds by Lemmas 44, 39, and 40, because 𝑢≤⌊4𝑛/5⌋−3 implies 𝑛≥⌈5𝑢/4⌉+4.

Actually, Lemmas 44, 39, and 40 can be used to prove a stronger result when 𝑢≥35.

Theorem 46. There exists an HPMD(2^𝑛𝑢¹) if 𝑢≥35 and 𝑛≥⌈5𝑢/4⌉+1 or 𝑢≥95 and 𝑛≥⌈5𝑢/4⌉−2.

Proof. When 𝑢≥35, 𝑛≥⌈5𝑢/4⌉+1≥45, so Lemmas 44, 39, and 40 apply. From 𝑢≤⌊4𝑛/5⌋−1, we obtain 𝑛≥⌈5𝑢/4⌉+1. When 𝑢≥95, 𝑛≥⌈5𝑢/4⌉−2≥117, so Lemma 40 applies. From 𝑢≤⌊4𝑛/5⌋+2, we obtain 𝑛≥⌈5𝑢/4⌉−2.

6. Conclusions

We have investigated the existence of HPMD(2^𝑛𝑢¹) for 1≤𝑢≤16. We also provided a general result for all 𝑢≥1 and 𝑛≥⌈5𝑢/4⌉+4. Most recursive constructions used in this paper are standard in combinatorial designs and many of the direct constructions of HPMDs in this paper are carried out by computer. From the previous sections, we obtain the main theorem of this paper while the existence problem of HPMD(2^𝑛𝑢¹) for 17≤𝑢≤94 and 𝑢+1≤𝑛≤⌈5𝑢/4⌉+3, and 𝑢≥95 and 𝑢+1≤𝑛≤⌈5𝑢/4⌉−3 remains open. The main result of this paper can be summarized in the following theorem.

Theorem 47. (a) For 1≤𝑢≤16, an HPMD(2^𝑛𝑢¹) exists if and only if 𝑛≥4 and 𝑛≥𝑢+1, with the possible exceptions of (𝑛,𝑢)∈{(7,5),(7,6),(11,9),(11,10)}.
(b) For 𝑢≥1, an HPMD(2^𝑛𝑢¹) exists if 𝑛≥⌈5𝑢/4⌉+4. Moreover, if 𝑢≥35, then there exists an HPMD(2^𝑛𝑢¹) for 𝑛≥⌈5𝑢/4⌉+1; if 𝑢≥95, then there exists an HPMD(2^𝑛𝑢¹) for 𝑛≥⌈5𝑢/4⌉−2.

Proof. (a) is a restatement of Theorem 41; (b) is a combination of Theorems 45 and 46.

Appendices

Here we list some HPMDs which are used in the previous sections. Most of them are obtained by computer. In the following list, the point set of an HPMD(2^𝑛𝑢¹) consists of 𝑍_2𝑛 and 𝑢 infinite points which are denoted by alphabet. For simplicity, we only list the starter blocks. We also use the +𝑘 method to develop blocks.

A. HPMD(2^𝑛1¹) for 9≤𝑛≤15

𝑛=9(+2mod18): (0,5,6,𝑥₁), (1,12,13,𝑥₁), (0,2,10,4), (0,3,11,5), (0,4,15,11), (0,6,4,1), (0,7,5,8), (0,13,1,3), (0,17,3,13).

𝑛=10(+2mod20): (0,15,8,𝑥₁), (1,4,13,𝑥₁), (0,1,6,2), (0,2,16,3), (0,3,11,5), (0,4,17,19), (0,5,9,1), (0,6,18,9), (0,8,7,13), (0,17,13,11).

𝑛=11(+2mod22): (0,20,16,𝑥₁), (1,11,7,𝑥₁), (0,1,4,6), (0,3,19,17), (0,4,13,14), (0,5,14,7), (0,6,20,5), (0,10,1,3), (0,12,7,15), (0,19,9,1), (0,21,5,9).

𝑛=12(+2mod24): (0,2,16,𝑥₁), (1,3,9,𝑥₁), (0,1,5,8), (0,3,22,2), (0,5,13,6), (0,6,14,9), (0,7,4,13), (0,10,21,19), (0,13,7,17), (0,15,11,1), (0,17,18,15), (0,20,19,11).

𝑛=13(+2mod26): (0,9,4,𝑥₁), (1,8,23,𝑥₁), (0,1,21,22), (0,2,9,8), (0,3,12,11), (0,5,15,12), (0,6,25,23), (0,8,24,9), (0,10,7,21), (0,12,8,2), (0,17,3,7), (0,21,23,15), (1,7,15,5).

𝑛=15(+2mod30): (0,24,2,𝑥₁), (1,23,29,𝑥₁), (0,1,4,23), (0,2,6,9), (0,5,13,2), (0,6,18,29), (0,7,17,10), (1,3,17,13), (0,9,20,13), (0,10,1,5), (0,13,22,14), (0,14,9,25), (0,17,16,12), (0,27,21,3), (0,29,19,17).

B. HPMD(2^𝑛4¹) for 𝑛≤19

𝑛=10(+2mod20): (0,17,4,𝑥₁), (1,14,19,𝑥₁), (0,12,19,𝑥₂), (1,19,18,𝑥₂), (0,19,1,𝑥₃), (1,6,14,𝑥₃), (0,1,2,𝑥₄), (1,12,7,𝑥₄), (0,2,13,16), (0,3,17,14), (0,9,15,4), (0,13,8,2), (1,5,17,13).

𝑛=11(+1mod22): (0,18,17,𝑥₁), (0,14,2,𝑥₂), (0,2,18,𝑥₃), (0,4,7,𝑥₄), (0,1,6,15), (0,6,19,5), (0,12,10,3).

𝑛=12(+2mod24): (0,11,18,𝑥₁), (1,12,7,𝑥₁), (0,4,9,𝑥₂), (1,23,0,𝑥₂), (0,17,16,𝑥₃), (1,18,15,𝑥₃), (0,16,15,𝑥₄), (1,11,20,𝑥₄), (0,1,3,9), (0,2,22,16), (0,3,19,14), (0,6,20,3), (0,9,23,2), (0,13,17,11), (0,15,11,19).

𝑛=13(+1mod26): (0,18,7,𝑥₁), (0,9,6,𝑥₂), (0,8,2,𝑥₃), (0,12,11,𝑥₄), (0,1,8,24), (0,3,9,7), (0,4,14,9), (0,5,1,15).

𝑛=14(+2mod28): (0,3,1,𝑥₁), (1,18,22,𝑥₁), (0,18,19,𝑥₂), (1,19,26,𝑥₂), (0,21,12,𝑥₃), (1,2,27,𝑥₃), (0,27,26,𝑥₄), (1,16,11,𝑥₄), (0,2,13,22), (0,5,21,6), (0,7,18,2), (0,8,24,5), (0,10,6,25), (0,13,23,7), (0,15,17,23), (0,17,11,8), (1,5,25,21).

