# Non-Kochen–Specker Contextuality

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Formalism

**Definition**

**1.**

- 1.
- Every vertex belongs to at least one hyperedge;
- 2.
- Every hyperedge contains at least two and at most n vertices;
- 3.
- No hyperedge shares only one vertex with another hyperedge;
- 4.
- Hyperedges may intersect each other in at most $n-2$ vertices
- 5.
- Graphically, vertices are represented as dots, and hyperedges are (curved) lines passing through them.

**Definition**

**2.**

- 1.
- No two vertices within any of its edges are both assigned a value of 1;
- 2.
- In any of its edges, not all of the vertices are assigned a value of 0.

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

- 1.
- No two vertices within any of its edges are both assigned a value of 1;
- 2.
- In any of its edges, not all of the vertices are assigned a value of 0.

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

#### 2.2. Generation of Non-KS MMPHs

**M1**consists of dropping vertices contained in single hyperedges (multiplicity $m=1$) [34] of either NBMMPHs or BMMPHs and a possible subsequent stripping of their hyperedges. The obtained smaller MMPHs are often non-KS, although never KS.**M2**consists of a random addition of hyperedges to MMPHs so as to obtain bigger ones, which then serve us to generate smaller non-KS NBMMPHs by stripping hyperedges randomly again;**M3**consists of the random deletion of vertices in either NBMMPHs or a BMMPHs until a non-KS NBMMPH is reached.

#### 2.3. Dimensions Three to Five and the Three Classes of Non-KS Contextual Sets from the Literature

`ALK`corresponds to a graph clique with three edges:

`AL`,

`LK`, and

`KA`.

**M1**. The 13-16 MMPH is not critical, and it contains four critical sub-MMPHs, the smallest of which is 10-9 [41].

**M1**and

**M2**. Therefore, below, we give distributions and samples of just four- and five-dim critical non-KS NBMMPHs presented in Figure 2a,f. Here, we only point out that the KS “bug,” the 8-7 non-KS NBMMPH shown in ([41] Figure 3a), is the smallest three-dim non-KS NBMMPH that satisfies our requirement that at least one of the hyperedges must contain n vertices (n being the dimension of the considered MMPH), none of which has the multiplicity $m=1$. Its string, the string of its filled MMPH, and their coordinatizations are given in Appendix A, as are the strings and coordinatizations of any other MMPH considered in the paper.

**M1**, we first generate the supermasters from the vector components. In the four-dim space, we obtain the 24-24 supermaster from the $\{0,\pm 1\}$ components and the 60-72 supermaster from the $\{0,\pm \varphi ,\varphi -1\}$ components, where $\varphi =\frac{1+\sqrt{5}}{2}$ (the golden ratio). Their strings and coordinatizations are given in Appendix A. Then, we randomly strip hyperedges from them, e.g., 14 from 24-24 and 21 from the 60-72 supermaster, so as to obtain the 20-10 and 58-51 masters, respectively. From the latter masters, we remove $m=1$ vertices, and from any of them, we generate the classes of critical MMPHs by stripping them further until we obtain critical MMPHs that form the 20-10 and 58-51 non-KS classes. In the five-dim space, we obtain the 105-136 supermaster from the $\{0,\pm 1\}$ components. Its string and coordinatization are given in Appendix A. Further, we randomly strip 86 hyperedges to obtain a 66-50 master and eventually obtain its class of critical non-KS NBMMPHs.

`1234`is of such a kind. Its filled MMPH shown in Figure 2c provides a coordinatization necessary for the implementation of the 4-3. The 16-9 critical of the 20-10 master shown in Figure 2d contains two $m=1$ vertices (

`9,B`), because $m=1$ vertices were stripped only once (from the master) when we started the generation of the 20-10 class. We can remove one or both of these vertices and still have a critical non-KS MMPH (15-9 or 14-9, respectively) if we want to for some reason. The 16-9 critical shown in Figure 2e has a parity proof, since in it, each vertex shares exactly two hyperedges, while there is an odd number of them (9). Strings and coordinatizations are given in Appendix A.

#### 2.4. Dimensions Six to Eight

**M1**to the 81-162 class. Their distribution is shown in Figure 3a in black. The $\{0,\pm 1\}$ set generates a 236-1216 master. Its non-KS NBMMPHs are also obtained via

**M1**and are shown in Figure 3a in green.

**M1**several times to obtain small non-KS critical NBMMPHs. As a result, hyperedges of all small NBMMPHs may contain some $m=1$ vertices essential for criticality, as shown in Figure 3f (the removal of vertex

`6`would terminate the criticality of the MMPH). In dimensions greater than nine, such vertices do not appear, although even here we can avoid their generation by applying

**M3**to KS NBMMPHs, as shown in Figure 3g.

**M1**. However, the MMPHs with $m=1$ vertices are also big, and obtaining small criticals with up to 40 hyperedges would require roughly one week on a supercomputer with 200 2.5 GHz CPUs working in parallel. We may be able to work around this problem by exploiting previously generated small KS criticals [52] so as to use them as masters for non-KS MMPHs while applying

**M3**, as shown in Figure 3h–j (cf. the six-dim star in Figure 3b). Notice the graphical similarity of the four-dim 18-9 ([51] Figure 3a) and eight-dim 36-9 (shown in Figure 3h) for each vertex from the 18-9 vs. a pair of vertices in the 36-9. Since the distribution of eight-dim KS MMPHs in Ref. [52] is abundant, we can arbitrarily generate many non-KS NBMMPHs in this manner via

**M3**.

#### 2.5. Dimensions Nine to Eleven

**M1**applicable. Thus, after the random stripping of 12,068,200 hyperedges, we obtained submasters with 505 hyperedges. By requiring that at least one of the hyperedges contains n vertices and that some of them can have the multiplicity $m=1$, our program STATES01 yields a series of critical NBMMPHs, the smallest of which is 13-6, as shown in Figure 4a. The hyperedge

`4ac7efhK2`contains nine vertices. (Notice also that the 13-6 NBMMPH remains a critical non-KS NBMMPH with any, some, or all of

`a,c,e,f,h,K`removed.) The filled 13-16, i.e., 44-6, also shown in Figure 4a, obtains the coordinatization directly from the supermaster, since the programs preserve the names of the vertices in the process of stripping and yielding sub-MMPHs. Obtaining a coordinatization via VECFIND takes too many CPU hours. The latter feature also makes

**M2**inapplicable.

**M3**, so as to apply it on KS NBMMPHs obtained via dimensional upscaling [55,56], as follows. We removed several vertices from the smallest critical 47-16 obtained in [56] until it was not critical any more. Then, STATES01 yielded the 19-8 critical shown in Figure 4b. (The removal of vertex

`L`would terminate the criticality of the MMPH as with the seven-dim one shown in Figure 3f, but that would not affect the full n-vertex requirement.)

**M3**. The procedure consists of removing vertices and/or hyperedges in such a way that an NBMMPH stops being critical, which enables us to generate smaller critical non-KS NBMMPHs from it via STATES01.

**M3**.

#### 2.6. Dimensions 12 to 16

## 3. Discussion

## 4. Methods

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MMPH | McKay–Megill–Pavičić hypergraph (Definition 1) |

NBMMPH | Non-binary McKay–Megill–Pavičić hypergraph (Definition 2) |

BMMPH | Binary McKay–Megill–Pavičić hypergraph (Definition 5) |

KS | Kochen–Specker (Definition 3) |

non-KS | Non-Kochen–Specker (Definition 4) |

M1, M2, M3 | Methods 1,2,3 (Section 2.2) |

## Appendix A. SCII Strings of Non-KS MMPH Classes and Their Masters and Supermasters

#### Appendix A.1. Three-dim MMPHs

**8-7**(KS “bug”)

`123,34,45,567,78,81,26.`

**13-7**(filled

**8-7**)

`123,394,4A5,567,7B8,8C1,2D6. 1`=(0,0,1),

`2`=(0,1,0),

`3`=(1,0,0),

`4`=(0,1,1),

`5`=(1,1,−1),

`6`=(1,0,1),

`7`=(−1,2,1),

`8`=(2,1,0),

`9`=(0,1,−1),

`A`=(2,−1,1),

`B`=(1,−2,5),

`C`=(1,−2,0),

`D`=(1,0,−1)

#### Appendix A.2. Four-dim MMPHs

**4-3**

`12,34,1234.`

**8-3**(filled

**4-3**)

`1562,3784,1234. 1`=(0,0,0,1),

`2`=(0,0,1,0),

`3`=(0,1,0,0),

`4`=(1,0,0,0),

`5`=(1,1,0,0),

`6`=(1,−1,0,0),

`7`=(0,0,1,1),

`8`=(0,0,1,−1)

**16-9**

`3124,49A8,8567,7BC3,DE,FG,GE62,FD51,FECA.`

**20-9**(filled

**16-9**)

`1`=(1,0,0,−1),

`2`=(0,1,1,0),

`3`=(1,1,−1,1),

`4`=(1,−1,1,1),

`5`=(1,0,0,1),

`6`=(0,1,−1,0),

`7`=(1,1,1,−1),

`8`=(−1,1,1,1),

`9`=(0,0,1,−1),

`A`=(1,1,0,0),

`B`=(0,0,1,1),

`C`=(1,−1,0,0),

`D`=(0,1,0,0),

`E`=(0,0,0,1),

`F`=(0,0,1,0),

`G`=(1,0,0,0),

`H`=(1,0,1,0),

`I`=(1,0,−1,0),

`J`=(0,1,0,1),

`K`=(0,1,0,−1)

**24-24**(Peres’ supermaster)

`LMNO,HIJK,DEFG,BCFG,9ADE,78EG,56DF,5678,9ABC,68JK,57HI,ACIK,9BHJ,1234,4DGO,3EFN,258M,167L,19CM,2ABL,3HKO,4IJN,34NO,12LM. 1`=(0,0,0,1),

`2`=(0,0,1,0),

`3`=(1,1,0,0),

`4`=(1,−1,0,0),

`5`=(0,1,0,−1),

`6`=(1,0,−1,0),

`7`=(1,0,1,0),

`8`=(0,1,0,1),

`9`=(0,1,−1,0),

`A`=(1,0,0,−1),

`B`=(1,0,0,1),

`C`=(0,1,1,0),

`D`=(1,1,1,1),

`E`=(1,−1,−1,1),

`F`=(1,−1,1,−1),

`G`=(1,1,−1,−1),

`H`=(−1,1,1,1),

`I`=(1,1,−1,1),

`J`=(1,1,1,−1),

`K`=(1,−1,1,1),

`L`=(0,1,0,0),

`M`=(1,0,0,0),

`N`=(0,0,1,1),

`O`=(0,0,1,−1)

**16-9**

`231,1BC6,654,45GF,FGED,DE32,789A,7BC8,9A.`

**20-9**(filled

**16-9**)

`2H31,1BC6,6I54,45GF,FGED,DE32,789A,7BC8,9JKA. 1`=(0,0,0,$\varphi $),

`2`=($\varphi $−1,0,−$\varphi $,0),

`3`=($\varphi $,0,$\varphi $−1,0),

`4`=(0,$\varphi $,0,$\varphi $),

`5`=(0,$\varphi $,0,−$\varphi $),

`6`=(0,0,$\varphi $,0),

`7`=(0,0,$\varphi $,$\varphi $),

`8`=(0,0,$\varphi $,−$\varphi $),

`9`=($\varphi $,$\varphi $−1,0,0),

`A`=($\varphi $−1,−$\varphi $,0,0),

`B`=($\varphi $,$\varphi $,0,0),

`C`=($\varphi $,−$\varphi $,0,0),

`D`=(0,$\varphi $−1,0,−$\varphi $),

`E`=(0,$\varphi $,0,$\varphi $−1),

`F`=($\varphi $,0,$\varphi $,0),

`G`=($\varphi $,0,−$\varphi $,0),

`H`=(0,$\varphi $,0,0),

`I`=($\varphi $,0,0,0),

`J`=(0,0,$\varphi $,$\varphi $−1),

`K`=(0,0,$\varphi $−1,−$\varphi $)

**60-72**(supermaster)

`1234,1256,1278,129A,13BC,13DE,13FG,1HI4,1JK4,1LM4,23NO,23PQ,23RS,2TU4,2VW4,2XY4,Za34,Za56,Za78,Za9A,Z5bc,Zde6,fg34,fg56,fg78,fg9A,hi34,hi56,hi78,hi9A,ajk6,a5lm,ano6,apq6,TUBC,TUDE,TUFG,TBbo,TCmd,VWBC,VWDE,VWFG,XYBC,XYDE,XYFG,HINO,HIPQ,HIRS,HNco,HOme,JKNO,JKPQ,JKRS,LMNO,LMPQ,LMRS,UBle,UnCc,UrsC,UtCu,jkbc,INld,InOb,IvwO,IxOy,nbco,rsbo,pbcq,vwco,tbuo,xyco,lmde.`

