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❚r✐♣❧❡ ❛♥❞ ◗✉❛❞r✉♣❧❡ ❊♥❝r②♣t✐♦♥✿ ❇r✐❞❣✐♥❣ t❤❡ ●❛♣s ❇❛rt ▼❡♥♥✐♥❦ ❛♥❞ ❇❛rt Pr❡♥❡❡❧ ❑❯ ▲❡✉✈❡♥ ✭❇❡❧❣✐✉♠✮ ❉❛❣st✉❤❧ ✖ ❏❛♥✉❛r② ✻✱ ✷✵✶✹ ✶ ✴ ✶✷

✲ ✲ ✶✾✼✽ ❚r✐♣❧❡✲❉❊❙ ✶✾✽✹ ❉❊❙❳ ✶✾✾✶ ■❉❊❆ ✷✵✵✶ ❆❊❙ ❚r✐♣❧❡✲❉❊❙ st✐❧❧ ❜r♦❛❞❧② ✉s❡❞ ❃✶✻✵✵ ❜② ◆■❙❚ ✈❛❧✐❞❛t❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❆❚▼s✱ ❊▼❱✱ ❚▲❙✱ ▼✐❝r♦s♦❢t✱ ✳✳✳ ■♥tr♦❞✉❝t✐♦♥ k m c E ✶✾✼✼ ❉❊❙ κ = 56 n = 64 ✷ ✴ ✶✷

✲ ✲ ✶✾✽✹ ❉❊❙❳ ✶✾✾✶ ■❉❊❆ ✷✵✵✶ ❆❊❙ ❚r✐♣❧❡✲❉❊❙ st✐❧❧ ❜r♦❛❞❧② ✉s❡❞ ❃✶✻✵✵ ❜② ◆■❙❚ ✈❛❧✐❞❛t❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❆❚▼s✱ ❊▼❱✱ ❚▲❙✱ ▼✐❝r♦s♦❢t✱ ✳✳✳ ■♥tr♦❞✉❝t✐♦♥ k m c E ✶✾✼✼ ❉❊❙ κ = 56 n = 64 ✶✾✼✽ ❚r✐♣❧❡✲❉❊❙ κ = 168 n = 64 ✷ ✴ ✶✷

✲ ✲ ✶✾✾✶ ■❉❊❆ ✷✵✵✶ ❆❊❙ ❚r✐♣❧❡✲❉❊❙ st✐❧❧ ❜r♦❛❞❧② ✉s❡❞ ❃✶✻✵✵ ❜② ◆■❙❚ ✈❛❧✐❞❛t❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❆❚▼s✱ ❊▼❱✱ ❚▲❙✱ ▼✐❝r♦s♦❢t✱ ✳✳✳ ■♥tr♦❞✉❝t✐♦♥ k m c E ✶✾✼✼ ❉❊❙ κ = 56 n = 64 ✶✾✼✽ ❚r✐♣❧❡✲❉❊❙ κ = 168 n = 64 ✶✾✽✹ ❉❊❙❳ κ = 184 n = 64 ✷ ✴ ✶✷

✲ ✲ ❚r✐♣❧❡✲❉❊❙ st✐❧❧ ❜r♦❛❞❧② ✉s❡❞ ❃✶✻✵✵ ❜② ◆■❙❚ ✈❛❧✐❞❛t❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ❆❚▼s✱ ❊▼❱✱ ❚▲❙✱ ▼✐❝r♦s♦❢t✱ ✳✳✳ ■♥tr♦❞✉❝t✐♦♥ k m c E ✶✾✼✼ ❉❊❙ κ = 56 n = 64 ✶✾✼✽ ❚r✐♣❧❡✲❉❊❙ κ = 168 n = 64 ✶✾✽✹ ❉❊❙❳ κ = 184 n = 64 ✶✾✾✶ ■❉❊❆ κ = 128 n = 64 ✷✵✵✶ ❆❊❙ κ ≥ 128 n = 128 ✷ ✴ ✶✷