𝑛=15(+1mod30): (0,8,1,𝑥₁), (0,18,9,𝑥₂), (0,4,28,𝑥₃), (0,2,22,𝑥₄), (0,1,17,13), (0,3,10,19), (0,5,19,11), (0,6,3,2), (0,10,23,5).

𝑛=17(+1mod34): (0,3,30,𝑥₁), (0,8,23,𝑥₂), (0,1,27,𝑥₃), (0,4,22,𝑥₄), (0,2,24,22), (0,5,16,13), (0,6,31,21), (0,7,21,6), (0,9,25,11), (0,10,6,5).

𝑛=18(+2mod36): (0,12,31,𝑥₁), (1,27,14,𝑥₁), (0,23,34,𝑥₂), (1,4,7,𝑥₂), (0,15,3,𝑥₃), (1,34,2,𝑥₃), (0,2,12,𝑥₄), (1,29,9,𝑥₄), (0,1,8,17), (0,5,15,16), (0,6,33,11), (0,7,11,31), (0,8,22,21), (0,11,9,4), (0,13,6,27), (0,16,4,1), (0,17,29,15), (0,22,17,23), (0,25,16,6), (0,28,26,19), (1,3,11,7).

𝑛=19(+1mod38): (0,29,34,𝑥₁), (0,13,8,𝑥₂), (0,18,28,𝑥₃), (0,20,6,𝑥₄), (0,1,23,27), (0,2,25,23), (0,3,11,1), (0,6,20,13), (0,7,3,29), (0,12,1,17), (0,17,9,3).

C. HPMD(2^𝑛5¹) for 𝑛≤23

𝑛=8(+1mod16): (0,11,5,𝑥₁), (0,12,2,𝑥₂), (0,13,15,𝑥₃), (0,14,7,𝑥₄), (0,15,3,𝑥₅), (0,1,4,11).

𝑛=10(+1mod20): (0,15,8,𝑥₁), (0,16,5,𝑥₂), (0,17,4,𝑥₃), (0,18,2,𝑥₄), (0,19,1,𝑥₅), (0,1,9,14), (0,3,14,6).

𝑛=11(+2mod22): (0,13,8,𝑥₁), (1,8,15,𝑥₁), (0,18,19,𝑥₂),(1,15,16,𝑥₂), (0,16,18,𝑥₃), (1,11,19,𝑥₃), (0,21,15,𝑥₄), (1,14,20,𝑥₄), (0,14,1,𝑥₅), (1,19,6,𝑥₅), (0,3,9,12), (0,4,2,10), (0,5,10,7), (0,15,5,3), (0,17,21,1).

𝑛=13(+2mod26): (0,4,25,𝑥₁), (1,21,12,𝑥₁), (0,25,6,𝑥₂), (1,24,5,𝑥₂), (0,1,3,𝑥₃), (1,10,16,𝑥₃), (0,14,22,𝑥₄), (1,7,19,𝑥₄), (0,23,24,𝑥₅), (1,0,3,𝑥₅), (0,2,21,16), (0,5,19,8), (0,9,17,2), (0,11,16,6), (0,12,1,4), (0,17,9,7), (1,5,15,11).

𝑛=14(+1mod28): (0,7,17,𝑥₁), (0,20,19,𝑥₂), (0,26,21,𝑥₃), (0,16,3,𝑥₄), (0,13,24,𝑥₅), (0,1,6,9), (0,2,10,3), (0,4,16,6), (0,6,23,19).

𝑛=17(+2mod34): (0,16,29,𝑥₁), (1,23,8,𝑥₁), (0,5,11,𝑥₂), (1,16,0,𝑥₂), (0,25,22,𝑥₃), (1,24,23,𝑥₃), (0,14,16,𝑥₄), (1,25,5,𝑥₄), (0,21,15,𝑥₅), (1,4,12,𝑥₅), (0,1,19,14), (0,3,8,12), (0,6,33,10), (0,7,28,4), (0,9,2,25), (0,11,24,2), (0,15,14,8), (0,19,31,27), (0,29,5,9), (0,31,13,33), (1,3,11,9).

𝑛=19(+2mod38): (0,7,37,𝑥₁), (1,2,26,𝑥₁), (0,6,7,𝑥₂), (1,13,0,𝑥₂), (0,4,30,𝑥₃), (1,37,13,𝑥₃), (0,14,11,𝑥₄), (1,33,22,𝑥₄), (0,22,15,𝑥₅), (1,35,34,𝑥₅), (0,2,14,23), (0,3,32,10), (0,5,9,25), (0,8,35,18), (0,10,33,6), (0,11,20,33), (0,15,3,5), (0,17,25,15), (0,18,16,3), (0,21,27,7), (0,29,36,35), (0,30,26,21), (1,11,35,19).

𝑛=22(+1mod44): (0,40,38,𝑥₁), (0,1,27,𝑥₂), (0,13,40,𝑥₃), (0,34,8,𝑥₄), (0,11,19,𝑥₅), (0,2,18,13), (0,3,15,24), (0,4,37,14), (0,5,24,3), (0,6,16,9), (0,7,39,19), (0,14,43,27), (0,15,9,1).

𝑛=23(+2mod46): (0,11,13,𝑥₁), (1,18,10,𝑥₁), (0,35,28,𝑥₂), (1,22,7,𝑥₂), (0,8,45,𝑥₃), (1,7,6,𝑥₃), (0,10,35,𝑥₄), (1,33,42,𝑥₄), (0,20,10,𝑥₅), (1,11,39,𝑥₅), (0,1,17,21), (0,2,6,33), (0,3,25,39), (0,5,3,40), (0,7,1,32), (0,9,4,28), (0,12,9,24), (0,13,43,17), (0,15,2,43), (0,16,15,3), (0,17,29,19), (0,19,8,41), (0,21,39,35), (0,26,22,6), (0,28,16,14), (0,29,7,45), (0,39,19,27).

D. HPMD(2^𝑛6¹) for 𝑛≤29

𝑛=8(+2mod16): (0,11,15,𝑥₁), (1,4,10,𝑥₁), (0,12,10,𝑥₂), (1,13,7,𝑥₂), (0,13,4,𝑥₃), (1,14,5,𝑥₃), (0,4,13,𝑥₄), (1,15,14,𝑥₄), (0,15,1,𝑥₅), (1,12,6,𝑥₅), (0,1,2,𝑥₆), (1,10,15,𝑥₆), (0,2,5,11).

𝑛=9(+1mod18): (0,12,15,𝑥₁), (0,13,6,𝑥₂), (0,14,4,𝑥₃), (0,15,2,𝑥₄), (0,16,8,𝑥₅), (0,17,1,𝑥₆), (0,1,7,14).