`1`=(0,0,0,$\varphi $),

`2`=(0,0,$\varphi $,0),

`3`=(0,$\varphi $,0,0),

`4`=($\varphi $,0,0,0),

`5`=($\varphi $,$\varphi $,0,0),

`6`=($\varphi $,−$\varphi $,0,0),

`7`=($\varphi $,$\varphi $−1,0,0),

`8`=($\varphi $−1,−$\varphi $,0,0),

`9`=(−$\varphi $,$\varphi $−1,0,0),

`A`=($\varphi $−1,$\varphi $,0,0),

`B`=($\varphi $,0,$\varphi $,0),

`C`=($\varphi $,0,−$\varphi $,0),

`D`=(−$\varphi $,0,$\varphi $−1,0),

`E`=($\varphi $−1,0,$\varphi $,0),

`F`=($\varphi $,0,$\varphi $−1,0),

`G`=($\varphi $−1,0,−$\varphi $,0),

`H`=(0,$\varphi $,$\varphi $,0),

`I`=(0,$\varphi $,−$\varphi $,0),

`J`=(0,$\varphi $−1,$\varphi $,0),

`K`=(0,−$\varphi $,$\varphi $−1,0),

`L`=(0,$\varphi $−1,−$\varphi $,0),

`M`=(0,$\varphi $,$\varphi $−1,0),

`N`=($\varphi $,0,0,$\varphi $),

`O`=($\varphi $,0,0,−$\varphi $),

`P`=($\varphi $,0,0,$\varphi $−1),

`Q`=($\varphi $−1,0,0,−$\varphi $),

`R`=(−$\varphi $,0,0,$\varphi $−1),

`S`=($\varphi $−1,0,0,$\varphi $),

`T`=(0,$\varphi $,0,$\varphi $),

`U`=(0,$\varphi $,0,−$\varphi $),

`V`=(0,$\varphi $,0,$\varphi $−1),

`W`=(0,$\varphi $−1,0,−$\varphi $),

`X`=(0,−$\varphi $,0,$\varphi $−1),

`Y`=(0,$\varphi $−1,0,$\varphi $),

`Z`=(0,0,$\varphi $,$\varphi $),

`a`=(0,0,$\varphi $,−$\varphi $),

`b`=($\varphi $,−$\varphi $,−$\varphi $,$\varphi $),

`c`=($\varphi $,−$\varphi $,$\varphi $,−$\varphi $),

`d`=($\varphi $,$\varphi $,$\varphi $,−$\varphi $),

`e`=($\varphi $,$\varphi $,−$\varphi $,$\varphi $),

`f`=(0,0,$\varphi $,$\varphi $−1),

`g`=(0,0,$\varphi $−1,−$\varphi $),

`h`=(0,0,−$\varphi $,$\varphi $−1),

`i`=(0,0,$\varphi $−1,$\varphi $),

`j`=($\varphi $,$\varphi $,$\varphi $−1,$\varphi $−1),

`k`=($\varphi $−1,$\varphi $−1,−$\varphi $,−$\varphi $),

`l`=(−$\varphi $,$\varphi $,$\varphi $,$\varphi $),

`m`=($\varphi $,−$\varphi $,$\varphi $,$\varphi $),

`n`=($\varphi $,$\varphi $,$\varphi $,$\varphi $),

`o`=($\varphi $,$\varphi $,−$\varphi $,−$\varphi $),

`p`=(−$\varphi $,−$\varphi $,$\varphi $−1,$\varphi $−1),

`q`=($\varphi $−1,$\varphi $−1,$\varphi $,$\varphi $),

`r`=($\varphi $,$\varphi $−1,$\varphi $,$\varphi $−1),

`s`=($\varphi $−1,−$\varphi $,$\varphi $−1,−$\varphi $),

`t`=(−$\varphi $,$\varphi $−1,−$\varphi $,$\varphi $−1),

`u`=($\varphi $−1,$\varphi $,$\varphi $−1,$\varphi $),

`v`=($\varphi $,$\varphi $−1,$\varphi $−1,$\varphi $),

`w`=($\varphi $−1,−$\varphi $,−$\varphi $,$\varphi $−1),

`x`=(−$\varphi $,$\varphi $−1,$\varphi $−1,−$\varphi $),

`y`=($\varphi $−1,$\varphi $,$\varphi $,$\varphi $−1)

#### Appendix A.3. Five-dim MMPHs

**7-5**

`41235,56,674,234,714.`

**16-5**(

**7-5**filled)

`41235,589A6,6BC74,2DE34,7FG14. 1`=(0,0,1,0,0),

`2`=(1,−1,0,0,0),

`3`=(1,1,0,0,0),

`4`=(0,0,0,0,1),

`5`=(0,0,0,1,0),

`6`=(0,1,1,0,0),

`7`=(1,0,0,1,0),

`8`=(1,0,0,0,−1),

`9`=(0,1,−1,0,0),

`A`=(1,0,0,0,1),

`B`=(1,−1,1,−1,0),

`C`=(1,1,−1,−1,0),

`D`=(0,0,1,1,0),

`E`=(0,0,1,−1,0),

`F`=(0,1,0,0,0),

`G`=(1,0,0,−1,0)

**16-9**

`63457,75B9E,EFGD,DC,CA86,12345,89125,AB,FGE.`

**26-9**(

**16-9**filled)

`63457,75B9E,EFHGD,DIJKC,CAL86,12345,89125,AMNOB,FPQGE. 1`=(1,1,1,−1,0),

`2`=(1,1,−1,1,0),

`3`=(1,−1,1,1,0),

`4`=(−1,1,1,1,0),

`5`=(0,0,0,0,1),

`6`=(0,0,1,−1,0),

`7`=(1,1,0,0,0),

`8`=(0,0,1,1,0),

`9`=(1,−1,0,0,0),

`A`=(0,1,0,0,1),

`B`=(0,0,1,0,0),

`C`=(1,0,0,0,0),

`D`=(0,1,1,0,0),

`E`=(0,0,0,1,0),

`F`=(1,0,0,0,1),

`G`=(0,1,−1,0,0),

`H`=(1,0,0,0,−1),

`I`=(0,0,0,1,1),

`J`=(0,1,−1,1,−1),

`K`=(0,1,−1,−1,1),

`L`=(0,1,0,0,−1),

`M`=(1,1,0,−1,−1),

`N`=(1,−1,0,−1,1),

`O`=(1,0,0,1,0),

`P`=(1,1,1,0,−1),

`Q`=(−1,1,1,0,1)

**105-136**(supermaster)

`12345,12367,12489,12AB5,134CD,13EF5,1GH45,1GH67,1G6IJ`,1GKL7,1H6MN,1HOP7,1EF89,1E8IP,1E9KN,1F8ML,1F9OJ,1ABCD,1ACJP,1ADLN,1QRST,1QUVW,1XYSZ,1XabW,1BCMK,1BDOI,1cYVd,1caeT,1fRbd,1fUeZ,1OIJP,1MKLN,234gh,23ij5,2kl45,2kl67,2k6mn,2kop7,2l6qr,2lst7,2ij89,2i8mt,2i9or,2j8qp,2j9sn,2ABgh,2Agtn,2Ahpr,2uvSw,2uxyW,2z!S",2z#$W, 2Bgoq,2Bhsm,2%!yd,2%#ew,2&v$d,2&xe",2smtn,2oqpr,’(345,’(367,’(489,’(AB5,’36)*,’3-/7, ’48:;,’4<=9,’A>?5,’@[B5,(36\],(3^_7,(48‘{,(4|}9,(A∼+15,(+2+3B5,3ijCD,3iC_*,3iD-\,3jC/],3jD^),3EFgh,3Eg_),3Eh/\,3+4+5Vw,3+4+6yT,3+7+8V",3+7+9$T,3Fg-],3Fh^*,3+A+8yZ, 3+A+9bw,3+B+5$Z,3+B+6b",3^_)*,3-/\],kl4CD,klEF5,k4C};,k4D<‘,kE+3?5,kF@∼5, l4C={,l4D|:, lE[+15,lF+2>5,GH4gh,GHij5,G4g}:,G4h=‘,Gi+3>5,Gj[∼5,+C4+5Rx,+C4+6vU,+C+7X%5,+C+Azc5, +D4+8R#,+D4+9!U,+D+4X&5,+D+Buc5,H4g<{,H4h|;,Hi@+15, Hj+2?5,+E4+8va,+E4+9Yx,+E+4zf5, +E+BQ%5,+F4+5!a,+F4+6Y#,+F+7uf5,+F+AQ&5,4|}:;,4<=‘{,+2+3>?5,@[∼+15.

`1`=(0,0,0,0,1),

`2`=(0,0,0,1,0),

`’`=(0,0,0,1,1),

`(`=(0,0,0,1,−1),

`3`=(0,0,1,0,0),

`k`=(0,0,1,0,1),

`l`=(0,0,1,0,−1),

`G`=(0,0,1,1,0),

`+C`=(0,0,1,1,1),

`+D`=(0,0,1,1,−1),

`H`=(0,0,1,−1,0),

`+E`=(0,0,1,−1,1),

`+F`=(0,0,−1,1,1),

`4`=(0,1,0,0,0),

`i`=(0,1,0,0,1),

`j`=(0,1,0,0,−1),

`E`=(0,1,0,1,0),

`+4`=(0,1,0,1,1),

`+7`=(0,1,0,1,−1),

`F`=(0,1,0,−1,0),

`+A`=(0,1,0,−1,1),

`+B`=(0,−1,0,1,1),

`A`=(0,1,1,0,0),

`u`=(0,1,1,0,1),

`z`=(0,1,1,0,−1),

`Q`=(0,1,1,1,0),

`+2`=(0,1,1,1,1),

`@`=(0,1,1,1,−1),

`X`=(0,1,1,−1,0),

`[`=(0,1,1,−1,1),

`+3`=(0,1,1,−1,−1),

`B`=(0,1,−1,0,0),

`%`=(0,1,−1,0,1),

`&`=(0,−1,1,0,1),

`c`=(0,1,−1,1,0),

`∼`=(0,1,−1,1,1),

`>`=(0,1,−1,1,−1),

`f`=(0,−1,1,1,0),

`?`=(0,1,−1,−1,1),

`+1`=(0,−1,1,1,1),

`5`=(1,0,0,0,0),

`g`=(1,0,0,0,1),

`h`=(1,0,0,0,−1),

`C`=(1,0,0,1,0),

`+8`=(1,0,0,1,1),

`+5`=(1,0,0,1,−1),

`D`=(1,0,0,−1,0),

`+6`=(1,0,0,−1,1),

`+9`=(−1,0,0,1,1),

`8`=(1,0,1,0,0),

`!`=(1,0,1,0,1),

`v`=(1,0,1,0,−1),

`Y`=(1,0,1,1,0),

`|`=(1,0,1,1,1),

`<`=(1,0,1,1,−1),

`R`=(1,0,1,−1,0),

`=`=(1,0,1,−1,1),

`}`=(1,0,1,−1,−1),

`9`=(1,0,−1,0,0),

`x`=(1,0,−1,0,1),

`#`=(−1,0,1,0,1),

`U`=(1,0,−1,1,0),

`‘`=(1,0,−1,1,1),

`:`=(1,0,−1,1,−1),

`a`=(−1,0,1,1,0),

`;`=(1,0,−1,−1,1),

`{`=(−1,0,1,1,1),

`6`=(1,1,0,0,0),

`$`=(1,1,0,0,1),

`y`=(1,1,0,0,−1),

`b`=(1,1,0,1,0), ^=(1,1,0,1,1),

`−`=(1,1,0,1,−1),

`V`=(1,1,0,−1,0),

`/`=(1,1,0,−1,1),

`_`=(1,1,0,−1,−1),

`e`=(1,1,1,0,0),

`s`=(1,1,1,0,1),

`o`=(1,1,1,0,−1),

`O`=(1,1,1,1,0),

`M`=(−1,1,1,1,0),

`I`=(1,−1,−1,1,0),

`K`=(1,1,1,−1,0),

`q`=(−1,1,1,0,1),

`m`=(1,−1,−1,0,1),

`S`=(1,1,−1,0,0),

`p`=(1,1,−1,0,1),

`t`=(1,1,−1,0,−1),

`L`=(1,1,−1,1,0),

`d`=(−1,1,1,0,0),

`J`=(1,−1,1,−1,0),

`P`=(1,1,−1,−1,0),

`N`=(1,−1,1,1,0),

`n`=(1,−1,1,0,−1),

`7`=(1,−1,0,0,0),

`w`=(1,−1,0,0,1),

`"`=(−1,1,0,0,1),

`T`=(1,−1,0,1,0),

`\`=(1,−1,0,1,1),

`)`=(1,−1,0,1,−1),

`Z`=(−1,1,0,1,0),

`*`=(1,−1,0,−1,1),

`]`=(−1,1,0,1,1),

`W`=(1,−1,1,0,0),

`r`=(1,−1,1,0,1)

**31-9**(

**10-9**filled)

`17835,27846,9BCD1,9EFG7,9HIJ2,8KLMA,3NOP4,6QRSA,5TUVA. 1`=(0,0,1,0,−1),

`2`=(0,0,0,1,−1),

`3`=(0,0,0,1,0),

`4`=(0,0,1,0,0),

`5`=(0,0,1,0,1),

`6`=(0,0,0,1,1),

`7`=(0,1,0,0,0),

`8`=(1,0,0,0,0),

`9`=(0,0,1,1,1),

`A`=(0,1,1,1,−1),

`B`=(0,0,−1,2,−1),

`C`=(1,2,0,0,0),

`D`=(2,−1,0,0,0),

`E`=(0,0,−1,−1,2),

`F`=(1,0,1,−1,0),

`G`=(2,0,−1,1,0),

`H`=(0,0,2,−1,−1),

`I`=(1,1,0,0,0),

`J`=(1,−1,0,0,0),

`K`=(0,0,1,−1,0),

`L`=(0,1,0,0,1),

`M`=(0,−1,1,1,1),

`N`=(0,0,0,0,1),

`O`=(−1,2,0,0,0),

`P`=(2,1,0,0,0),

`Q`=(0,1,1,−1,1),

`R`=(1,1,−1,0,0),

`S`=(2,−1,1,0,0),

`T`=(0,1,0,−1,0),

`U`=(2,1,−1,1,1),

`V`=(2,−1,1,−1,−1)