✲ ✲ ■♥tr♦❞✉❝t✐♦♥ k m c E ✶✾✼✼ ❉❊❙ κ = 56 n = 64 ✶✾✼✽ ❚r✐♣❧❡✲❉❊❙ κ = 168 n = 64 • ❚r✐♣❧❡✲❉❊❙ st✐❧❧ ❜r♦❛❞❧② ✉s❡❞ ✶✾✽✹ ❉❊❙❳ κ = 184 n = 64 • ❃✶✻✵✵ ❜② ◆■❙❚ ✈❛❧✐❞❛t❡❞ ✐♠♣❧❡♠❡♥t❛t✐♦♥s ✶✾✾✶ ■❉❊❆ κ = 128 n = 64 ✷✵✵✶ ❆❊❙ κ ≥ 128 n = 128 • ❆❚▼s✱ ❊▼❱✱ ❚▲❙✱ ▼✐❝r♦s♦❢t✱ ✳✳✳ ✷ ✴ ✶✷

■♥tr♦❞✉❝t✐♦♥✿ ❚r✐♣❧❡✲❉❊❙ k 1 k 2 k 3 m c E E E • ❉♦✉❜❧❡✲❉❊❙✿ ♦♥❧② ♠❛r❣✐♥❛❧ s❡❝✉r✐t② ✐♥❝r❡❛s❡ • ❚r✐♣❧❡✲❉❊❙ • E ◦ D ◦ E ✈❡rs✉s E ◦ E ◦ E • k 1 = k 3 ✈❡rs✉s k 1 � = k 3 ✸ ✴ ✶✷

✲ ■❞❡❛❧ ❝✐♣❤❡r ♠♦❞❡❧ ■♥❢♦r♠❛t✐♦♥✲t❤❡♦r❡t✐❝ ❞✐st✐♥❣✉✐s❤❡r ❤❛s ❛❝❝❡ss t♦ ■♥tr♦❞✉❝t✐♦♥✿ ❈❛s❝❛❞❡ ❊♥❝r②♣t✐♦♥ k 1 k 2 k r m c E E E · · · · · · • κ, n ❛r❜✐tr❛r② • r ′ := ⌈ r/ 2 ⌉ ✹ ✴ ✶✷

✲ ■♥tr♦❞✉❝t✐♦♥✿ ❈❛s❝❛❞❡ ❊♥❝r②♣t✐♦♥ k 1 k 2 k r m c E E E · · · · · · • κ, n ❛r❜✐tr❛r② • r ′ := ⌈ r/ 2 ⌉ • ■❞❡❛❧ ❝✐♣❤❡r ♠♦❞❡❧ • ■♥❢♦r♠❛t✐♦♥✲t❤❡♦r❡t✐❝ ❞✐st✐♥❣✉✐s❤❡r ❤❛s ❛❝❝❡ss t♦ E ✹ ✴ ✶✷

✲ ❋♦r ✿ ❜♦✉♥❞s ♥♦♥✲t✐❣❤t ❢♦r ❚r✐♣❧❡✲❉❊❙✿ ▼❛✐♥ ❣♦❛❧✿ t✐❣❤t s❡❝✉r✐t② ❢♦r tr✐♣❧❡ ❡♥❝r②♣t✐♦♥ ❙t❛t❡ ♦❢ t❤❡ ❆rt r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ t✐❣❤t r = 1 , 2 κ κ ❬❉❍✼✼❪ ✓ r = 3 , 4 κ + min { κ/ 2 , n/ 2 } ❬❇❘✵✻✱●▼✵✾❪ κ + n/ 2 ❬▲✉❝✾✽✱●❛➸✶✸❪ ✗ � � ( r ′ − 1) κ + r ′ − 1 r ≥ 5 κ + min κ, n/ 2 r ′ n ❬●❛➸✶✸❪ ✗ ❬●▼✵✾❪ r ′ • ❬▲❡❡✶✸❪ ✿ κ + min { κ, n } − 16 r ( n 2 + 2) s❡❝✉r✐t② ✐❢ r ≥ 16 ✺ ✴ ✶✷