𝑛=10(+2mod20): (0,15,19,𝑥₁), (1,12,10,𝑥₁), (0,16,8,𝑥₂), (1,17,5,𝑥₂), (0,17,4,𝑥₃), (1,16,9,𝑥₃), (0,14,17,𝑥₄), (1,19,18,𝑥₄), (0,19,1,𝑥₅), (1,10,14,𝑥₅), (0,1,2,𝑥₆), (1,14,19,𝑥₆), (0,2,9,15), (0,6,15,12), (0,11,5,17).

𝑛=11(+1mod22): (0,4,7,𝑥₁), (0,7,2,𝑥₂), (0,20,12,𝑥₃), (0,18,4,𝑥₄), (0,1,6,𝑥₅), (0,16,13,𝑥₆), (0,2,17,16), (0,9,19,10).

𝑛=12(+2mod24): (0,16,11,𝑥₁), (1,3,20,𝑥₁), (0,18,3,𝑥₂), (1,15,6,𝑥₂), (0,15,13,𝑥₃), (1,8,16,𝑥₃), (0,17,23,𝑥₄), (1,2,8,𝑥₄), (0,3,2,𝑥₅), (1,12,19,𝑥₅), (0,11,20,𝑥₆), (1,4,9,𝑥₆), (0,1,6,4), (0,2,15,10), (0,4,1,14), (0,23,7,3), (1,5,15,7).

𝑛=13(+1mod26): (0,2,8,𝑥₁), (0,16,2,𝑥₂), (0,15,10,𝑥₃), (0,10,9,𝑥₄), (0,22,4,𝑥₅), (0,14,19,𝑥₆), (0,1,25,6), (0,3,12,9), (0,4,15,7).

𝑛=14(+2mod28): (0,25,1,𝑥₁), (1,14,26,𝑥₁), (0,22,11,𝑥₂), (1,13,16,𝑥₂), (0,8,7,𝑥₃), (1,3,4,𝑥₃), (0,9,8,𝑥₄), (1,18,25,𝑥₄), (0,4,27,𝑥₅), (1,11,20,𝑥₅), (0,21,18,𝑥₆), (1,6,11,𝑥₆), (0,1,23,21), (0,2,26,18), (0,3,22,10), (0,6,19,2), (0,11,17,5), (0,15,5,13), (0,19,15,7).

𝑛=15(+1mod30): (0,19,18,𝑥₁), (0,18,25,𝑥₂), (0,5,22,𝑥₃), (0,27,10,𝑥₄), (0,6,29,𝑥₅), (0,26,16,𝑥₆), (0,1,17,28), (0,3,24,22), (0,4,28,8), (0,9,21,16).

𝑛=17(+1mod34): (0,10,19,𝑥₁), (0,3,29,𝑥₂), (0,33,18,𝑥₃), (0,7,25,𝑥₄), (0,13,14,𝑥₅), (0,25,31,𝑥₆), (0,2,33,4), (0,4,27,14), (0,8,30,20), (0,11,23,5), (0,15,8,2).

𝑛=18(+2mod36): (0,3,7,𝑥₁), (1,0,14,𝑥₁), (0,1,32,𝑥₂), (1,10,27,𝑥₂), (0,35,31,𝑥₃), (1,22,20,𝑥₃), (0,26,33,𝑥₄), (1,35,0,𝑥₄), (0,4,25,𝑥₅), (1,29,26,𝑥₅), (0,2,26,𝑥₆), (1,7,17,𝑥₆), (0,5,12,20), (0,6,2,14), (0,9,3,16), (0,10,5,19), (0,11,13,33), (0,13,6,21), (0,19,27,13), (0,23,35,25), (0,25,14,6), (0,27,15,31), (0,29,20,17).

𝑛=19(+1mod38): (0,20,23,𝑥₁), (0,21,20,𝑥₂), (0,33,11,𝑥₃), (0,27,4,𝑥₄), (0,25,1,𝑥₅), (0,11,17,𝑥₆), (0,1,14,31), (0,2,25,9), (0,4,35,33), (0,8,32,3), (0,10,36,26), (0,18,12,8).

𝑛=22(+2mod44): (0,9,39,𝑥₁), (1,2,18,𝑥₁), (0,12,16,𝑥₂), (1,33,15,𝑥₂), (0,6,38,𝑥₃), (1,17,19,𝑥₃), (0,5,13,𝑥₄), (1,20,16,𝑥₄), (0,42,15,𝑥₅), (1,15,40,𝑥₅), (0,38,41,𝑥₆), (1,39,0,𝑥₆), (0,1,7,24), (0,2,10,21), (0,7,20,3), (0,10,3,14), (0,13,11,20), (0,14,35,1), (0,15,42,16), (0,19,23,10), (0,23,12,7), (0,25,17,37), (0,26,18,15), (0,29,25,9), (0,31,8,41), (0,35,30,29), (1,13,37,11).

𝑛=23(+1mod46): (0,24,3,𝑥₁), (0,18,25,𝑥₂), (0,5,1,𝑥₃), (0,1,33,𝑥₄), (0,45,37,𝑥₅), (0,27,39,𝑥₆), (0,2,32,20), (0,3,34,32), (0,4,20,40), (0,8,5,24), (0,9,42,11), (0,10,38,29), (0,11,24,17), (0,21,15,10).

𝑛=29(+1mod58): (0,53,22,𝑥₁), (0,42,34,𝑥₂), (0,46,39,𝑥₃), (0,19,20,𝑥₄), (0,15,6,𝑥₅), (0,13,53,𝑥₆), (0,2,16,15), (0,3,10,49), (0,4,25,11), (0,5,40,48), (0,6,37,4), (0,11,44,42), (0,12,35,3), (0,17,47,34), (0,18,55,17), (0,22,50,26), (0,36,30,10).

E. HPMD(2^𝑛7¹) for 𝑛≤23

𝑛=8(+1mod16): (0,5,9,𝑥₁), (0,10,11,𝑥₂), (0,11,14,𝑥₃), (0,12,3,𝑥₄), (0,13,6,𝑥₅), (0,14,4,𝑥₆), (0,15,1,𝑥₇).

𝑛=9(+2mod18): (0,14,12,𝑥₁), (1,13,3,𝑥₁), (0,13,17,𝑥₂), (1,14,8,𝑥₂), (0,2,8,𝑥₃), (1,15,7,𝑥₃), (0,15,4,𝑥₄), (1,6,11,𝑥₄), (0,4,7,𝑥₅), (1,17,16,𝑥₅), (0,17,1,𝑥₆), (1,4,14,𝑥₆), (0,1,2,𝑥₇), (1,12,5,𝑥₇), (0,7,13,10).