#### Appendix A.4. Six-dim MMPHs

**19-7**

`7!8gw,woO6i,i;)EB,B<:b7,’!)Jb6,’o:8Eu,;<OJgu.`

**11-7**

`w!g,wi6,’!)Jb6,’u,B)i,Jgu,Bb.`

**21-7**(

**11-7, 19-7**filled)

`w!78#g,woOi6%,’!)Jb6,’o:8Eu,;B)i#E,;<OJgu,B<7:b%.w`=($\omega $,1,1,${\omega}^{2}$,1,$\omega $),

`’`=(${\omega}^{2}$,$\omega $,$\omega $,1,1,1),

`;`=($\omega $,1,1,${\omega}^{2}$,$\omega $,1),

`!`=($\omega $,1,${\omega}^{2}$,1,$\omega $,1),

`B`=(1,${\omega}^{2}$,$\omega $,1,$\omega $,1),

`<`=($\omega $,1,$\omega $,1,1,${\omega}^{2}$),

`o`=(1,${\omega}^{2}$,1,1,1,${\omega}^{2}$),

`)`=($\omega $,${\omega}^{2}$,1,1,1,$\omega $),

`7`=(1,$\omega $,${\omega}^{2}$,1,1,$\omega $),

`O`=(1,$\omega $,$\omega $,${\omega}^{2}$,1,1),

`:`=(${\omega}^{2}$,1,1,1,$\omega $,$\omega $),

`8`=(1,$\omega $,1,${\omega}^{2}$,$\omega $,1),

`i`=(${\omega}^{2}$,1,${\omega}^{2}$,1,1,1),

`J`=(1,$\omega $,1,1,$\omega $,${\omega}^{2}$),

`#`=(1,1,1,1,${\omega}^{2}$,${\omega}^{2}$),

`b`=(1,1,1,$\omega $,1,1),

`6`=(1,1,$\omega $,1,${\omega}^{2}$,$\omega $),

`g`=(${\omega}^{2}$,${\omega}^{2}$,1,1,1,1),

`E`=(1,1,$\omega $,${\omega}^{2}$,1,$\omega $),

`%`=($\omega $,$\omega $,1,1,${\omega}^{2}$,1),

`u`=(1,1,${\omega}^{2}$,1,${\omega}^{2}$,1)

**79-162**(81-162 stripped; %,#))

`123456,12789A,1BCD5E,1B7FGH,1ICJ9K,1I3LGM,1NODAP,`1NQ4HR, 1SOJ6T,1SU8MR,1VQLET,1VUFKP,WXY45Z,WX7abA,Wcde5E,Wc7Ffg,WIdJbh,WIYifM,WjkeAP,WjQ4gl,WSkJZm,WSnaMl,WoQiEm,WonFhP,pqY89Z,pq3ab6,pcre9K,pc3Lsg,pBrDbh,pBYisH,pjte6T,pjU8gu,pNtDZm,pNnaHu,pvnLhT,pvUiKm,wxyD5Z,wx7azH,w!"e56,w!78g,wI"Lzh,wIyiK,w$keHR,w$ODgl,wVkLZ,wV&aKl,woOi6,wo&8hR,’(yDbA,’(Y4zH,’!)Jb6,’!Y8*M,’c)LzE,’cyF*K,’-kJHu,’-tDMl,’vkLA/,’v:4Kl,’otF6/,’o:8Eu,;("e9A,;(34g,;x)J9Z,;x3a*M,;B)iE,;B"F*h,;<teMR,;<OJgu,;vOiA=,;v>4hR,;VtFZ=,;V>aEu,?!reGM,?!CJsg,?qyFGZ,?qCazE,?2yisA,?2r4zh,?$:eET,?$UFg/,?-&JhT,?-UiM,?N:4Z,?N&aA/,@(deGH,@(CDfg,@X)LGZ,@XCa*K,@2)if6,@2d8*h,@<:eKP,@<QLg/,@->DhP,@-QiH=,@S:8Z=,@S>a6/,[xdJsH,[xrDfM,[X"LsA,[Xr4K,[q"Ff6,[qd8E,[<&JKm,[<nLM,[$>DEm,[$nFH=,[j>46,[j&8A=,(S"Uzm,(SynT,(VdUb,(VY&fT,(oCn9,(o3&Gm,xj)UzP,xjyQ*T,xvdU5/,xv7:fT,xorQ9/,xo3:sP,!N)nP,!N"Q*m,!vCn5=,!v7>Gm,!VrQb=,!VY>sP,X$)UbR,X$YO*T, X-"U5u,X-7tT,XorOGu,XoCtsR,q<yQbR,q<YOzP,q-"Q9l,q-3kP,qvdOGl,qvCkfR,2<yn5u,2<7tzm, 2$)n9l,2$3k*m,2Vdtsl,2Vrkfu,c-3&5=,c-7>9,cN)&fR,cNdO*,cSy>sR,cSrOz=,B<Y&5/,B<7:b,Bj)&Gl,BjCk*,BS":sl,BSrk/,I$Y>9/,I$3:b=,Ijy>Gu,IjCtz=,IN":fu,INdt/.

**81-162**

`123456,12789A,1BCD5E,1B7FGH,1ICJ9K,1I3LGM,1NODAP,1NQ4HR,1SOJ6T,1SU8MR,1VQLET,1VUFKP,WXY45Z,WX7abA,Wcde5E,Wc7Ffg,WIdJbh,WIYifM,WjkeAP,WjQ4gl,WSkJZm,WSnaMl,WoQiEm,WonFhP,pqY89Z,pq3ab6,pcre9K,pc3Lsg,pBrDbh,pBYisH,pjte6T,pjU8gu,pNtDZm,pNnaHu,pvnLhT,pvUiKm,wxyD5Z,wx7azH,w!"e56,w!78#g,wI"Lzh,wIyi#K,w$keHR,w$ODgl,wVkLZ%,wV&aKl,woOi6%,wo&8hR,’(yDbA,’(Y4zH,’!)Jb6,’!Y8*M,’c)LzE,’cyF*K,’-kJHu,’-tDMl,’vkLA/,’v:4Kl,’otF6/,’o:8Eu,;("e9A,;(34#g,;x)J9Z,;x3a*M,;B)i#E,;B"F*h,;<teMR,;<OJgu,;vOiA=,;v>4hR,;VtFZ=,;V>aEu,?!reGM,?!CJsg,?qyFGZ,?qCazE,?2yisA,?2r4zh,?$:eET,?$UFg/,?-&JhT,?-UiM%,?N:4Z%,?N&aA/,@(deGH,@(CDfg,@X)LGZ,@XCa*K,@2)if6,@2d8*h,@<:eKP,@<QLg/,@->DhP, @-QiH=,@S:8Z=,@S>a6/,[xdJsH,[xrDfM,[X"LsA,[Xr4#K,[q"Ff6,[qd8#E,[<&JKm,[<nLM%, [$>DEm,[$nFH=,[j>46%,[j&8A=,(S"Uzm,(Syn#T,(VdUb%,(VY&fT,(oCn9%,(o3&Gm,xj)UzP,xjyQ*T, xvdU5/,xv7:fT,xorQ9/,xo3:sP,!N)n#P,!N"Q*m,!vCn5=,!v7>Gm,!VrQb=,!VY>sP,X$)UbR,X$YO*T,X-"U5u,X-7t#T,XorOGu,XoCtsR,q<yQbR,q<YOzP,q-"Q9l,q-3k#P,qvdOGl,qvCkfR,2<yn5u,2<7tzm,2$)n9l,2$3k*m,2Vdtsl,2Vrkfu,c-3&5=,c-7>9%,cN)&fR,cNdO*%,cSy>sR,cSrOz=,B<Y&5/,B<7:b%,Bj)&Gl,BjCk*%,BS":sl,BSrk#/,I$Y>9/,I$3:b=,Ijy>Gu,IjCtz=,IN":fu,INdt#/. 1=($\omega $,1,1,1,1,1), W=($\omega $,1,1,1,${\omega}^{2}$,$\omega $),p=($\omega $,1,1,1,$\omega $,${\omega}^{2}$), w=($\omega $,1,1,${\omega}^{2}$,1,$\omega $),’`=(${\omega}^{2}$,$\omega $,$\omega $,1,1,1),

`;`=($\omega $,1,1,${\omega}^{2}$,$\omega $,1),

`?`=($\omega $,1,1,$\omega $,1,${\omega}^{2}$),

`@`=($\omega $,1,1,$\omega $,${\omega}^{2}$,1),

`[`=(1,${\omega}^{2}$,${\omega}^{2}$,1,1,1),

`(`=($\omega $,1,${\omega}^{2}$,1,1,$\omega $),

`x`=(${\omega}^{2}$,$\omega $,1,$\omega $,1,1),

`!`=($\omega $,1,${\omega}^{2}$,1,$\omega $,1),

`X`=(${\omega}^{2}$,$\omega $,1,1,$\omega $,1),

`q`=(${\omega}^{2}$,$\omega $,1,1,1,$\omega $),

`2`=(1,${\omega}^{2}$,$\omega $,$\omega $,1,1),

`c`=($\omega $,1,${\omega}^{2}$,$\omega $,1,1),

`B`=(1,${\omega}^{2}$,$\omega $,1,$\omega $,1),

`I`=(1,${\omega}^{2}$,$\omega $,1,1,$\omega $),

`<`=($\omega $,1,$\omega $,1,1,${\omega}^{2}$),

`$`=($\omega $,1,$\omega $,1,${\omega}^{2}$,1),

`-`=(1,${\omega}^{2}$,1,${\omega}^{2}$,1,1),

`j`=($\omega $,1,$\omega $,${\omega}^{2}$,1,1),

`N`=(1,${\omega}^{2}$,1,$\omega $,$\omega $,1),

`S`=(1,${\omega}^{2}$,1,$\omega $,1,$\omega $),

`v`=(1,${\omega}^{2}$,1,1,${\omega}^{2}$,1),

`V`=(1,${\omega}^{2}$,1,1,$\omega $,$\omega $),

`o`=(1,${\omega}^{2}$,1,1,1,${\omega}^{2}$),

`)`=($\omega $,${\omega}^{2}$,1,1,1,$\omega $),

`"`=(${\omega}^{2}$,1,$\omega $,$\omega $,1,1),

`y`=($\omega $,${\omega}^{2}$,1,1,$\omega $,1),

`d`=(${\omega}^{2}$,1,$\omega $,1,$\omega $,1),

`r`=(${\omega}^{2}$,1,$\omega $,1,1,$\omega $),

`C`=(1,$\omega $,${\omega}^{2}$,$\omega $,1,1),

`Y`=($\omega $,${\omega}^{2}$,1,$\omega $,1,1),

`3`=(1,$\omega $,${\omega}^{2}$,1,$\omega $,1),

`7`=(1,$\omega $,${\omega}^{2}$,1,1,$\omega $),

`k`=(${\omega}^{2}$,1,1,$\omega $,$\omega $,1),

`t`=(${\omega}^{2}$,1,1,$\omega $,1,$\omega $),

`O`=(1,$\omega $,$\omega $,${\omega}^{2}$,1,1),

`:`=(${\omega}^{2}$,1,1,1,$\omega $,$\omega $),

`>`=(${\omega}^{2}$,1,1,1,1,${\omega}^{2}$),

`&`=(${\omega}^{2}$,1,1,1,${\omega}^{2}$,1),

`Q`=(1,$\omega $,$\omega $,1,${\omega}^{2}$,1),

`n`=(${\omega}^{2}$,1,1,${\omega}^{2}$,1,1),

`U`=(1,$\omega $,$\omega $,1,1,${\omega}^{2}$),

`a`=($\omega $,${\omega}^{2}$,$\omega $,1,1,1),

`8`=(1,$\omega $,1,${\omega}^{2}$,$\omega $,1),

`4`=(1,$\omega $,1,${\omega}^{2}$,1,$\omega $),

`F`=(1,$\omega $,1,$\omega $,${\omega}^{2}$,1),

`i`=(${\omega}^{2}$,1,${\omega}^{2}$,1,1,1),

`L`=(1,$\omega $,1,$\omega $,1,${\omega}^{2}$),

`D`=(1,$\omega $,1,1,${\omega}^{2}$,$\omega $),

`J`=(1,$\omega $,1,1,$\omega $,${\omega}^{2}$),

`*`=(1,1,1,1,1,$\omega $),

`z`=(1,1,1,1,$\omega $,1),

`#`=(1,1,1,1,${\omega}^{2}$,${\omega}^{2}$),

`e`=(1,$\omega $,1,1,1,1),

`b`=(1,1,1,$\omega $,1,1),

`5`=(1,1,1,$\omega $,$\omega $,${\omega}^{2}$),

`9`=(1,1,1,$\omega $,${\omega}^{2}$,$\omega $),

`f`=(1,1,1,${\omega}^{2}$,1,${\omega}^{2}$),

`G`=(1,1,1,${\omega}^{2}$,$\omega $,$\omega $),

`s`=(1,1,1,${\omega}^{2}$,${\omega}^{2}$,1),

`Z`=(1,1,$\omega $,1,1,1),

`A`=(1,1,$\omega $,1,$\omega $,${\omega}^{2}$),

`6`=(1,1,$\omega $,1,${\omega}^{2}$,$\omega $),

`H`=(1,1,$\omega $,$\omega $,1,${\omega}^{2}$),

`g`=(${\omega}^{2}$,${\omega}^{2}$,1,1,1,1),

`M`=(1,1,$\omega $,$\omega $,${\omega}^{2}$,1),

`E`=(1,1,$\omega $,${\omega}^{2}$,1,$\omega $),

`K`=(1,1,$\omega $,${\omega}^{2}$,$\omega $,1),

`h`=($\omega $,$\omega $,${\omega}^{2}$,1,1,1),

`=`=($\omega $,$\omega $,1,1,1,${\omega}^{2}$),

`%`=($\omega $,$\omega $,1,1,${\omega}^{2}$,1),

`l`=(1,1,${\omega}^{2}$,1,1,${\omega}^{2}$),

`R`=(1,1,${\omega}^{2}$,1,$\omega $,$\omega $),

`u`=(1,1,${\omega}^{2}$,1,${\omega}^{2}$,1),

`/`=(1,1,${\omega}^{2}$,${\omega}^{2}$,1,1),

`m`=($\omega $,$\omega $,1,${\omega}^{2}$,1,1),

`P`=(1,1,${\omega}^{2}$,$\omega $,1,$\omega $),

`T`=(1,1,${\omega}^{2}$,$\omega $,$\omega $,1)