✲ ❚r✐♣❧❡✲❉❊❙✿ ▼❛✐♥ ❣♦❛❧✿ t✐❣❤t s❡❝✉r✐t② ❢♦r tr✐♣❧❡ ❡♥❝r②♣t✐♦♥ ❙t❛t❡ ♦❢ t❤❡ ❆rt r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ t✐❣❤t r = 1 , 2 κ κ ❬❉❍✼✼❪ ✓ r = 3 , 4 κ + min { κ/ 2 , n/ 2 } ❬❇❘✵✻✱●▼✵✾❪ κ + n/ 2 ❬▲✉❝✾✽✱●❛➸✶✸❪ ✗ � � ( r ′ − 1) κ + r ′ − 1 r ≥ 5 κ + min κ, n/ 2 r ′ n ❬●❛➸✶✸❪ ✗ ❬●▼✵✾❪ r ′ • ❬▲❡❡✶✸❪ ✿ κ + min { κ, n } − 16 r ( n 2 + 2) s❡❝✉r✐t② ✐❢ r ≥ 16 • ❋♦r r = 3 , 4 ✿ ❜♦✉♥❞s ♥♦♥✲t✐❣❤t ❢♦r κ ≤ n ✺ ✴ ✶✷

✲ ▼❛✐♥ ❣♦❛❧✿ t✐❣❤t s❡❝✉r✐t② ❢♦r tr✐♣❧❡ ❡♥❝r②♣t✐♦♥ ❙t❛t❡ ♦❢ t❤❡ ❆rt r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ t✐❣❤t r = 1 , 2 κ κ ❬❉❍✼✼❪ ✓ r = 3 , 4 κ + min { κ/ 2 , n/ 2 } ❬❇❘✵✻✱●▼✵✾❪ κ + n/ 2 ❬▲✉❝✾✽✱●❛➸✶✸❪ ✗ � � ( r ′ − 1) κ + r ′ − 1 r ≥ 5 κ + min κ, n/ 2 r ′ n ❬●❛➸✶✸❪ ✗ ❬●▼✵✾❪ r ′ • ❬▲❡❡✶✸❪ ✿ κ + min { κ, n } − 16 r ( n 2 + 2) s❡❝✉r✐t② ✐❢ r ≥ 16 • ❋♦r r = 3 , 4 ✿ ❜♦✉♥❞s ♥♦♥✲t✐❣❤t ❢♦r κ ≤ n ❚r✐♣❧❡✲❉❊❙✿ 2 84 ≤ 2 88 ✺ ✴ ✶✷

✲ ❙t❛t❡ ♦❢ t❤❡ ❆rt r♦✉♥❞s s❡❝✉r✐t② ❛tt❛❝❦ t✐❣❤t r = 1 , 2 κ κ ❬❉❍✼✼❪ ✓ r = 3 , 4 κ + min { κ/ 2 , n/ 2 } ❬❇❘✵✻✱●▼✵✾❪ κ + n/ 2 ❬▲✉❝✾✽✱●❛➸✶✸❪ ✗ � � ( r ′ − 1) κ + r ′ − 1 r ≥ 5 κ + min κ, n/ 2 r ′ n ❬●❛➸✶✸❪ ✗ ❬●▼✵✾❪ r ′ • ❬▲❡❡✶✸❪ ✿ κ + min { κ, n } − 16 r ( n 2 + 2) s❡❝✉r✐t② ✐❢ r ≥ 16 • ❋♦r r = 3 , 4 ✿ ❜♦✉♥❞s ♥♦♥✲t✐❣❤t ❢♦r κ ≤ n ❚r✐♣❧❡✲❉❊❙✿ 2 84 ≤ 2 88 ▼❛✐♥ ❣♦❛❧✿ t✐❣❤t s❡❝✉r✐t② ❢♦r tr✐♣❧❡ ❡♥❝r②♣t✐♦♥ ✺ ✴ ✶✷

✲ ✿ ❉✐st✐♥❣✉✐s❤❛❜❧❡ ❢r♦♠ r❛♥❞♦♠ ✐♥ ❝♦♥st❛♥t ★q✉❡r✐❡s ■♠♣r♦✈✐♥❣ ❆tt❛❝❦s k 1 k 2 k r m c E E E · · · · · · • ❬●❛➸✶✸❪ ✿ ❛tt❛❝❦ ✐♥ 2 κ + r ′− 1 n q✉❡r✐❡s r ′ ✻ ✴ ✶✷