𝑛=10(+1mod20): (0,13,16,𝑥₁), (0,14,5,𝑥₂), (0,15,7,𝑥₃), (0,16,2,𝑥₄), (0,17,6,𝑥₅), (0,18,3,𝑥₆), (0,19,1,𝑥₇), (0,1,9,13).

𝑛=11(+2mod22): (0,16,8,𝑥₁), (1,17,3,𝑥₁), (0,17,21,𝑥₂), (1,18,16,𝑥₂), (0,4,12,𝑥₃), (1,19,11,𝑥₃), (0,19,4,𝑥₄), (1,10,15,𝑥₄), (0,12,19,𝑥₅), (1,21,20,𝑥₅), (0,21,1,𝑥₆), (1,14,10,𝑥₆), (0,1,2,𝑥₇), (1,6,19,𝑥₇), (0,2,17,7), (0,3,9,19), (0,6,15,12).

𝑛=12(+1mod24): (0,7,20,𝑥₁), (0,18,21,𝑥₂), (0,19,23,𝑥₃), (0,20,5,𝑥₄), (0,11,10,𝑥₅), (0,8,6,𝑥₆), (0,21,11,𝑥₇), (0,1,16,18), (0,5,22,14).

𝑛=13(+2mod26): (0,21,11,𝑥₁), (1,2,4,𝑥₁), (0,17,2,𝑥₂), (1,22,21,𝑥₂), (0,6,22,𝑥₃), (1,19,25,𝑥₃), (0,7,1,𝑥₄), (1,16,24,𝑥₄), (0,23,21,𝑥₅), (1,8,18,𝑥₅), (0,18,19,𝑥₆), (1,3,22,𝑥₆), (0,15,14,𝑥₇), (1,18,11,𝑥₇), (0,3,17,22), (0,5,8,17), (0,11,15,12), (0,12,6,2), (1,9,19,5).

𝑛=14(+1mod28): (0,10,25,𝑥₁), (0,1,8,𝑥₂), (0,13,19,𝑥₃), (0,19,24,𝑥₄), (0,23,21,𝑥₅), (0,17,13,𝑥₆), (0,9,2,𝑥₇), (0,2,18,1), (0,3,23,20), (0,4,22,16).

𝑛=15(+2mod30): (0,14,21,𝑥₁), (1,5,0,𝑥₁), (0,4,26,𝑥₂), (1,9,29,𝑥₂), (0,12,25,𝑥₃), (1,19,26,𝑥₃), (0,2,12,𝑥₄), (1,11,27,𝑥₄), (0,6,7,𝑥₅), (1,13,14,𝑥₅), (0,20,9,𝑥₆), (1,7,20,𝑥₆), (0,27,24,𝑥₇), (1,20,7,𝑥₇), (0,3,27,14), (0,5,8,2), (0,8,29,1), (0,9,2,25), (0,11,20,19), (0,18,13,4), (1,15,11,3).

𝑛=17(+2mod34): (0,6,2,𝑥₁), (1,15,33,𝑥₁), (0,1,23,𝑥₂), (1,4,28,𝑥₂), (0,20,25,𝑥₃), (1,7,4,𝑥₃), (0,21,11,𝑥₄), (1,20,8,𝑥₄), (0,12,1,𝑥₅), (1,29,2,𝑥₅), (0,33,8,𝑥₆), (1,28,23,𝑥₆), (0,8,18,𝑥₇), (1,27,25,𝑥₇), (0,2,5,30), (0,7,22,20), (0,9,19,6), (0,11,24,16), (0,13,14,5), (0,15,4,1), (0,16,9,29), (0,19,15,23), (1,3,7,19).

𝑛=18(+1mod36): (0,15,17,𝑥₁), (0,35,11,𝑥₂), (0,34,31,𝑥₃), (0,19,32,𝑥₄), (0,4,7,𝑥₅), (0,28,9,𝑥₆), (0,11,1,𝑥₇), (0,1,26,14), (0,5,14,20), (0,7,30,4), (0,8,28,6), (0,21,16,9).

𝑛=19(+2mod38): (0,24,18,𝑥₁), (1,29,25,𝑥₁), (0,31,1,𝑥₂), (1,30,36,𝑥₂), (0,11,26,𝑥₃), (1,36,3,𝑥₃), (0,8,36,𝑥₄), (1,11,9,𝑥₄), (0,16,29,𝑥₅), (1,17,22,𝑥₅), (0,18,15,𝑥₆), (1,13,8,𝑥₆), (0,37,16,𝑥₇), (1,26,29,𝑥₇), (0,1,33,2), (0,2,11,25), (0,4,14,1), (0,7,27,29), (0,12,4,18), (0,15,8,35), (0,17,28,12), (0,21,9,15), (0,23,7,11), (0,29,21,4), (0,33,13,37).

𝑛=22(+1mod44): (0,13,43,𝑥₁), (0,8,42,𝑥₂), (0,24,31,𝑥₃), (0,17,12,𝑥₄), (0,16,9,𝑥₅), (0,18,6,𝑥₆), (0,12,33,𝑥₇), (0,1,26,29), (0,2,29,25), (0,4,14,8), (0,5,34,13), (0,6,41,11), (0,9,7,33), (0,20,17,1).

𝑛=23(+2mod46): (0,44,12,𝑥₁), (1,25,19,𝑥₁), (0,2,41,𝑥₂), (1,11,30,𝑥₂), (0,36,21,𝑥₃), (1,23,18,𝑥₃), (0,16,4,𝑥₄), (1,29,35,𝑥₄), (0,38,45,𝑥₅), (1,15,8,𝑥₅), (0,12,13,𝑥₆), (1,43,40,𝑥₆), (0,11,28,𝑥₇), (1,34,15,𝑥₇), (0,3,8,45), (0,4,3,1), (0,5,37,20), (0,6,35,25), (0,8,26,22), (0,9,44,39), (0,10,40,26), (0,13,33,35), (0,15,27,6), (0,17,1,19), (0,19,22,9), (0,21,36,15), (0,22,11,37), (0,28,25,33), (1,5,39,31).

F. HPMD(2^𝑛8¹) for 𝑛≤23

𝑛=10(+4mod20): (0,2,14,𝑥₁), (1,15,19,𝑥₁), (2,0,9,𝑥₁), (3,5,16,𝑥₁), (0,16,15,𝑥₂), (1,13,2,𝑥₂), (2,7,5,𝑥₂), (3,18,12,𝑥₂), (0,15,12,𝑥₃),(1,0,13,𝑥₃), (2,18,7,𝑥₃), (3,9,14,𝑥₃), (0,5,1,𝑥₄), (1,16,4,𝑥₄), (2,10,3,𝑥₄), (3,11,10,𝑥₄), (0,6,3,𝑥₅),(1,5,8,𝑥₅), (2,8,6,𝑥₅), (3,15,9,𝑥₅), (0,4,18,𝑥₆), (1,9,16,𝑥₆), (2,3,19,𝑥₆), (3,14,17,𝑥₆), (0,1,2,𝑥₇), (1,7,12,𝑥₇), (2,6,13,𝑥₇), (3,16,7,𝑥₇), (0,17,19,𝑥₈), (1,19,6,𝑥₈), (2,4,16,𝑥₈), (3,6,1,𝑥₈), (0,3,4,11), (1,14,5,2).