**31-16**

`237,7HG,GTUVRP,PRNQSM,MIJKL2,235,7235,3NOKLM,WXJRSM,WYZGRV,aOQbcM,YdHce,XIfbcM,fgUHce,gTUGH,YZdGH.`

**44-16**(master for

**31-16**)

`123456,72389A,723B5C,7DEFGH,2IJKLM,3NOKLM,PNQRSM,PTUGRV,WXJRSM,WYZGRV,aOQbcM,aYdHce,XIfbcM,fgUHce,gTUhGH,iYZdGH. 1`=(0,0,1,−1,1,0),

`7`=(0,0,−1,1,1,0),

`2`=(0,1,0,0,0,1),

`3`=(0,1,0,0,0,−1),

`P`=(0,1,0,0,1,0),

`W`=(0,1,0,0,−1,0),

`a`=(0,1,0,1,0,0),

`X`=(0,1,0,1,1,1),

`D`=(0,1,0,1,−1,0),

`N`=(0,1,0,1,−1,1),

`I`=(0,1,0,1,−1,−1),

`f`=(0,1,0,−1,0,0),

`O`=(0,1,0,−1,1,1),

`J`=(0,1,0,−1,1,−1),

`Q`=(0,−1,0,1,1,1),

`E`=(0,1,1,0,1,0),

`g`=(0,1,1,1,1,0),

`i`=(0,1,1,1,−1,0),

`Y`=(0,1,1,−1,1,0),

`T`=(0,1,1,−1,−1,0),

`Z`=(0,1,−1,1,1,0),

`U`=(0,1,−1,1,−1,0),

`F`=(0,−1,1,1,0,0),

`h`=(0,1,−1,−1,1,0),

`d`=(0,−1,1,1,1,0),

`G`=(1,0,0,0,0,1),

`H`=(1,0,0,0,0,−1),

`8`=(1,0,0,1,−1,0),

`B`=(1,0,0,−1,1,0),

`4`=(−1,0,0,1,1,0),

`9`=(1,0,1,0,1,0),

`b`=(1,0,1,0,1,−1),

`c`=(1,0,1,0,−1,1),

`5`=(1,0,1,1,0,0),

`R`=(1,0,1,1,0,−1),

`K`=(1,0,1,1,1,0),

`S`=(1,0,1,−1,0,1),

`L`=(1,0,1,−1,−1,0),

`M`=(1,0,−1,0,0,0),

`6`=(1,0,−1,0,1,0),

`e`=(1,0,−1,0,1,1),

`C`=(−1,0,1,0,1,0),

`A`=(−1,0,1,1,0,0),

`V`=(−1,0,1,1,0,1)

**117-116**(stripmaster of the 236-1216 supermaster)

`123456,123789,1AB4CD,1AB7EF,1GHI5F,1GHJC9,1KLI8D,1KLJE6,MN8COP,MNF6QR,STE5OP,ST9DQR,UVWXYZ,UVabcd,UefgYZ,Uefabh,Uijgcd,UijWXh,UBk4XZ,UBk7ac,U3l4bd,U3l7WY,ULmIXY,ULmJbc,UHnIad,UHnJWZ,opVgYZ,opVabh,opijgh,opabqR,opYZPr,oistuv,oiwxyz,oj!"#$,oj%&’(,)*Vgcd,)*VWXh,)*efgh,)*WXqR,)*cdPr,)e-/uv,)e:;yz,)f!"<=,)f%&>?,*e@[#$,*e\]’(,*fst^_,*fwx‘{,pi@[<=,pi\]>?,pj-/^_,pj:;‘{,2|V4XZ,2|V7ac,2|3l47,2|ac}∼,2|XZ+1+2,2389u{,2356^z,2l\"+3+4,2l%[+5+6,A+7V4bd,A+7V7WY, A+7Bk47,A+7WY}∼,A+7bd+1+2,ABEFu{,ABCD^z,Ak\"+8+9,Ak%[+A+B,G+CVIXY,G+CVJbc,G+CHnIJ,G+Cbc+D+E,G+CXY+F+G,GHC9#?,GH5F<(,Gn:t+3+B,Gnw/+8+6,K+HVIad,K+HVJWZ,K+HLmIJ,K+HWZ+D+E,K+Had+F+G,KLE6#?,KL8D<(,Km:t+5+9,Kmw/+A+4,+7B@&+3+4,+7B!]+5+6,+7k89y_,+7k56‘v,|3@&+8+9,|3!]+A+B,|lEFy_,|lCD‘v,+HL-x+3+B,+HLs;+8+6,+HmC9’=,+Hm5F>$,+CH-x+5+9,+CHs;+A+4,+CnE6’=,+Cn8D>$,efab+IO,efYZQ+J,ijWX+IO,ijcdQ+J,Bkac+K+L, BkXZ+M+N,3lWY+K+L,3lbd+M+N,Lmbc+O+P,LmXY+Q+R,HnWZ+O+P,Hnad+Q+R. 1`=(0,1,0,−1,0,0),

`M`=(0,1,0,−1,1,1),

`S`=(0,1,0,−1,1,−1),

`T`=(0,1,0,−1,−1,1),

`N`=(0,−1,0,1,1,1),

`U`=(0,1,1,0,0,0),

`o`=(0,1,1,0,1,1),

`)`=(0,1,1,0,1,−1),

`*`=(0,1,1,0,−1,1),

`p`=(0,1,1,0,−1,−1),

`2`=(0,1,1,1,0,1),

`A`=(0,1,1,1,0,−1),

`G`=(0,1,1,1,1,0),

`K`=(0,1,1,1,−1,0),

`+7`=(0,1,1,−1,0,1),

`|`=(0,1,1,−1,0,−1),

`+H`=(0,1,1,−1,1,0),

`+C`=(0,1,1,−1,−1,0),

`V`=(0,1,−1,0,0,0),

`e`=(0,1,−1,0,1,1),

`i`=(0,1,−1,0,1,−1),

`j`=(0,1,−1,0,−1,1),

`f`=(0,−1,1,0,1,1),

`B`=(0,1,−1,1,0,1),

`3`=(0,1,−1,1,0,−1),

`L`=(0,1,−1,1,1,0),

`H`=(0,1,−1,1,−1,0),

`l`=(0,1,−1,−1,0,1),

`k`=(0,−1,1,1,0,1),

`n`=(0,1,−1,−1,1,0),

`m`=(0,−1,1,1,1,0),

`I`=(1,0,0,0,0,1),

`J`=(1,0,0,0,0,−1),

`4`=(1,0,0,0,1,0),

`7`=(1,0,0,0,−1,0),

`g`=(1,0,0,1,0,0),

`W`=(1,0,0,1,1,1),

`a`=(1,0,0,1,1,−1),

`b`=(1,0,0,1,−1,1),

`X`=(1,0,0,1,−1,−1),

`h`=(1,0,0,−1,0,0),

`c`=(1,0,0,−1,1,1),

`Y`=(1,0,0,−1,1,−1),

`Z`=(1,0,0,−1,−1,1),

`d`=(−1,0,0,1,1,1),

`E`=(1,0,1,0,1,1),

`8`=(1,0,1,0,1,−1),

`C`=(1,0,1,0,−1,1),

`5`=(1,0,1,0,−1,−1),

`−`=(1,0,1,1,0,1),

`s`=(1,0,1,1,0,−1),

`@`=(1,0,1,1,1,0),

`!`=(1,0,1,1,−1,0),

`:`=(1,0,1,−1,0,1),

`w`=(1,0,1,−1,0,−1),

`\`=(1,0,1,−1,1,0),

`%`=(1,0,1,−1,−1,0),

`9`=(1,0,−1,0,1,1),

`F`=(1,0,−1,0,1,−1),

`6`=(1,0,−1,0,−1,1),

`D`=(−1,0,1,0,1,1),

`t`=(1,0,−1,1,0,1),

`/`=(1,0,−1,1,0,−1),

`"`=(1,0,−1,1,1,0),

`[`=(1,0,−1,1,−1,0),

`x`=(1,0,−1,−1,0,1),

`;`=(−1,0,1,1,0,1),

`&`=(1,0,−1,−1,1,0),

`]`=(−1,0,1,1,1,0),

`+A`=(1,1,0,0,1,1),

`+5`=(1,1,0,0,1,−1),

`+8`=(1,1,0,0,−1,1),

`+3`=(1,1,0,0,−1,−1),

`>`=(1,1,0,1,0,1),

`’`=(1,1,0,1,0,−1),

`‘`=(1,1,0,1,1,0),

`y`=(1,1,0,1,−1,0),

`<`=(1,1,0,−1,0,1),

`#`=(1,1,0,−1,0,−1), ^=(1,1,0,−1,1,0),

`u`=(1,1,0,−1,−1,0),

`+Q`=(1,1,1,0,0,1),

`+O`=(1,1,1,0,0,−1),

`+M`=(1,1,1,0,1,0),

`+K`=(1,1,1,0,−1,0),

`Q`=(1,1,1,1,0,0),

`+I`=(1,1,1,−1,0,0),

`O`=(−1,1,1,1,0,0),

`+F`=(1,1,−1,0,0,1),

`+D`=(1,1,−1,0,0,−1),

`+1`=(1,1,−1,0,1,0),

`}`=(1,1,−1,0,−1,0),

`P`=(1,1,−1,1,0,0),

`+J`=(1,−1,−1,1,0,0),

`q`=(1,1,−1,−1,0,0),

`+L`=(−1,1,1,0,1,0),

`+N`=(1,−1,−1,0,1,0),

`+6`=(1,−1,0,0,1,1),

`+B`=(1,−1,0,0,1,−1),

`+4`=(1,−1,0,0,−1,1),

`+9`=(−1,1,0,0,1,1),

`(`=(1,−1,0,1,0,1),

`?`=(1,−1,0,1,0,−1),

`z`=(1,−1,0,1,1,0),

`{`=(1,−1,0,1,−1,0),

`$`=(1,−1,0,−1,0,1),

`=`=(−1,1,0,1,0,1),

`v`=(1,−1,0,−1,1,0),

`_`=(−1,1,0,1,1,0),

`+G`=(1,−1,1,0,0,1),

`+E`=(1,−1,1,0,0,−1),

`+2`=(1,−1,1,0,1,0),

`∼`=(1,−1,1,0,−1,0),

`r`=(1,−1,1,1,0,0),

`+P`=(−1,1,1,0,0,1),

`+R`=(1,−1,−1,0,0,1),

`R`=(1,−1,1,−1,0,0)

#### Appendix A.5. Seven-dim MMPHs

**14-8**

`12567,189A5BC,189DE7,189HJ,189HB,2D,2EC,AJ.`

**31-13**

`1234,189DEF,189GHIJ,189KHBL,2MNDOIP,2MNEOCL,2MNGKF,QRNSAJP,QT4U,RTV9,WXMS,WYV8AJP,XY3U.`

**34-14**(master for

**14-8**and

**31-13**)

`1234567,189A5BC,189DE7F,189GHIJ,189KHBL,2MNDOIP,2MNEOCL,2MNGK6F,QRNSAJP,QT4U567,RTV9567,WXMS567,WYV8AJP,XY3U567. 1`=(0,0,0,1,0,0,0);

`2`=(0,0,1,0,0,0,0);

`3`=(1,−1,0,0,0,0,0);

`4`=(1,1,0,0,0,0,0);

`5`=(0,0,0,0,0,0,1);

`6`=(0,0,0,0,1,1,0);

`7`=(0,0,0,0,1,−1,0);

`8`=(0,1,−1,0,0,0,0);