✲ ■♠♣r♦✈✐♥❣ ❆tt❛❝❦s k 1 k 2 k r m c E E E · · · · · · • ❬●❛➸✶✸❪ ✿ ❛tt❛❝❦ ✐♥ 2 κ + r ′− 1 n q✉❡r✐❡s r ′ • κ = 0 ✿ π π π m c · · · · · · • ❉✐st✐♥❣✉✐s❤❛❜❧❡ ❢r♦♠ r❛♥❞♦♠ ✐♥ ❝♦♥st❛♥t ★q✉❡r✐❡s ✻ ✴ ✶✷

❆tt❛❝❦ ✐❞❡❛✿ ❋♦r♠❛❧✐③❛t✐♦♥ ♦❢ ♠❡❡t✲✐♥✲t❤❡✲♠✐❞❞❧❡ ❛tt❛❝❦ ❬❉❉❑❙✶✷❪ ✿ ❛tt❛❝❦ ✐♥ ✐♥ ✐♥❝♦♠♣❛r❛❜❧❡ ♠♦❞❡❧ ❈♦r♦❧❧❛r②✿ ❛tt❛❝❦ ✐♥ q✉❡r✐❡s ■♠♣r♦✈✐♥❣ ❆tt❛❝❦s k 1 k 2 k r m c E E E · · · · · · • ❬●❛➸✶✸❪ ✿ ❛tt❛❝❦ ✐♥ 2 κ + r ′− 1 n q✉❡r✐❡s r ′ ❘❡s✉❧t ✶✿ ❛tt❛❝❦ ✐♥ 2 r ′ κ q✉❡r✐❡s ✼ ✴ ✶✷

❈♦r♦❧❧❛r②✿ ❛tt❛❝❦ ✐♥ q✉❡r✐❡s ■♠♣r♦✈✐♥❣ ❆tt❛❝❦s k 1 k 2 k r m c E E E · · · · · · • ❬●❛➸✶✸❪ ✿ ❛tt❛❝❦ ✐♥ 2 κ + r ′− 1 n q✉❡r✐❡s r ′ ❘❡s✉❧t ✶✿ ❛tt❛❝❦ ✐♥ 2 r ′ κ q✉❡r✐❡s • ❆tt❛❝❦ ✐❞❡❛✿ • ❋♦r♠❛❧✐③❛t✐♦♥ ♦❢ ♠❡❡t✲✐♥✲t❤❡✲♠✐❞❞❧❡ ❛tt❛❝❦ √ 2 r ) κ ✐♥ ✐♥❝♦♠♣❛r❛❜❧❡ ♠♦❞❡❧ • ❬❉❉❑❙✶✷❪ ✿ ❛tt❛❝❦ ✐♥ 2 ( r − ✼ ✴ ✶✷

■♠♣r♦✈✐♥❣ ❆tt❛❝❦s k 1 k 2 k r m c E E E · · · · · · • ❬●❛➸✶✸❪ ✿ ❛tt❛❝❦ ✐♥ 2 κ + r ′− 1 n q✉❡r✐❡s r ′ ❘❡s✉❧t ✶✿ ❛tt❛❝❦ ✐♥ 2 r ′ κ q✉❡r✐❡s • ❆tt❛❝❦ ✐❞❡❛✿ • ❋♦r♠❛❧✐③❛t✐♦♥ ♦❢ ♠❡❡t✲✐♥✲t❤❡✲♠✐❞❞❧❡ ❛tt❛❝❦ √ 2 r ) κ ✐♥ ✐♥❝♦♠♣❛r❛❜❧❡ ♠♦❞❡❧ • ❬❉❉❑❙✶✷❪ ✿ ❛tt❛❝❦ ✐♥ 2 ( r − ❈♦r♦❧❧❛r②✿ ❛tt❛❝❦ ✐♥ 2 κ + r ′− 1 min { r ′ κ,n } q✉❡r✐❡s r ′ ✼ ✴ ✶✷