𝑛=11(+1mod22): (0,14,20,𝑥₁), (0,15,16,𝑥₂), (0,16,4,𝑥₃), (0,17,8,𝑥₄), (0,18,5,𝑥₅), (0,19,9,𝑥₆), (0,20,3,𝑥₇), (0,21,1,𝑥₈), (0,3,10,18).

𝑛=13(+1mod26): (0,18,7,𝑥₁), (0,19,5,𝑥₂), (0,20,11,𝑥₃), (0,21,4,𝑥₄), (0,2,18,𝑥₅), (0,23,1,𝑥₆), (0,24,23,𝑥₇), (0,1,9,𝑥₈), (0,3,10,15), (0,6,2,12).

𝑛=14(+2mod28): (0,21,20,𝑥₁), (1,16,21,𝑥₁), (0,24,18,𝑥₂), (1,19,3,𝑥₂), (0,1,5,𝑥₃), (1,12,18,𝑥₃), (0,8,23,𝑥₄), (1,23,2,𝑥₄), (0,4,2,𝑥₅), (1,3,23,𝑥₅), (0,16,25,𝑥₆), (1,9,0,𝑥₆), (0,12,3,𝑥₇), (1,27,20,𝑥₇), (0,23,24,𝑥₈), (1,14,25,𝑥₈), (0,2,19,8), (0,3,13,19), (0,7,12,25), (0,10,7,3), (0,18,17,5).

𝑛=17(+1mod34): (0,22,16,𝑥₁), (0,31,20,𝑥₂), (0,5,8,𝑥₃), (0,30,9,𝑥₄), (0,14,4,𝑥₅), (0,33,6,𝑥₆), (0,16,27,𝑥₇), (0,21,12,𝑥₈), (0,1,3,22), (0,4,10,2), (0,8,23,7), (0,9,29,24).

𝑛=19(+1mod38): (0,9,27,𝑥₁), (0,23,37,𝑥₂), (0,17,22,𝑥₃), (0,11,2,𝑥₄), (0,13,35,𝑥₅), (0,6,8,𝑥₆), (0,16,12,𝑥₇), (0,15,9,𝑥₈), (0,1,28,26), (0,3,7,17), (0,7,4,12), (0,20,15,14), (0,25,17,7).

𝑛=22(+2mod44): (0,9,33,𝑥₁), (1,20,14,𝑥₁), (0,18,11,𝑥₂), (1,43,26,𝑥₂), (0,14,17,𝑥₃), (1,39,24,𝑥₃), (0,15,35,𝑥₄), (1,14,40,𝑥₄), (0,16,6,𝑥₅), (1,29,33,𝑥₅), (0,10,42,𝑥₆), (1,15,25,𝑥₆), (0,41,3,𝑥₇), (1,42,6,𝑥₇), (0,19,8,𝑥₈), (1,10,37,𝑥₈), (0,1,40,38), (0,2,41,20), (0,4,37,14), (0,5,20,7), (0,7,14,13), (0,11,29,32), (0,13,43,4), (0,17,25,16), (0,20,12,33), (0,23,15,27), (0,25,27,43), (0,29,28,19), (1,27,17,13).

𝑛=23(+1mod46): (0,42,24,𝑥₁), (0,33,6,𝑥₂), (0,4,5,𝑥₃), (0,30,9,𝑥₄), (0,45,11,𝑥₅), (0,38,21,𝑥₆), (0,22,15,𝑥₇), (0,44,1,𝑥₈), (0,2,33,28), (0,5,19,39), (0,6,38,3), (0,8,18,6), (0,9,36,25), (0,13,4,30), (0,15,32,22).

G. HPMD(2^𝑛9¹) for 𝑛≤19

𝑛=10(+4mod20): (0,17,2,𝑥₁), (1,18,5,𝑥₁), (2,0,16,𝑥₁), (3,11,15,𝑥₁), (0,13,17,𝑥₂), (1,12,6,𝑥₂), (2,19,8,𝑥₂), (3,10,19,𝑥₂), (0,4,19,𝑥₃), (1,9,8,𝑥₃), (2,15,13,𝑥₃), (3,18,14,𝑥₃), (0,2,1,𝑥₄), (1,17,12,𝑥₄), (2,7,3,𝑥₄), (3,4,10,𝑥₄), (0,7,4,𝑥₅), (1,14,18,𝑥₅), (2,4,7,𝑥₅), (3,9,1,𝑥₅), (0,8,7,𝑥₆), (1,10,13,𝑥₆), (2,3,14,𝑥₆), (3,5,12,𝑥₆), (0,1,15,𝑥₇), (1,19,2,𝑥₇), (2,10,4,𝑥₇), (3,16,5,𝑥₇), (0,11,5,𝑥₈), (1,2,14,𝑥₈), (2,13,19,𝑥₈), (3,8,0,𝑥₈), (0,5,8,𝑥₉), (1,3,15,𝑥₉), (2,8,6,𝑥₉), (3,2,17,𝑥₉).

𝑛=13(+2mod26): (0,14,4,𝑥₁), (1,3,19,𝑥₁), (0,6,10,𝑥₂), (1,23,15,𝑥₂), (0,1,2,𝑥₃), (1,16,23,𝑥₃), (0,2,21,𝑥₄), (1,9,8,𝑥₄), (0,23,3,𝑥₅), (1,24,20,𝑥₅), (0,25,11,𝑥₆), (1,10,22,𝑥₆), (0,10,1,𝑥₇), (1,21,2,𝑥₇), (0,20,18,𝑥₈), (1,25,3,𝑥₈), (0,8,17,𝑥₉), (1,15,18,𝑥₉), (0,3,14,9), (0,5,15,8), (0,11,6,21).

𝑛=14(+1mod28): (0,27,12,𝑥₁), (1,0,5,𝑥₁), (0,6,1,𝑥₂), (1,3,14,𝑥₂), (0,26,22,𝑥₃), (1,9,19,𝑥₃), (0,12,4,𝑥₄), (1,19,23,𝑥₄), (0,10,7,𝑥₅), (1,23,2,𝑥₅), (0,16,3,𝑥₆), (1,21,18,𝑥₆), (0,22,13,𝑥₇), (1,27,20,𝑥₇), (0,7,26,𝑥₈), (1,24,17,𝑥₈), (0,4,17,𝑥₉), (1,7,10,𝑥₉), (0,1,18,27), (0,2,20,23), (0,8,25,13), (0,11,23,19).