`9`=(0,1,1,0,0,0,0);

`A`=(0,0,0,0,1,0,0);

`B`=(1,0,0,0,0,−1,0);

`C`=(1,0,0,0,0,1,0);

`D`=(1,0,0,0,1,1,−1);

`E`=(−1,0,0,0,1,1,1);

`F`=(1,0,0,0,0,0,1);

`G`=(1,0,0,0,1,−1,−1);

`H`=(1,0,0,0,1,1,1);

`I`=(1,0,0,0,−1,0,0);

`J`=(0,0,0,0,0,1,−1);

`K`=(1,0,0,0,−1,1,−1);

`L`=(0,0,0,0,1,0,−1);

`M`=(0,1,0,1,0,0,0);

`N`=(0,1,0,−1,0,0,0);

`O`=(1,0,0,0,1,−1,1);

`P`=(0,0,0,0,0,1,1);

`Q`=(−1,1,1,1,0,0,0);

`R`=(1,1,−1,1,0,0,0);

`S`=(1,0,1,0,0,0,0);

`T`=(1,−1,1,1,0,0,0);

`U`=(0,0,1,−1,0,0,0);

`V`=(1,0,0,−1,0,0,0);

`W`=(1,−1,−1,1,0,0,0);

`X`=(1,1,−1,−1,0,0,0);

`Y`=(1,1,1,1,0,0,0)

#### Appendix A.6. Eight-dim MMPHs

**15-9**

`17426538,8E,E2M,M3N,N4O,O5U,U7Q,Q6H,H1.`

**36-9**(master for

**15-9**)

`17426538,8ABDF9CE,ELaR2YVM,M3WSDKZN,NCJXR4TO,OV5PSBIU,UZ9GXa7Q,QTY6PWAH,HIKFGJL1.`

`1`=(0,0,0,0,0,0,0,1),

`2`=(0,0,0,0,0,0,1,0),

`3`=(0,0,0,0,0,1,0,0),

`4`=(0,0,0,0,1,0,0,0),

`5`=(0,0,1,1,0,0,0,0),

`6`=(0,0,−1,1,0,0,0,0),

`7`=(1,1,0,0,0,0,0,0),

`8`=(−1,1,0,0,0,0,0,0),

`9`=(0,0,0,0,0,0,1,1),

`A`=(0,0,1,1,1,−1,0,0),

`B`=(1,1,0,0,0,0,−1,1),

`C`=(1,1,0,0,0,0,1,−1),

`D`=(0,0,−1,0,1,0,0,0),

`E`=(0,0,0,1,0,1,0,0),

`F`=(0,0,1,−1,1,1,0,0),

`G`=(0,0,0,1,1,0,0,0),

`H`=(0,0,1,1,−1,1,0,0),

`I`=(1,0,0,0,0,0,1,0),

`J`=(0,0,−1,0,0,1,0,0),

`K`=(−1,0,0,0,0,0,1,0),

`L`=(0,1,0,0,0,0,0,0),

`M`=(0,0,1,0,1,0,0,0),

`N`=(0,0,0,1,0,0,0,0),

`O`=(−1,1,0,0,0,0,1,1),

`P`=(0,0,0,0,1,1,0,0),

`Q`=(−1,1,0,0,0,0,1,−1),

`R`=(1,0,0,0,0,0,0,1),

`S`=(0,−1,0,0,0,0,0,1),

`T`=(0,−1,0,0,0,0,1,0),

`U`=(0,0,−1,1,−1,1,0,0),

`V`=(0,0,−1,1,1,−1,0,0),

`W`=(1,1,0,0,0,0,1,1),

`X`=(0,0,1,0,0,1,0,0),

`Y`=(−1,0,0,0,0,0,0,1),

`Z`=(−1,1,0,0,0,0,−1,1),

`a`=(0,0,−1,−1,1,1,0,0)

#### Appendix A.7. Nine-dim MMPHs

**13-6**

`SU,1G42U,1S,472acefhK,G72,4U.`

**44-6**(

**13-6**filled)

`SUCDEFOQ6,1G42U8H95,1SAIMSVXbi,472acefhK,G72LWYZdg,4UBJ3NPRT. 1`=(0,0,0,0,0,0,0,1,0);

`2`=(0,0,0,0,0,0,1,0,0);

`3`=(0,0,0,0,1,1,0,0,0);

`4`=(0,0,0,1,0,0,0,0,1);

`5`=(0,0,0,1,0,0,0,0,−1);

`6`=(0,0,0,1,0,0,−1,1,0);

`7`=(0,0,1,0,0,0,0,1,0);

`8`=(0,0,1,0,0,1,0,0,0);

`9`=(0,0,1,0,0,−1,0,0,0);

`A`=(0,0,1,0,−1,0,1,0,0);

`B`=(0,0,1,0,−1,1,1,0,0);

`C`=(0,0,−1,−1,1,1,0,1,1);

`D`=(0,0,1,−1,−1,−1,0,1,1);

`H`=(0,1,0,0,1,0,0,0,0);

`E`=(0,1,0,0,1,−1,1,1,−1);

`F`=(0,1,0,0,1,−1,−1,−1,1);

`G`=(0,1,0,0,−1,0,0,0,0);

`I`=(0,1,0,1,1,1,1,0,1);

`J`=(0,1,0,−1,0,0,0,−1,1);

`K`=(1,−1,1,1,−1,1,0,−1,−1);

`L`=(0,1,0,−1,1,1,0,0,0);

`M`=(0,1,0,−1,1,−1,1,0,−1);

`N`=(0,1,−1,0,0,0,1,1,0);

`O`=(0,1,1,1,0,1,1,0,1);

`P`=(0,1,1,1,1,−1,1,−1,−1);

`Q`=(0,1,1,−1,0,1,−1,0,−1);

`R`=(0,1,−1,0,0,0,1,1,0);

`S`=(0,−1,1,0,1,0,0,0,0);

`U`=(1,0,0,0,0,0,0,0,0);

`V`=(1,0,0,0,0,−1,0,0,1);

`W`=(1,0,0,1,0,1,0,0,1);

`X`=(1,0,0,−1,0,1,0,0,0);

`Y`=(1,0,0,−1,0,−1,0,0,1);

`Z`=(1,0,1,0,0,0,0,−1,−1);

`a`=(1,1,0,0,−1,−1,0,0,0);

`b`=(1,1,1,1,0,0,−1,0,−1);

`c`=(1,1,1,−1,1,1,0,−1,1);

`d`=(−1,1,1,1,1,−1,0,−1,1);

`e`=(1,−1,−1,−1,−1,1,0,1,1);

`f`=(1,1,−1,1,1,1,0,1,−1);

`g`=(1,1,−1,1,1,−1,0,1,−1);

`h`=(1,−1,0,0,1,−1,0,0,0);

`i`=(1,−1,−1,1,0,0,1,0,−1).

**19-8**

`1234567,129ABCDE,13FGH5IJE,GB6E,AJE,97E,LH4DE,FIE.`

**47-16**(master for

**19-8**)

`123456789,12ABCDEF,13HIJ5KL,1AMCLNOPQ,1BRSETUVQ,1H4567VWX,`1ICDEFYOX,1ZJ4FTUW9,1ZCDEFYP8,23abRSET,cdIeMD6f,cgaZMCLN,dghA4567,ijhBRk7f,ilbZJ4FT,jlHeSkKN.

`1`=(1,0,0,0,0,0,0,0,0),

`2`=(0,1,0,0,0,0,0,0,0),

`3`=(0,0,1,0,0,0,0,0,0),

`c`=(1,1,1,1,0,0,0,0,0),

`d`=(1,−1,1,−1,0,0,0,0,0),

`g`=(1,−1,−1,1,0,0,0,0,0),

`i`=(1,−1,−1,−1,0,0,0,0,0),

`j`=(1,−1,1,1,0,0,0,0,0),

`l`=(1,1,1,−1,0,0,0,0,0),

`h`=(1,1,0,0,0,0,0,0,0),

`A`=(0,0,1,1,0,0,0,0,0),

`B`=(0,0,1,−1,0,0,0,0,0),

`H`=(0,1,0,1,0,0,0,0,0),

`I`=(0,1,0,−1,0,0,0,0,0),

`e`=(1,0,−1,0,0,0,0,0,0),

`a`=(1,0,0,−1,0,0,0,0,0),

`b`=(1,0,0,1,0,0,0,0,0),

`Z`=(0,1,−1,0,0,0,0,0,0),

`J`=(0,0,0,0,1,0,0,0,0),

`4`=(0,0,0,0,0,1,0,0,0),

`5`=(0,0,0,0,0,0,1,0,0),

`M`=(0,0,0,0,1,1,1,1,0),

`C`=(0,0,0,0,1,−1,1,−1,0),

`D`=(0,0,0,0,1,−1,−1,1,0),

`R`=(0,0,0,0,1,−1,−1,−1,0),

`S`=(0,0,0,0,1,−1,1,1,0),

`k`=(0,0,0,0,1,1,1,−1,0),

`E`=(0,0,0,0,1,1,0,0,0),

`F`=(0,0,0,0,0,0,1,1,0),

`T`=(0,0,0,0,0,0,1,−1,0),

`K`=(0,0,0,0,0,1,0,1,0),

`L`=(0,0,0,0,0,1,0,−1,0),

`N`=(0,0,0,0,1,0,−1,0,0),

`6`=(0,0,0,0,1,0,0,−1,0),

`7`=(0,0,0,0,1,0,0,1,0),

`f`=(0,0,0,0,0,1,−1,0,0),

`Y`=(0,1,1,1,0,0,0,0,1),

`O`=(0,1,−1,1,0,0,0,0,−1),

`P`=(0,1,1,−1,0,0,0,0,−1),

`U`=(0,1,1,1,0,0,0,0,−1),

`V`=(0,1,−1,−1,0,0,0,0,−1),

`W`=(0,1,1,−1,0,0,0,0,1),

`Q`=(0,1,0,0,0,0,0,0,1),

`X`=(0,0,1,0,0,0,0,0,−1),

`8`=(0,0,0,1,0,0,0,0,−1),

`:9`=(0,0,0,1,0,0,0,0,1)

#### Appendix A.8. Ten-dim MMPHs

**18-9**

`1BC5DEFGH9,1BCKL9A,T5DEU,TPR,TbL9A,CbKP,bKG,bKFR,bKUHA.`

**50-15**(master for

**18-9**)

`12BCUVfgik,1DEJXYceoq,1DELMVWajk,1DEMNRSblm,1DEOPRTajk`,1DEOQYZajk,2GHLNXZajk,2GHPQUWblm,45EFUVcdmn,46GIJSTajk,46GIUVceoq,56ABUVdeij,78ACUVfhpq,79HIUVabop,89DFUVghln.

`1`=(1,0,0,0,0,0,0,0,0,0),

`2`=(0,1,0,0,0,0,0,0,0,0),

`3`=(0,0,1,0,0,0,0,0,0,0),

`4`=(1,1,1,1,0,0,0,0,0,0),

`5`=(1,−1,1,−1,0,0,0,0,0,0),

`6`=(1,−1,−1,1,0,0,0,0,0,0),

`7`=(1,−1,−1,−1,0,0,0,0,0,0),

`8`=(1,−1,1,1,0,0,0,0,0,0),

`9`=(1,1,1,−1,0,0,0,0,0,0),

`A`=(1,1,0,0,0,0,0,0,0,0),

`B`=(0,0,1,1,0,0,0,0,0,0),

`C`=(0,0,1,−1,0,0,0,0,0,0),

`D`=(0,1,0,1,0,0,0,0,0,0),

`E`=(0,1,0,−1,0,0,0,0,0,0),

`F`=(1,0,−1,0,0,0,0,0,0,0),

`G`=(1,0,0,−1,0,0,0,0,0,0),

`H`=(1,0,0,1,0,0,0,0,0,0),

`I`=(0,1,−1,0,0,0,0,0,0,0),

`J`=(0,0,0,0,1,0,0,0,0,0),

`K`=(0,0,0,0,0,1,0,0,0,0),

`L`=(0,0,1,0,1,1,1,0,0,0),

`M`=(0,0,1,0,−1,1,−1,0,0,0),

`N`=(0,0,1,0,−1,−1,1,0,0,0),

`O`=(0,0,1,0,−1,−1,−1,0,0,0),

`P`=(0,0,1,0,−1,1,1,0,0,0),

`Q`=(0,0,1,0,1,1,−1,0,0,0),

`R`=(0,0,1,0,1,0,0,0,0,0),

`S`=(0,0,0,0,0,1,1,0,0,0),

`T`=(0,0,0,0,0,1,−1,0,0,0),

`U`=(0,0,0,0,1,0,1,0,0,0),

`V`=(0,0,0,0,1,0,−1,0,0,0),

`W`=(0,0,1,0,0,−1,0,0,0,0),

`X`=(0,0,1,0,0,0,−1,0,0,0),

`Y`=(0,0,1,0,0,0,1,0,0,0),

`Z`=(0,0,0,0,1,−1,0,0,0,0),

`a`=(0,0,0,0,0,0,0,1,0,0),

`b`=(0,0,0,0,0,0,0,0,1,0),

`c`=(0,0,0,0,0,1,0,1,1,1),

`d`=(0,0,0,0,0,1,0,−1,1,−1),

`e`=(0,0,0,0,0,1,0,−1,−1,1),

`f`=(0,0,0,0,0,1,0,−1,−1,−1),

`g`=(0,0,0,0,0,1,0,−1,1,1),

`h`=(0,0,0,0,0,1,0,1,1,−1),

`i`=(0,0,0,0,0,1,0,1,0,0),

`j`=(0,0,0,0,0,0,0,0,1,1),

`k`=(0,0,0,0,0,0,0,0,1,−1),

`l`=(0,0,0,0,0,0,0,1,0,1),

`m`=(0,0,0,0,0,0,0,1,0,−1),

`n`=(0,0,0,0,0,1,0,0,−1,0),

`o`=(0,0,0,0,0,1,0,0,0,−1),

`p`=(0,0,0,0,0,1,0,0,0,1),

`q`=(0,0,0,0,0,0,0,1,−1,0)

#### Appendix A.9. Eleven-dim MMPHs

**19-8**

`123456789AB,1234567CDF,1GHKLMDA,27KL9,567MC,567B,H8F,G8F.`

**50-14**(master for

**19-8**)

`123456789AB,1234567CDEF,1GHIJKLMDAN,1GHIJKLOPQR,27STUVKL8FW,27STUVKL9QX,347YZabMDAN,567cdefMCXR,567cdefgOEW,567cdefgPBN,cdhijkJV8FW,eYlmnIUk9QX,fZoHTjmn8FW,abGShilo8FW. 1`=(0,0,1,1,1,1,0,0,0,0,0),

`2`=(0,0,1,−1,1,−1,0,0,0,0,0),

`3`=(0,0,0,1,0,−1,0,0,0,0,0),

`4`=(0,0,1,0,−1,0,0,0,0,0,0),

`5`=(0,1,0,0,0,0,0,0,0,0,0),

`6`=(1,0,0,0,0,0,0,0,0,0,0),

`7`=(0,0,0,0,0,0,1,0,0,0,0),

`8`=(0,0,0,1,0,0,0,0,0,0,0),

`9`=(0,0,1,0,0,0,0,0,0,0,0),

`A`=(0,0,0,0,0,1,0,0,0,0,0),

`B`=(0,0,0,0,1,0,0,0,0,0,0),

`C`=(1,−1,1,0,1,0,0,0,0,0,0),

`D`=(1,1,0,1,0,1,0,0,0,0,0),

`E`=(1,1,0,−1,0,−1,0,0,0,0,0),

`F`=(−1,1,1,0,1,0,0,0,0,0,0),

`G`=(0,1,−1,1,0,0,1,0,0,0,0),

`H`=(1,0,1,1,0,0,0,−1,0,0,0),

`I`=(1,0,0,0,1,1,0,1,0,0,0),

`J`=(0,1,0,0,−1,1,−1,0,0,0,0),

`K`=(0,0,1,0,−1,0,1,1,0,0,0),

`L`=(0,0,0,1,0,−1,−1,1,0,0,0),

`M`=(1,0,1,0,0,−1,1,0,0,0,0),

`N`=(0,−1,1,0,0,1,0,1,0,0,0),

`O`=(−1,1,0,0,0,0,1,1,0,0,0),

`P`=(1,0,−1,−1,0,0,0,1,0,0,0),

`Q`=(0,1,1,−1,0,0,−1,0,0,0,0),

`R`=(1,0,0,1,−1,0,−1,0,0,0,0),

`S`=(0,1,0,1,1,0,0,1,0,0,0),

`T`=(1,1,0,0,0,0,1,−1,0,0,0),

`U`=(0,1,0,0,1,−1,−1,0,0,0,0),

`V`=(1,0,0,0,−1,−1,0,1,0,0,0),

`W`=(1,1,0,−1,0,1,0,0,0,0,0),

`X`=(1,−1,−1,0,1,0,0,0,0,0,0),

`Y`=(0,0,0,0,0,0,0,0,1,0,0),

`Z`=(0,0,0,0,0,0,0,0,0,1,0),

`a`=(0,0,0,0,0,0,0,1,1,1,1),

`b`=(0,0,0,0,0,0,0,1,−1,1,−1),

`c`=(0,0,0,0,0,0,0,1,−1,−1,1),

`d`=(0,0,0,0,0,0,0,1,−1,−1,−1),

`e`=(0,0,0,0,0,0,0,1,−1,1,1),

`f`=(0,0,0,0,0,0,0,1,1,1,−1),

`g`=(0,0,0,0,0,0,0,1,1,0,0),

`h`=(0,0,0,0,0,0,0,0,0,1,1),

`i`=(0,0,0,0,0,0,0,0,0,1,−1),

`j`=(0,0,0,0,0,0,0,0,1,0,1),

`k`=(0,0,0,0,0,0,0,0,1,0,−1),

`l`=(0,0,0,0,0,0,0,1,0,−1,0),

`m`=(0,0,0,0,0,0,0,1,0,0,−1),

`n`=(0,0,0,0,0,0,0,1,0,0,1),

`o`=(0,0,0,0,0,0,0,0,1,−1,0)

#### Appendix A.10. Twelve-dim MMPHs

**19-9**

`123456789ABC,17DJBL,28PQA,PJ,3478ST,5678Q,SC,T9,DL`

**52-9**(master for

**19-9**)

`123456789ABC,17DEFGHIJBKL,28MNOPHIQAKR,3478STUVWXYR,5678ZabcdWQe,ZafghiGPjJke,bSlmnFOijdoC,cTpENhmn9qYo,UVDMfglpkXqL. 1`=(0,0,1,1,1,1,0,0,0,0,0,0),

`2`=(0,0,1,−1,1,−1,0,0,0,0,0,0),

`3`=(0,0,0,1,0,−1,0,0,0,0,0,0),

`4`=(0,0,1,0,−1,0,0,0,0,0,0,0),

`5`=(0,1,0,0,0,0,0,0,0,0,0,0),

`6`=(1,0,0,0,0,0,0,0,0,0,0,0),

`7`=(0,0,0,0,0,0,0,1,0,0,0,0),

`8`=(0,0,0,0,0,0,1,0,0,0,0,0),

`Z`=(0,0,0,1,0,0,0,0,0,0,0,0),

`a`=(0,0,1,0,0,0,0,0,0,0,0,0),

`b`=(0,0,0,0,0,1,0,0,0,0,0,0),

`c`=(0,0,0,0,1,0,0,0,0,0,0,0),

`S`=(1,−1,1,0,1,0,0,0,0,0,0,0),

`T`=(1,1,0,1,0,1,0,0,0,0,0,0),

`U`=(1,1,0,−1,0,−1,0,0,0,0,0,0),

`V`=(−1,1,1,0,1,0,0,0,0,0,0,0),

`D`=(0,1,−1,1,0,0,1,0,0,0,0,0),

`M`=(1,0,1,1,0,0,0,−1,0,0,0,0),

`f`=(1,0,0,0,1,1,0,1,0,0,0,0),

`g`=(0,1,0,0,−1,1,−1,0,0,0,0,0),

`l`=(0,0,1,0,−1,0,1,1,0,0,0,0),

`p`=(0,0,0,1,0,−1,−1,1,0,0,0,0),

`E`=(1,0,1,0,0,−1,1,0,0,0,0,0),

`N`=(0,−1,1,0,0,1,0,1,0,0,0,0),

`h`=(−1,1,0,0,0,0,1,1,0,0,0,0),

`m`=(1,0,−1,−1,0,0,0,1,0,0,0,0),

`n`=(0,1,1,−1,0,0,−1,0,0,0,0,0),

`F`=(1,0,0,1,−1,0,−1,0,0,0,0,0),

`O`=(0,1,0,1,1,0,0,1,0,0,0,0),

`i`=(1,1,0,0,0,0,1,−1,0,0,0,0),

`G`=(0,1,0,0,1,−1,−1,0,0,0,0,0),

`P`=(1,0,0,0,−1,−1,0,1,0,0,0,0),

`H`=(1,1,0,−1,0,1,0,0,0,0,0,0),

`I`=(1,−1,−1,0,1,0,0,0,0,0,0,0),

`j`=(0,0,0,0,0,0,0,0,1,0,0,0),

`d`=(0,0,0,0,0,0,0,0,0,1,0,0),

`J`=(0,0,0,0,0,0,0,0,0,1,1,0),

`k`=(0,0,0,0,0,0,0,0,0,1,−1,0),

`W`=(0,0,0,0,0,0,0,0,1,0,1,0),

`Q`=(0,0,0,0,0,0,0,0,1,0,−1,0),

`9`=(0,0,0,0,0,0,0,0,1,−1,0,0),

`e`=(0,0,0,0,0,0,0,0,0,0,0,1),

`A`=(0,0,0,0,0,0,0,0,1,1,1,1),

`B`=(0,0,0,0,0,0,0,0,1,1,−1,−1),

`K`=(0,0,0,0,0,0,0,0,1,−1,1,−1),

`X`=(0,0,0,0,0,0,0,0,1,−1,−1,−1),

`q`=(0,0,0,0,0,0,0,0,1,1,1,−1),

`Y`=(0,0,0,0,0,0,0,0,1,1,−1,1),

`L`=(0,0,0,0,0,0,0,0,1,0,0,1),

`o`=(0,0,0,0,0,0,0,0,0,0,1,1),

`C`=(0,0,0,0,0,0,0,0,0,0,1,−1),

`R`=(0,0,0,0,0,0,0,0,0,1,0,−1)

#### Appendix A.11. Thirteen-dim MMPHs

**19-8**

`123456789ABCD,123456789EFG,12345678ILM,289EBM,34789C,56789LG,5678ABM,`9FD.

**63-16**(master for

**19-8**)

`123456789ABCD,123456789EFGH,12345678IJKLM,17NOPQRSTUVWM`,28XYZaRS9EbBM,3478cdef9bghH,3478cdef9WhiC,3478cdefIjVkM, 5678lmno9LpGq, 5678lmnoATrBM,lmstuvQa9WpFD,lmstuvQaEKwxM,ncyz!PZv9AkLM,od"OYuz!9kgiq,od"OYuz!bUj xM,efNXsty"rJwWM.

`1`=(0,0,1,1,1,1,0,0,0,0,0,0,0),

`2`=(0,0,1,−1,1,−1,0,0,0,0,0,0,0),

`3`=(0,0,0,1,0,−1,0,0,0,0,0,0,0),

`4`=(0,0,1,0,−1,0,0,0,0,0,0,0,0),

`5`=(0,1,0,0,0,0,0,0,0,0,0,0,0),

`6`=(1,0,0,0,0,0,0,0,0,0,0,0,0),

`7`=(0,0,0,0,0,0,0,1,0,0,0,0,0),

`8`=(0,0,0,0,0,0,1,0,0,0,0,0,0),

`l`=(0,0,0,1,0,0,0,0,0,0,0,0,0),

`m`=(0,0,1,0,0,0,0,0,0,0,0,0,0),

`n`=(0,0,0,0,0,1,0,0,0,0,0,0,0),

`o`=(0,0,0,0,1,0,0,0,0,0,0,0,0),

`c`=(1,−1,1,0,1,0,0,0,0,0,0,0,0),

`d`=(1,1,0,1,0,1,0,0,0,0,0,0,0),

`e`=(1,1,0,−1,0,−1,0,0,0,0,0,0,0),

`f`=(−1,1,1,0,1,0,0,0,0,0,0,0,0),

`N`=(0,1,−1,1,0,0,1,0,0,0,0,0,0),

`X`=(1,0,1,1,0,0,0,−1,0,0,0,0,0),

`s`=(1,0,0,0,1,1,0,1,0,0,0,0,0),

`t`=(0,1,0,0,−1,1,−1,0,0,0,0,0,0),

`y`=(0,0,1,0,−1,0,1,1,0,0,0,0,0),

`"`=(0,0,0,1,0,−1,−1,1,0,0,0,0,0),

`O`=(1,0,1,0,0,−1,1,0,0,0,0,0,0),

`Y`=(0,−1,1,0,0,1,0,1,0,0,0,0,0),

`u`=(−1,1,0,0,0,0,1,1,0,0,0,0,0),

`z`=(1,0,−1,−1,0,0,0,1,0,0,0,0,0),

`!`=(0,1,1,−1,0,0,−1,0,0,0,0,0,0),

`P`=(1,0,0,1,−1,0,−1,0,0,0,0,0,0),

`Z`=(0,1,0,1,1,0,0,1,0,0,0,0,0),

`v`=(1,1,0,0,0,0,1,−1,0,0,0,0,0),

`Q`=(0,1,0,0,1,−1,−1,0,0,0,0,0,0),

`a`=(1,0,0,0,−1,−1,0,1,0,0,0,0,0),

`R`=(1,1,0,−1,0,1,0,0,0,0,0,0,0),

`S`=(1,−1,−1,0,1,0,0,0,0,0,0,0,0),

`9`=(0,0,0,0,0,0,0,0,1,0,0,0,0),

`A`=(0,0,0,0,0,0,0,0,0,1,0,0,0),

`E`=(0,0,0,0,0,0,0,0,0,1,1,0,0),

`b`=(0,0,0,0,0,0,0,0,0,1,−1,0,0),

`T`=(0,0,0,0,0,0,0,0,1,0,1,0,0),

`r`=(0,0,0,0,0,0,0,0,1,0,−1,0,0),

`I`=(0,0,0,0,0,0,0,0,1,−1,0,0,0),

`B`=(0,0,0,0,0,0,0,0,0,0,0,1,0),

`J`=(0,0,0,0,0,0,0,0,1,1,1,1,0),

`K`=(0,0,0,0,0,0,0,0,1,1,−1,−1,0),

`w`=(0,0,0,0,0,0,0,0,1,−1,1,−1,0),

`U`=(0,0,0,0,0,0,0,0,1,−1,−1,−1,0),

`j`=(0,0,0,0,0,0,0,0,1,1,1,−1,0),

`V`=(0,0,0,0,0,0,0,0,1,1,−1,1,0),

`x`=(0,0,0,0,0,0,0,0,1,0,0,1,0),

`k`=(0,0,0,0,0,0,0,0,0,0,1,1,0),

`L`=(0,0,0,0,0,0,0,0,0,0,1,−1,0),

`W`=(0,0,0,0,0,0,0,0,0,1,0,−1,0),

`M`=(0,0,0,0,0,0,0,0,0,0,0,0,1),

`p`=(0,0,0,0,0,0,0,0,0,1,1,1,1),

`F`=(0,0,0,0,0,0,0,0,0,1,−1,1,−1),

`G`=(0,0,0,0,0,0,0,0,0,1,−1,−1,1),

`g`=(0,0,0,0,0,0,0,0,0,1,1,−1,1),

`h`=(0,0,0,0,0,0,0,0,0,1,1,1,−1),

`i`=(0,0,0,0,0,0,0,0,0,1,−1,1,1),

`H`=(0,0,0,0,0,0,0,0,0,0,0,1,1),

`C`=(0,0,0,0,0,0,0,0,0,0,1,0,1),

`D`=(0,0,0,0,0,0,0,0,0,0,1,0,−1),

`q`=(0,0,0,0,0,0,0,0,0,1,0,0,−1)