𝑛=15(+2mod30): (0,18,1,𝑥₁), (1,15,10,𝑥₁), (0,5,12,𝑥₂), (1,18,25,𝑥₂), (0,21,3,𝑥₃), (1,6,20,𝑥₃), (0,2,25,𝑥₄), (1,29,12,𝑥₄), (0,6,26,𝑥₅), (1,11,17,𝑥₅), (0,26,8,𝑥₆), (1,19,9,𝑥₆), (0,8,24,𝑥₇), (1,17,13,𝑥₇), (0,1,23,𝑥₈), (1,4,14,𝑥₈), (0,29,10,𝑥₉), (1,24,27,𝑥₉), (0,4,29,1), (0,9,13,2), (0,11,2,21), (0,17,14,8), (0,27,5,29).

𝑛=19(+2mod38): (0,6,1,𝑥₁), (1,19,14,𝑥₁), (0,35,2,𝑥₂), (1,4,21,𝑥₂), (0,9,20,𝑥₃), (1,14,35,𝑥₃), (0,25,37,𝑥₄), (1,16,30,𝑥₄), (0,34,24,𝑥₅), (1,27,23,𝑥₅), (0,8,10,𝑥₆), (1,11,9,𝑥₆), (0,11,15,𝑥₇), (1,18,6,𝑥₇), (0,4,11,𝑥₈), (1,17,26,𝑥₈), (0,3,4,𝑥₉), (1,24,25,𝑥₉), (0,5,30,2), (0,12,32,17), (0,13,35,3), (0,15,29,22), (0,18,7,6), (0,22,21,11), (0,24,17,8), (0,29,23,31), (1,3,33,15).

H. HPMD(2^𝑛10¹) for 𝑛=13,14 and 10≤𝑢≤13

𝑛=13, 𝑢=10(+1mod26): (0,16,19,𝑥₁), (0,17,23,𝑥₂), (0,18,11,𝑥₃), (0,15,25,𝑥₄), (0,20,2,𝑥₅), (0,21,9,𝑥₆), (0,22,8,𝑥₇), (0,23,22,𝑥₈), (0,4,6,𝑥₉), (0,24,5,𝑦₀), (0,1,10,15).

𝑛=13, 𝑢=12(+1mod26): (0,19,17,𝑥₁), (0,3,23,𝑥₂), (0,1,18,𝑥₃), (0,8,7,𝑥₄), (0,7,12,𝑥₅), (0,22,11,𝑥₆), (0,23,25,𝑥₇), (0,4,16,𝑥₈), (0,10,5,𝑥₉), (0,11,20,𝑦₀), (0,16,4,𝑦₁), (0,18,24,𝑦₂).

𝑛=14, 𝑢=10(+2mod28): (0,15,18,𝑥₁), (1,22,25,𝑥₁), (0,16,26,𝑥₂), (1,3,9,𝑥₂), (0,9,17,𝑥₃), (1,20,26,𝑥₃), (0,13,8,𝑥₄), (1,0,7,𝑥₄), (0,23,5,𝑥₅), (1,18,14,𝑥₅), (0,12,1,𝑥₆), (1,13,22,𝑥₆), (0,19,24,𝑥₇), (1,12,11,𝑥₇), (0,11,27,𝑥₈), (1,14,16,𝑥₈), (0,20,21,𝑥₉), (1,23,10,𝑥₉), (0,21,11,𝑦₀), (1,26,20,𝑦₀), (0,4,22,20), (0,5,25,21), (0,25,23,27).

𝑛=14, 𝑢=12(+4mod28): (0,9,16,𝑥₁), (1,26,21,𝑥₁), (2,15,14,𝑥₁), (3,24,7,𝑥₁), (0,18,11,𝑥₂), (1,3,26,𝑥₂), (2,17,4,𝑥₂), (3,16,9,𝑥₂), (0,10,20,𝑥₃), (1,20,11,𝑥₃), (2,5,6,𝑥₃), (3,27,1,𝑥₃), (0,20,19,𝑥₄), (1,25,17,𝑥₄), (2,19,20,𝑥₄), (3,18,2,𝑥₄), (0,8,9,𝑥₅), (1,9,4,𝑥₅), (2,26,23,𝑥₅), (3,15,26,𝑥₅), (0,22,2,𝑥₆), (1,12,8,𝑥₆), (2,7,27,𝑥₆), (3,21,25,𝑥₆), (0,26,27,𝑥₇), (1,13,2,𝑥₇), (2,8,21,𝑥₇), (3,19,16,𝑥₇), (0,2,24,𝑥₈), (1,27,18,𝑥₈), (2,0,25,𝑥₈), (3,9,15,𝑥₈), (0,7,1,𝑥₉), (1,14,16,𝑥₉), (2,20,15,𝑥₉), (3,13,22,𝑥₉), (0,5,23,𝑦₀), (1,22,14,𝑦₀), (2,11,9,𝑦₀), (3,20,24,𝑦₀), (0,17,5,𝑦₁), (1,23,27,𝑦₁), (2,6,22,𝑦₁), (3,12,0,𝑦₁), (0,15,22,𝑦₂), (1,6,5,𝑦₂), (2,21,24,𝑦₂), (3,8,11,𝑦₂), (0,6,17,16), (1,11,19,22).

𝑛=14, 𝑢=13(+4mod28): (𝑥₁,0,24,2), (𝑥₁,1,11,5), (𝑥₁,2,26,3), (𝑥₁,3,21,12), (𝑥₂,0,2,9), (𝑥₂,1,27,4), (𝑥₂,2,5,22), (𝑥₂,3,20,7), (𝑥₃,0,19,11), (𝑥₃,1,9,13), (𝑥₃,2,18,20), (𝑥₃,3,0,10), (𝑥₄,0,25,20), (𝑥₄,1,7,23), (𝑥₄,2,24,9), (𝑥₄,3,2,14), (𝑥₅,0,3,15), (𝑥₅,1,26,9), (𝑥₅,2,21,8), (𝑥₅,3,4,22), (𝑥₆,0,26,13), (𝑥₆,1,19,12), (𝑥₆,2,8,7), (𝑥₆,3,13,6), (𝑥₇,0,20,3), (𝑥₇,1,17,24), (𝑥₇,2,27,5), (𝑥₇,3,26,18), (𝑥₈,0,21,26), (𝑥₈,1,4,8), (𝑥₈,2,11,13), (𝑥₈,3,10,23), (𝑥₉,0,16,23), (𝑥₉,1,23,14), (𝑥₉,2,1,25), (𝑥₉,3,6,4), (𝑦₀,0,5,4), (𝑦₀,1,14,3), (𝑦₀,2,3,18), (𝑦₀,3,12,13), (𝑦₁,0,22,12), (𝑦₁,1,12,7), (𝑦₁,2,23,6), (𝑦₁,3,1,21), (𝑦₂,0,9,18), (𝑦₂,1,3,27), (𝑦₂,2,6,1), (𝑦₂,3,16,24), (𝑦₃,0,17,1), (𝑦₃,1,2,10), (𝑦₃,2,12,24), (𝑦₃,3,7,15).