#### Appendix A.12. Fourteen-dim MMPHs

**19-9**

`123456789ABCDE,12345679ABFGD,1OPF,27a,347E,3479ABP,567CG,9a,O89AB.`

**66-15**(master for

**19-9**)

`123456789ABCDE,12345679ABFGDH,1IJKLMNOPFQRST,27UVWXMN YZTabc,347defghijkElm,347defg9ABPHnm,347defgFijkGDH,567opqrOBPFsZt,567opqr9ABaunl,567opqrijbkCGu,opvwxyLX9iQYab,qdz!"KWyAt#Sab,re$JVx!"s%abck,fgIUvwz$Oh89AB,fgIUvwz$#R%jab. 1`=(0,0,1,1,1,1,0,0,0,0,0,0,0,0),

`2`=(0,0,1,−1,1,−1,0,0,0,0,0,0,0,0),

`3`=(0,0,0,1,0,−1,0,0,0,0,0,0,0,0),

`4`=(0,0,1,0,−1,0,0,0,0,0,0,0,0,0),

`5`=(0,1,0,0,0,0,0,0,0,0,0,0,0,0),

`6`=(1,0,0,0,0,0,0,0,0,0,0,0,0,0),

`7`=(0,0,0,0,0,0,1,0,0,0,0,0,0,0),

`o`=(0,0,0,1,0,0,0,0,0,0,0,0,0,0),

`p`=(0,0,1,0,0,0,0,0,0,0,0,0,0,0),

`q`=(0,0,0,0,0,1,0,0,0,0,0,0,0,0),

`r`=(0,0,0,0,1,0,0,0,0,0,0,0,0,0),

`d`=(1,−1,1,0,1,0,0,0,0,0,0,0,0,0),

`e`=(1,1,0,1,0,1,0,0,0,0,0,0,0,0),

`f`=(1,1,0,−1,0,−1,0,0,0,0,0,0,0,0),

`g`=(−1,1,1,0,1,0,0,0,0,0,0,0,0,0),

`I`=(0,1,−1,1,0,0,1,0,0,0,0,0,0,0),

`U`=(1,0,1,1,0,0,0,−1,0,0,0,0,0,0),

`v`=(1,0,0,0,1,1,0,1,0,0,0,0,0,0),

`w`=(0,1,0,0,−1,1,−1,0,0,0,0,0,0,0),

`z`=(0,0,1,0,−1,0,1,1,0,0,0,0,0,0),

`$`=(0,0,0,1,0,−1,−1,1,0,0,0,0,0,0),

`J`=(1,0,1,0,0,−1,1,0,0,0,0,0,0,0),

`V`=(0,−1,1,0,0,1,0,1,0,0,0,0,0,0),

`x`=(−1,1,0,0,0,0,1,1,0,0,0,0,0,0),

`!`=(1,0,−1,−1,0,0,0,1,0,0,0,0,0,0),

`"`=(0,1,1,−1,0,0,−1,0,0,0,0,0,0,0),

`K`=(1,0,0,1,−1,0,−1,0,0,0,0,0,0,0),

`W`=(0,1,0,1,1,0,0,1,0,0,0,0,0,0),

`y`=(1,1,0,0,0,0,1,−1,0,0,0,0,0,0),

`L`=(0,1,0,0,1,−1,−1,0,0,0,0,0,0,0),

`X`=(1,0,0,0,−1,−1,0,1,0,0,0,0,0,0),

`M`=(1,1,0,−1,0,1,0,0,0,0,0,0,0,0),

`N`=(1,−1,−1,0,1,0,0,0,0,0,0,0,0,0),

`O`=(0,0,0,0,0,0,0,0,0,0,1,0,0,0),

`h`=(0,0,0,0,0,0,0,0,1,−1,0,0,0,0),

`8`=(0,0,0,0,0,0,0,0,1,1,0,0,0,0),

`9`=(0,0,0,0,0,0,0,0,0,0,0,0,0,1),

`A`=(0,0,0,0,0,0,0,0,0,0,0,1,1,0),

`B`=(0,0,0,0,0,0,0,0,0,0,0,1,−1,0),

`P`=(0,0,0,0,0,0,0,1,0,−1,0,0,0,0),

`F`=(0,0,0,0,0,0,0,1,0,1,0,0,0,0),

`i`=(0,0,0,0,0,0,0,0,0,0,0,1,0,0),

`Q`=(0,0,0,0,0,0,0,0,1,0,0,0,−1,0),

`Y`=(0,0,0,0,0,0,0,0,1,0,0,0,1,0),

`s`=(0,0,0,0,0,0,0,0,1,0,0,1,1,−1),

`Z`=(0,0,0,0,0,0,0,0,1,0,0,−1,−1,−1),

`t`=(0,0,0,0,0,0,0,0,1,0,0,0,0,1),

`#`=(0,0,0,0,0,0,0,0,1,0,0,1,−1,−1),

`R`=(0,0,0,0,0,0,0,0,1,0,0,1,1,1),

`%`=(0,0,0,0,0,0,0,0,1,0,0,−1,0,0),

`j`=(0,0,0,0,0,0,0,0,0,0,0,0,1,−1),

`S`=(0,0,0,0,0,0,0,0,1,0,0,−1,1,−1),

`T`=(0,0,0,0,0,0,0,0,0,0,0,1,0,−1),

`a`=(0,0,0,0,0,0,0,0,0,1,1,0,0,0),

`b`=(0,0,0,0,0,0,0,0,0,1,−1,0,0,0),

`c`=(0,0,0,0,0,0,0,0,1,0,0,1,−1,1),

`k`=(0,0,0,0,0,0,0,0,0,0,0,0,1,1),

`C`=(0,0,0,0,0,0,0,1,−1,1,1,0,0,0),

`G`=(0,0,0,0,0,0,0,1,−1,−1,−1,0,0,0),

`u`=(0,0,0,0,0,0,0,1,1,0,0,0,0,0),

`D`=(0,0,0,0,0,0,0,1,1,−1,1,0,0,0),

`E`=(0,0,0,0,0,0,0,1,0,0,−1,0,0,0),

`H`=(0,0,0,0,0,0,0,0,1,0,−1,0,0,0),

`n`=(0,0,0,0,0,0,0,1,−1,1,−1,0,0,0),

`l`=(0,0,0,0,0,0,0,1,−1,−1,1,0,0,0),

`m`=(0,0,0,0,0,0,0,1,1,1,1,0,0,0)

#### Appendix A.13. Fifteen-dim MMPHs

**25-8**

`123456789ABCDEF,1234567GHIJKLD,1RS89ABCDEF,27TRSX9GH,347CL,347BK,347AIJ,T8X9.`

**66-14**(master for

**25-8**)

`123456789ABCDEF,1234567GHIJKLDM,1NOPQRS89ABCDEF,27TUVWRSX9YGHZa,347bcdeZfghiCjL,347bcdeaklBmKhi,347bcdenoApIJgl,567qrstuYpmjMEF, 567qrstvw9Yfkno,qrxyz!QW8uvwX9Y,qrxyz!QWX9YGHZa,sb"#$PV!8uvwX9Y,tc%OUz#$8uvwX9Y,deNTxy"%8uvwX9Y. 1`=(0,0,1,1,1,1.0,0,0,0,0,0,0,0,0),

`2`=(0,0,1,−1,1,−1.0,0,0,0,0,0,0,0,0),

`3`=(0,0,0,1,0,−1.0,0,0,0,0,0,0,0,0),

`4`=(0,0,1,0,−1,0.0,0,0,0,0,0,0,0,0),

`5`=(0,1,0,0,0,0.0,0,0,0,0,0,0,0,0),

`6`=(1,0,0,0,0,0.0,0,0,0,0,0,0,0,0),

`7`=(0,0,0,0,0,0.1,0,0,0,0,0,0,0,0),

`q`=(0,0,0,1,0,0.0,0,0,0,0,0,0,0,0),

`r`=(0,0,1,0,0,0.0,0,0,0,0,0,0,0,0),

`s`=(0,0,0,0,0,1.0,0,0,0,0,0,0,0,0),

`t`=(0,0,0,0,1,0.0,0,0,0,0,0,0,0,0),

`b`=(1,−1,1,0,1,0.0,0,0,0,0,0,0,0,0),

`c`=(1,1,0,1,0,1.0,0,0,0,0,0,0,0,0),

`d`=(1,1,0,−1,0,−1,0,0,0,0,0,0,0,0,0),

`e`=(−1,1,1,0,1,0.0,0,0,0,0,0,0,0,0),

`N`=(0,1,−1,1,0,0.1,0,0,0,0,0,0,0,0),

`T`=(1,0,1,1,0,0,0,−1,0,0,0,0,0,0,0),

`x`=(1,0,0,0,1,1.0,1,0,0,0,0,0,0,0),

`y`=(0,1,0,0,−1,1.−1,0,0,0,0,0,0,0,0),

`"`=(0,0,1,0,−1,0.1,1,0,0,0,0,0,0,0),

`%`=(0,0,0,1,0,−1.−1,1,0,0,0,0,0,0,0),

`O`=(1,0,1,0,0,−1.1,0,0,0,0,0,0,0,0),

`U`=(0,−1,1,0,0,1.0,1,0,0,0,0,0,0,0),

`z`=(−1,1,0,0,0,0.1,1,0,0,0,0,0,0,0),

`#`=(1,0,−1,−1,0,0.0,1,0,0,0,0,0,0,0),

`$`=(0,1,1,−1,0,0.−1,0,0,0,0,0,0,0,0),

`P`=(1,0,0,1,−1,0.−1,0,0,0,0,0,0,0,0),

`V`=(0,1,0,1,1,0.0,1,0,0,0,0,0,0,0),

`!`=(1,1,0,0,0,0.1,−1,0,0,0,0,0,0,0),

`Q`=(0,1,0,0,1,−1.−1,0,0,0,0,0,0,0,0),

`W`=(1,0,0,0,−1,−1.0,1,0,0,0,0,0,0,0),

`R`=(1,1,0,−1,0,1.0,0,0,0,0,0,0,0,0),

`S`=(1,−1,−1,0,1,0.0,0,0,0,0,0,0,0,0),

`8`=(0,0,0,0,0,0.0,0,0,1,1,1,1,0,0),

`u`=(0,0,0,0,0,0.0,0,0,1,−1,1,−1,0,0),

`v`=(0,0,0,0,0,0.0,0,0,0,1,0,−1,0,0),

`w`=(0,0,0,0,0,0.0,0,0,1,0,−1,0,0,0),

`X`=(0,0,0,0,0,0.0,0,1,0,0,0,0,0,0),

`9`=(0,0,0,0,0,0.0,0,0,0,0,0,0,0,1),

`Y`=(0,0,0,0,0,0.0,0,0,0,0,0,0,1,0),

`G`=(0,0,0,0,0,0.0,0,0,0,1,0,0,0,0),

`H`=(0,0,0,0,0,0.0,0,0,1,0,0,0,0,0),

`Z`=(0,0,0,0,0,0.0,0,0,0,0,0,1,0,0),

`a`=(0,0,0,0,0,0.0,0,0,0,0,1,0,0,0),

`f`=(0,0,0,0,0,0.0,1,−1,1,0,1,0,0,0),

`k`=(0,0,0,0,0,0.0,1,1,0,1,0,1,0,0),

`n`=(0,0,0,0,0,0.0,1,1,0,−1,0,−1,0,0),

`o`=(0,0,0,0,0,0.0,1,−1,−1,0,−1,0,0,0),

`A`=(0,0,0,0,0,0.0,0,1,−1,1,0,0,1,0),

`p`=(0,0,0,0,0,0.0,1,0,1,1,0,0,0,−1),

`I`=(0,0,0,0,0,0.0,1,0,0,0,1,1,0,1),

`J`=(0,0,0,0,0,0.0,0,1,0,0,−1,1,−1,0),

`g`=(0,0,0,0,0,0.0,0,0,1,0,−1,0,1,1),

`l`=(0,0,0,0,0,0.0,0,0,0,1,0,−1,−1,1),

`B`=(0,0,0,0,0,0.0,1,0,1,0,0,−1,1,0),

`m`=(0,0,0,0,0,0.0,0,1,−1,0,0,−1,0,−1),

`K`=(0,0,0,0,0,0.0,1,−1,0,0,0,0,−1,−1),

`h`=(0,0,0,0,0,0.0,1,0,−1,−1,0,0,0,1),

`i`=(0,0,0,0,0,0.0,0,1,1,−1,0,0,−1,0),

`C`=(0,0,0,0,0,0.0,1,0,0,1,−1,0,−1,0),

`j`=(0,0,0,0,0,0.0,0,1,0,1,1,0,0,1),

`L`=(0,0,0,0,0,0.0,1,1,0,0,0,0,1,−1),

`D`=(0,0,0,0,0,0.0,0,1,0,0,1,−1,−1,0),

`M`=(0,0,0,0,0,0.0,1,0,0,0,−1,−1,0,1),

`E`=(0,0,0,0,0,0.0,1,1,0,−1,0,1,0,0),

`F`=(0,0,0,0,0,0.0,1,−1,−1,0,1,0,0,0)

#### Appendix A.14. Sixteen-dim MMPHs

**22-9**

`123456789ABCDEFG,17HID,28UdG,3478efE,5678,Bd,e9,fICF,HUA.`

**70-9**(master for

**22-9**)

`123456789ABCDEFG,17HIJKLMNOPQRDST,28UVWXLMYZabScdG,3478efghijPbkElm,5678nopqrZsQtukT,novwxyKXzasBu!dm,pe"#$JWy9iY%tR&’,qf(IVx#$zOjC)lF&,ghHUvw"(ArN%)c!’`.

`1`=(0,0,1,1,1,1,0,0,0,0,0,0,0,0,0,0),

`2`=(0,0,1,−1,1,−1,0,0,0,0,0,0,0,0,0,0),

`3`=(0,0,0,1,0,−1,0,0,0,0,0,0,0,0,0,0),

`4`=(0,0,1,0,−1,0,0,0,0,0,0,0,0,0,0,0),

`5`=(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0),

`6`=(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0),

`7`=(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0),

`8`=(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0),

`n`=(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0),

`o`=(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0),

`p`=(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0),

`q`=(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0),

`e`=(1,−1,1,0,1,0,0,0,0,0,0,0,0,0,0,0),

`f`=(1,1,0,1,0,1,0,0,0,0,0,0,0,0,0,0),

`g`=(1,1,0,−1,0,−1,0,0,0,0,0,0,0,0,0,0),

`h`=(−1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0),

`H`=(0,1,−1,1,0,0,1,0,0,0,0,0,0,0,0,0),

`U`=(1,0,1,1,0,0,0,−1,0,0,0,0,0,0,0,0),

`v`=(1,0,0,0,1,1,0,1,0,0,0,0,0,0,0,0),

`w`=(0,1,0,0,−1,1,−1,0,0,0,0,0,0,0,0,0),

`"`=(0,0,1,0,−1,0,1,1,0,0,0,0,0,0,0,0),

`(`=(0,0,0,1,0,−1,−1,1,0,0,0,0,0,0,0,0),

`I`=(1,0,1,0,0,−1,1,0,0,0,0,0,0,0,0,0),

`V`=(0,−1,1,0,0,1,0,1,0,0,0,0,0,0,0,0),

`x`=(−1,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0),

`#`=(1,0,−1,−1,0,0,0,1,0,0,0,0,0,0,0,0),

`$`=(0,1,1,−1,0,0,−1,0,0,0,0,0,0,0,0,0),

`J`=(1,0,0,1,−1,0,−1,0,0,0,0,0,0,0,0,0),

`W`=(0,1,0,1,1,0,0,1,0,0,0,0,0,0,0,0),

`y`=(1,1,0,0,0,0,1,−1,0,0,0,0,0,0,0,0),

`K`=(0,1,0,0,1,−1,−1,0,0,0,0,0,0,0,0,0),

`X`=(1,0,0,0,−1,−1,0,1,0,0,0,0,0,0,0,0),

`L`=(1,1,0,−1,0,1,0,0,0,0,0,0,0,0,0,0),

`M`=(1,−1,−1,0,1,0,0,0,0,0,0,0,0,0,0,0),

`9`=(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0),

`A`=(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0),

`i`=(0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0),

`Y`=(0,0,0,0,0,0,0,0,0,1,−1,0,0,0,0,0),

`r`=(0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0),

`N`=(0,0,0,0,0,0,0,0,1,0,−1,0,0,0,0,0),

`z`=(0,0,0,0,0,0,0,0,1,−1,0,0,0,0,0,0),

`%`=(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0),

`O`=(0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0),

`j`=(0,0,0,0,0,0,0,0,1,1,−1,−1,0,0,0,0),

`P`=(0,0,0,0,0,0,0,0,1,−1,1,−1,0,0,0,0),

`Z`=(0,0,0,0,0,0,0,0,1,−1,−1,−1,0,0,0,0),

`a`=(0,0,0,0,0,0,0,0,1,1,1,−1,0,0,0,0),

`s`=(0,0,0,0,0,0,0,0,1,1,−1,1,0,0,0,0),

`b`=(0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0),

`B`=(0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0),

`C`=(0,0,0,0,0,0,0,0,0,0,1,−1,0,0,0,0),

`Q`=(0,0,0,0,0,0,0,0,0,1,0,−1,0,0,0,0),

`t`=(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0),

`R`=(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0),

`u`=(0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0),

`k`=(0,0,0,0,0,0,0,0,0,0,0,0,0,1,−1,0),

`D`=(0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0),

`S`=(0,0,0,0,0,0,0,0,0,0,0,0,1,0,−1,0),

`)`=(0,0,0,0,0,0,0,0,0,0,0,0,1,−1,0,0),

`T`=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1),

`c`=(0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1),

`!`=(0,0,0,0,0,0,0,0,0,0,0,0,1,1,−1,−1),

`d`=(0,0,0,0,0,0,0,0,0,0,0,0,1,−1,1,−1),

`E`=(0,0,0,0,0,0,0,0,0,0,0,0,1,−1,−1,−1),

`l`=(0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,−1),

`F`=(0,0,0,0,0,0,0,0,0,0,0,0,1,1,−1,1),

`m`=(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1),

`&`=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1),

`’`=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,−1),

`G`=(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,−1)

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**Figure 1.**(

**a**) Yu-Oh’s three-dim non-KS 13-16 non-KS NBMMPH ( [36] Figure 2); gray vertices that enlarge 13-16 to 25-16 are necessary for coordinatization and implementation; (

**b**) Howard, Wallman, Veitech, and Emerson’s four-dim 30-108 non-KS NBMMPH ([4] Figure 2); (

**c**) Cabello, Portillo, Solís, and Svozil’s five-dim 10-9 non-KS NBMMPH ([42] Figure 5a); the original symbols are presented in brackets (A,B).

**Figure 2.**(

**a**) Distributions of critical four-dim non-KS NBMMPHs obtained from submaster 20-10, which was obtained from (Peres’) 24-24 supermaster (generated by vector components $\{0,\pm 1\}$) by

**M1**(dots in red) and from submaster 58-51, itself obtained from the 60-72 supermaster (generated by vector components $\{0,\pm \varphi ,\varphi -1\}$, where $\varphi $ is the golden ratio: $\frac{1+\sqrt{5}}{2}$) by

**M1**(in black); the abscissa is l (number of hyperedges); and the ordinate is k (number of vertices). The dots represent $(k,l)$. Consecutive dots (same l) are shown as strips; (

**b**) the smallest non-KS in the distributions: 4-3; (

**c**) BMMPH 8-3—filled with 4-3—which one needs for obtaining the coordinatization and implementation of 4-3; (

**d**) the 16-9 critical obtained from the 20-10 master; (

**e**) the 16-9 critical obtained from the 58-51 master; (

**f**) distributions of critical five-dim non-KS NBMMPHs obtained from submaster 66-50 which was obtained from the 105-136 supermaster (generated by vector components $\{0,\pm 1\}$); (

**g**) the smallest critical; (

**h**) a 16-9 critical for the sake of comparison with four-dim 16-9s; strings and coordinatizations are given in Appendix A.

**Figure 3.**(

**a**) Distributions of 6-dim critical non-KS NBMMPHs obtained from two different submasters—see text; (

**b**) the smallest critical non-KS NBMMPH obtained from the former class by

**M3**; it has a parity proof; (

**c**) an even smaller critical non-KS NBMMPH obtained from it by hand; it has a parity proof; (

**d**) the smallest critical non-KS NBMMPH obtained from the latter class by

**M1**; (

**e**) distributions of 7-dim critical non-KS NBMMPHs—see text; (

**f**) 14-8 non-KS NBMMPH, one of the smallest non-KS NBMMPHs obtained via

**M3**from the smallest KS NBMMPH 34-14; (

**g**) 31-13 also obtained from the 34-14 (no m = 1 vertices essential for criticality); (

**h**,

**i**) two 8-dim KS MMPHs with the smallest number of hyperedges (9); (

**i**) serves us in generating the 15-9 non-KS NBMMPH in (

**j**); (

**h**–

**j**) MMPHs have parity proofs; strings and coordinatizations are given in Appendix A.

**Figure 4.**(

**a**) The 44-6 BMMPH and its critical subgraph 13-6 non-KS NBMMPH directly obtained from the supermaster via

**M1**; (

**b**) the critical nine-dim 19-8 obtained via

**M3**from the master 47-16; (

**c**) the critical ten-dim 18-9 non-KS NBMMPH obtained via

**M3**from the 50-15 master; (

**d**) the critical eleven-dim 19-8 non-KS NBMMPH obtained via

**M3**from the 50-14 master. Strings and coordinatizations are given in Appendix A.

**Figure 5.**(

**a**) Twelve-dim 19-9 critical non-KS NBMMPH directly obtained from the master 52-9 via

**M3**; (

**b**) 13-dim 19-8 critical non-KS NBMMPH obtained from the master 63-16, where the hyperedges do not form any loop with an order of three or higher; (

**c**) 14-dim critical obtained from 66-15, where the maximal loop also has an order of 2; (

**d**) 15-dim 25-8 critical from the 66-14 master; (

**e**) 16-dim 22-9 critical from the 70-9 master, where all criticals are obtained via

**M3**; all criticals and masters are given in the Appendix A.

**Table 1.**The smallest critical non-KS MMPHs obtained via the small vector component method and by the dimensional upscaling method via

**M1**and

**M3**. Notice the steady fluctuation in the number of hyperedges over dimensions which is consistent with our previous result showing that the minimum complexity of NBMMPHs does not grow with the dimensions. The MMPH strings and coordinatizations of both the criticals and their masters are given in Appendix A. $\varphi $ is the Golden ratio, and $\omega $ is the cube root of 1.

dim | Smallest Critical MMPHs | Master | Vector Components |
---|---|---|---|

3-dim | 8-7 (Kochen–Specker’s “bug”) | 49-36 (Bub’s KS MMPH) | $\{0,\pm 1,\pm 2,5\}$ |

4-dim | 4-3 | 8-3 | $\{0,\pm 1\}$ |

4-dim | 16-9 | 58-51 | $\{0,\pm \varphi ,\varphi -1\}$ |

5-dim | 7-5 | 16-5 | $\{0,\pm 1\}$ |

6-dim | 11-7 | 19-7 | $\{0,1,\omega ,{\omega}^{2}\}$ |

7-dim | 14-8 | 34-14 | $\{0,\pm 1\}$ |

8-dim | 15-9 | 2768-1346016 | $\{0,\pm 1\}$ |

9-dim | 13-6 | 9586-12068705 | $\{0,\pm 1\}$ |

9-dim | 19-8 | 47-16 | $\{0,\pm 1\}$ |

10-dim | 18-9 | 50-15 | $\{0,\pm 1\}$ |

11-dim | 19-8 | 50-14 | $\{0,\pm 1\}$ |

12-dim | 19-9 | 52-9 | $\{0,\pm 1\}$ |

13-dim | 19-8 | 63-16 | $\{0,\pm 1\}$ |

14-dim | 19-9 | 66-15 | $\{0,\pm 1\}$ |

15-dim | 25-8 | 66-14 | $\{0,\pm 1\}$ |

16-dim | 22-9 | 70-9 | $\{0,\pm 1\}$ |

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**MDPI and ACS Style**

Pavičić, M.
Non-Kochen–Specker Contextuality. *Entropy* **2023**, *25*, 1117.
https://doi.org/10.3390/e25081117

**AMA Style**

Pavičić M.
Non-Kochen–Specker Contextuality. *Entropy*. 2023; 25(8):1117.
https://doi.org/10.3390/e25081117

**Chicago/Turabian Style**

Pavičić, Mladen.
2023. "Non-Kochen–Specker Contextuality" *Entropy* 25, no. 8: 1117.
https://doi.org/10.3390/e25081117