I. Miscellaneous HPMD(2^𝑛𝑢¹)

𝑛=17, 𝑢=12(+1mod34): (0,13,28,𝑥₁), (0,1,23,𝑥₂), (0,2,21,𝑥₃), (0,24,19,𝑥₄), (0,27,26,𝑥₅), (0,18,10,𝑥₆), (0,11,2,𝑥₇), (0,30,27,𝑥₈), (0,5,12,𝑥₉), (0,14,1,𝑦₀), (0,6,4,𝑦₁), (0,28,31,𝑦₂), (0,4,14,22), (0,9,29,18).

𝑛=17, 𝑢=13(+2mod34): (0,18,20,𝑥₁), (1,11,13,𝑥₁), (0,10,16,𝑥₂), (1,19,27,𝑥₂), (0,30,31,𝑥₃), (1,33,6,𝑥₃), (0,22,13,𝑥₄), (1,31,0,𝑥₄), (0,27,26,𝑥₅), (1,28,15,𝑥₅), (0,32,7,𝑥₆), (1,25,14,𝑥₆), (0,26,30,𝑥₇), (1,7,11,𝑥₇), (0,14,27,𝑥₈), (1,27,24,𝑥₈), (0,28,33,𝑥₉), (1,13,32,𝑥₉), (0,3,19,𝑦₀), (1,12,20,𝑦₀), (0,33,12,𝑦₁), (1,10,7,𝑦₁), (0,20,2,𝑦₂), (1,29,17,𝑦₂), (0,12,23,𝑦₃), (1,15,10,𝑦₃), (0,7,28,9), (0,19,5,10), (0,23,24,19).

𝑛=17, 𝑢=15(+2mod34): (0,32,11,𝑥₁), (1,33,0,𝑥₁), (0,6,5,𝑥₂), (1,27,26,𝑥₂), (0,22,29,𝑥₃), (1,19,16,𝑥₃), (0,21,33,𝑥₄), (1,12,32,𝑥₄), (0,31,24,𝑥₅), (1,14,3,𝑥₅), (0,14,6,𝑥₆), (1,3,17,𝑥₆), (0,4,32,𝑥₇), (1,17,21,𝑥₇), (0,19,25,𝑥₈), (1,10,12,𝑥₈), (0,18,8,𝑥₉), (1,23,13,𝑥₉), (0,9,30,𝑦₀), (1,16,7,𝑦₀), (0,10,13,𝑦₁), (1,9,28,𝑦₁), (0,11,31,𝑦₂), (1,6,14,𝑦₂), (0,29,20,𝑦₃), (1,8,9,𝑦₃), (0,5,15,𝑦₄), (1,30,8,𝑦₄), (0,30,12,𝑦₅), (1,29,25,𝑦₅), (0,15,18,11).

𝑛=17, 𝑢=16(+1mod34): (0,3,𝑥₁,21), (0,16,𝑥₂,6), (0,1,𝑥₃,2), (0,12,𝑥₄,1), (0,5,𝑥₅,32), (0,15,𝑥₆,30), (0,10,𝑥₇,15), (0,7,𝑥₈,16), (0,31,𝑥₉,23), (0,9,𝑦₀,5), (0,27,𝑦₁,13), (0,26,𝑦₂,14), (0,24,𝑦₃,26), (0,30,𝑦₄,9), (0,23,𝑦₅,20), (0,22,𝑦₆,28).

𝑛=19, 𝑢=10(+1mod38): (0,28,5,𝑥₁), (0,33,32,𝑥₂), (0,27,30,𝑥₃), (0,20,17,𝑥₄), (0,7,16,𝑥₅), (0,8,24,𝑥₆), (0,23,35,𝑥₇), (0,13,34,𝑥₈), (0,4,36,𝑥₉), (0,6,37,𝑦₀), (0,1,11,13), (0,5,23,14), (0,11,7,24), (0,22,20,12).

𝑛=19, 𝑢=14(+1mod38): (0,30,𝑥₁,22), (0,31,𝑥₂,27), (0,35,𝑥₃,37), (0,8,𝑥₄,24), (0,25,𝑥₅,15), (0,29,𝑥₆,36), (0,12,𝑥₇,35), (0,28,𝑥₈,23), (0,6,𝑥₉,5), (0,36,𝑦₀,16), (0,21,𝑦₁,12), (0,32,𝑦₂,29), (0,34,𝑦₃,20), (0,37,𝑦₄,25), (0,20,6,33), (0,4,11,21).

𝑛=19, 𝑢=15(+2mod38): (0,28,𝑥₁,11), (1,17,𝑥₁,4), (0,14,𝑥₂,7), (1,25,𝑥₂,28), (0,16,𝑥₃,12), (1,9,𝑥₃,5), (0,32,𝑥₄,20), (1,19,𝑥₄,13), (0,10,𝑥₅,1), (1,31,𝑥₅,26), (0,8,𝑥₆,3), (1,21,𝑥₆,22), (0,15,𝑥₇,17), (1,4,𝑥₇,12), (0,20,𝑥₈,4), (1,29,𝑥₈,3), (0,9,𝑥₉,8), (1,18,𝑥₉,27), (0,21,𝑦₀,5), (1,24,𝑦₀,10), (0,25,𝑦₁,2), (1,10,𝑦₁,37), (0,37,𝑦₂,23), (1,6,𝑦₂,34), (0,12,𝑦₃,32), (1,5,𝑦₃,25), (0,33,𝑦₄,25), (1,2,𝑦₄,0), (0,23,𝑦₅,16), (1,30,𝑦₅,7), (0,31,3,14), (0,2,6,13), (0,3,25,31).

𝑛=19,𝑢=16(+1mod38): (0,28,𝑥₁,27,)(0,27,𝑥₂,4,)(0,35,𝑥₃,30,)(0,24,𝑥₄,31,), (0,12,𝑥₅,21,),(0,22,𝑥₆,8,), (0,2,𝑥₇,6,), (0,37,𝑥₈,17,), (0,20,𝑥₉,7,), (0,10,𝑦₀,2,), (0,29,𝑦₁,13,), (0,3,𝑦₂,29,), (0,26,𝑦₃,24,), (0,18,𝑦₄,15,), (0,1,𝑦₅,33,), (0,16,𝑦₆,5,), (0,4,10,25,).

𝑛=21, 𝑢=19(+2mod42): (0,9,𝑥₁,14), (1,6,𝑥₁,15), (0,27,𝑥₂,39), (1,40,𝑥₂,38), (0,22,𝑥₃,4), (1,35,𝑥₃,31), (0,15,𝑥₄,2), (1,30,𝑥₄,21), (0,8,𝑥₅,36), (1,3,𝑥₅,29), (0,12,𝑥₆,23), (1,7,𝑥₆,10), (0,16,𝑥₇,8), (1,9,𝑥₇,7), (0,32,𝑥₈,28), (1,41,𝑥₈,19), (0,37,𝑥₉,19), (1,34,𝑥₉,18), (0,10,𝑦₀,15), (1,11,𝑦₀,4), (0,2,𝑦₁,22), (1,17,𝑦₁,11), (0,41,𝑦₂,38), (1,18,𝑦₂,17), (0,24,𝑦₃,5), (1,21,𝑦₃,32), (0,18,𝑦₄,1), (1,31,𝑦₄,14), (0,19,𝑦₅,27), (1,10,𝑦₅,0), (0,31,𝑦₆,16), (1,12,𝑦₆,39), (0,7,𝑦₇,6), (1,32,𝑦₇,25), (0,30,𝑦₈,17), (1,39,𝑦₈,20), (0,3,𝑦₉,35), (1,14,𝑦₉,26), (0,13,6,41).

𝑛=22, 𝑢,=16,(+2mod44): (0,12,𝑥₁,11), (1,31,𝑥₁,20), (0,43,𝑥₂,10), (1,42,𝑥₂,3), (0,4,𝑥₃,17), (0,5,𝑥₄,13), (1,14,𝑥₄,4), (0,39,𝑥₅,4), (1,4,𝑥₅,31), (0,20,𝑥₆,23), (1,29,𝑥₆,36), (0,28,𝑥₇,20), (1,41,𝑥₇,37), (0,1,𝑥₈,33), (1,24,𝑥₈,18), (0,35,𝑥₉,15), (1,16,𝑥₉,14), (0,17,𝑦₀,42), (1,40,𝑦₀,35), (0,32,𝑦₁,18), (1,7,𝑦₁,41), (0,38,𝑦₂,19), (1,21,𝑦₂,24), (0,18,𝑦₃,30), (1,39,𝑦₃,13), (0,23,𝑦₄,8), (1,2,𝑦₄,9), (0,30,𝑦₅,7), (1,17,𝑦₅,8), (0,3,𝑦₆,9), (1,0,𝑦₆,16), (0,6,4,37), (0,15,17,36), (0,13,18,28), (0,11,20,27), (1,35,3,21), (1,27,𝑥₃,26).

𝑛=23, 𝑢=17(+2mod46): (0,4,19,𝑥₁), (1,9,30,𝑥₁), (0,40,5,𝑥₂), (1,17,8,𝑥₂), (0,6,35,𝑥₃), (1,45,44,𝑥₃), (0,13,37,𝑥₄), (1,12,0,𝑥₄), (0,5,45,𝑥₅), (1,26,22,𝑥₅), (0,35,41,𝑥₆), (1,10,36,𝑥₆), (0,37,27,𝑥₇), (1,42,40,𝑥₇), (0,39,21,𝑥₈), (1,34,18,𝑥₈), (0,21,36,𝑥₉), (1,18,5,𝑥₉), (0,24,38,𝑦₀), (1,3,45,𝑦₀), (0,8,26,𝑦₁), (1,39,19,𝑦₁), (0,16,33,𝑦₂), (1,19,38,𝑦₂), (0,43,40,𝑦₃), (1,14,33,𝑦₃), (0,45,34,𝑦₄), (1,32,39,𝑦₄), (0,27,32,𝑦₅), (1,4,35,𝑦₅), (0,22,42,𝑦₆), (1,5,17,𝑦₆), (0,28,18,𝑦₇), (1,11,25,𝑦₇), (0,1,31,7), (0,2,43,17), (0,3,30,39), (0,10,22,8), (0,25,13,45).

J. Miscellaneous HPMD(ℎ^𝑛𝑢¹)

ℎ=3, 𝑛=4, 𝑢=1: (0,1,2,3), (0,2,𝑥₁,11), (0,3,5,2), (0,5,7,6), (0,6,9,7), (0,7,1,𝑥₁), (0,9,11,10), (0,10,3,1), (0,11,6,5), (0,𝑥₁,10,9), (1,3,𝑥₁,6), (1,4,7,10), (1,6,8,𝑥₁), (1,7,4,2), (1,8,6,11), (1,10,11,8), (1,11,2,4), (2,5,4,𝑥₁), (2,7,5,8), (2,8,9,3), (2,9,8,7), (2,11,𝑥₁,9), (3,4,9,𝑥₁), (3,6,𝑥₁,4), (3,8,5,6), (3,9,10,8), (3,10,4,5), (4,6,7,9), (4,10,5,11), (4,11,9,6), (5,10,7,𝑥₁), (5,𝑥₁,8,11), (7,8,10,𝑥₁).

ℎ=3, 𝑛=4, 𝑢=2(+2mod12): (0,2,𝑥₁,5), (1,7,𝑥₁,2), (0,5,𝑥₂,3), (1,2,𝑥₂,0), (0,3,1,6), (0,7,6,1), (0,10,7,9).

ℎ=3, 𝑛=4, 𝑢=4(+1mod12): (0,6,𝑥₁,11), (0,11,𝑥₂,10), (0,10,𝑥₃,7), (0,7,𝑥₄,9), (0,9,6,3).

ℎ=8, 𝑛=4, 𝑢=2(+2mod32): (0,7,5,𝑥₁), (1,14,0,𝑥₁), (0,19,13,𝑥₂), (1,26,16,𝑥₂), (0,1,15,26), (0,2,23,1), (0,3,18,13), (0,5,27,18), (0,9,11,6), (0,10,25,31), (0,11,29,2), (0,13,22,15), (0,17,6,3), (0,23,30,29).

ℎ=8, 𝑛=5, 𝑢=14(+1mod40): (0,1,13,𝑥₁), (0,37,3,𝑥₂), (0,17,1,𝑥₃), (0,7,38,𝑥₄), (0,9,23,𝑥₅), (0,29,21,𝑥₆), (0,22,24,𝑥₇), (0,26,22,𝑥₈), (0,33,11,𝑥₉), (0,39,33,𝑦₀), (0,8,12,𝑦₁), (0,28,4,𝑦₂), (0,23,26,𝑦₃), (0,38,9,𝑦₄), (0,13,34,21).

Acknowledgments

The author would like to acknowledge the support of National Science Foundation under Grant no. CCR-0541070.