Some combinatorial aspects of perfect codes. rs tt - - PowerPoint PPT Presentation

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Some combinatorial aspects of perfect codes. rs tt - - PowerPoint PPT Presentation

Jan 20, 2024 470 likes •1.36k views

Some combinatorial aspects of perfect codes. rs tt rst s r s t r t st

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SLIDE 1

Some combinatorial aspects of perfect codes.

❈❧❛✉❞✐♦ ◗✉r❡s❤✐ ❙t❛t❡ ❯♥✐✈❡rs✐t② ♦❢ ❈❛♠♣✐♥❛s✱ ❇r❛③✐❧

❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❙✳❈♦st❛

❙♣❡❝✐❛❧ ❉❛②s ♦♥ ❈♦♠❜✐♥❛t♦r✐❛❧ ❈♦♥str✉❝t✐♦♥s ✉s✐♥❣ ❋✐♥✐t❡ ❋✐❡❧❞s ❛s ♣❛rt ♦❢

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶ ✴ ✺✾

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SLIDE 2

Codes in the Lee metric

❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥✿ ❈ ⊆ Z♥

q✳

■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥

♥ q ✭▲❡❡ ♠❡tr✐❝✮✱

❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ ✶✿ ❞ ① ② ♠✐♥ ① ② q ① ② ❢♦r ① ②

q

❋♦r ❡①❛♠♣❧❡✱ ❢♦r q ✾

✾

✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✿

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷ ✴ ✺✾

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SLIDE 3

Codes in the Lee metric

❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥✿ ❈ ⊆ Z♥

q✳

■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z♥

q ✭▲❡❡ ♠❡tr✐❝✮✱

❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ ✶✿ ❞ ① ② ♠✐♥ ① ② q ① ② ❢♦r ① ②

q

❋♦r ❡①❛♠♣❧❡✱ ❢♦r q ✾

✾

✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✿

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷ ✴ ✺✾

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SLIDE 4

Codes in the Lee metric

❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥✿ ❈ ⊆ Z♥

q✳

■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z♥

q ✭▲❡❡ ♠❡tr✐❝✮✱

❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞(①, ②) = ♠✐♥{|① − ②|, q − |① − ②|} ❢♦r ①, ② ∈ Zq ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q ✾

✾

✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✿

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷ ✴ ✺✾

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SLIDE 5

Codes in the Lee metric

❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥✿ ❈ ⊆ Z♥

q✳

■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z♥

q ✭▲❡❡ ♠❡tr✐❝✮✱

❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞(①, ②) = ♠✐♥{|① − ②|, q − |① − ②|} ❢♦r ①, ② ∈ Zq ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q = ✾ ⇒ Z✾ = {✵, ✶, ✷, ✸, ✹, ✺, ✻, ✼, ✽}✿

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷ ✴ ✺✾

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SLIDE 6

Codes in the Lee metric

❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥✿ ❈ ⊆ Z♥

q✳

■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z♥

q ✭▲❡❡ ♠❡tr✐❝✮✱

❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞(①, ②) = ♠✐♥{|① − ②|, q − |① − ②|} ❢♦r ①, ② ∈ Zq ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q = ✾ ⇒ Z✾ = {✵, ✶, ✷, ✸, ✹, ✺, ✻, ✼, ✽}✿ ■❢ ① = ✼ ❡ ② = ✷

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸ ✴ ✺✾

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SLIDE 7

Codes in the Lee metric

❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥✿ ❈ ⊆ Z♥

q✳

■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z♥

q ✭▲❡❡ ♠❡tr✐❝✮✱

❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞(①, ②) = ♠✐♥{|① − ②|, q − |① − ②|} ❢♦r ①, ② ∈ Zq ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q = ✾ ⇒ Z✾ = {✵, ✶, ✷, ✸, ✹, ✺, ✻, ✼, ✽}✿ ■❢ ① = ✼ ❡ ② = ✷✱ ⑤①✲②⑤❂✺

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹ ✴ ✺✾

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SLIDE 8

Codes in the Lee metric

❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥✿ ❈ ⊆ Z♥

q✳

■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z♥

q ✭▲❡❡ ♠❡tr✐❝✮✱

❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞(①, ②) = ♠✐♥{|① − ②|, q − |① − ②|} ❢♦r ①, ② ∈ Zq ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q = ✾ ⇒ Z✾ = {✵, ✶, ✷, ✸, ✹, ✺, ✻, ✼, ✽}✿ ■❢ ① = ✼ ❡ ② = ✷✱ ⑤①✲②⑤❂✺✱ q✲⑤①✲②⑤❂✹✱ ❞ ✼ ✷ ✹ ✭♠❡tr✐❝ ✐♥ t❤❡ ❣r❛♣❤✮✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺ ✴ ✺✾

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SLIDE 9

Codes in the Lee metric

❆ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥✿ ❈ ⊆ Z♥

q✳

■♥ ✶✾✺✽ ❈✳❨✳▲❡❡ ♣r♦♣♦s❡ t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z♥

q ✭▲❡❡ ♠❡tr✐❝✮✱

❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞(①, ②) = ♠✐♥{|① − ②|, q − |① − ②|} ❢♦r ①, ② ∈ Zq ❋♦r ❡①❛♠♣❧❡✱ ❢♦r q = ✾ ⇒ Z✾ = {✵, ✶, ✷, ✸, ✹, ✺, ✻, ✼, ✽}✿ ■❢ ① = ✼ ❡ ② = ✷✱ ⑤①✲②⑤❂✺✱ q✲⑤①✲②⑤❂✹✱ ❞(✼, ✷) =✹ ✭♠❡tr✐❝ ✐♥ t❤❡ ❣r❛♣❤✮✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺ ✴ ✺✾

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SLIDE 10

Codes in the Lee metric

❈♦♥s✐❞❡r ❛ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥✿ ❈ ⊆ Z♥

q✳

■♥ ✶✾✺✽ ❈✳ ❨✳ ▲❡❡ ♣r♦♣♦s❡s t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z♥

q ✭▲❡❡ ♠❡tr✐❝✮✱

❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞(①, ②) = ♠✐♥{|① − ②|, q − |① − ②|} ❢♦r ①, ② ∈ Zq ❋♦r ❛♥② ♥✱ ✐❢ ① = (①✶, . . . , ①♥) ∈ Z♥

q ❡ ② = (②✶, . . . , ②♥) ∈ Z♥ q✿

❞(①, ②) = ♥

✐=✶ ❞(①✐, ②✐)

✭q = ✷, ✸ ⇒ ▲❡❡❂❍❛♠♠✐♥❣✮✳ ❊①❛♠♣❧❡✿ ■♥

✷ ✾ ✇❡ ❤❛✈❡ ❞

✷ ✶ ✼ ✻ ✹ ✹ ✽✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✻ ✴ ✺✾

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SLIDE 11

Codes in the Lee metric

❈♦♥s✐❞❡r ❛ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥✿ ❈ ⊆ Z♥

q✳

■♥ ✶✾✺✽ ❈✳ ❨✳ ▲❡❡ ♣r♦♣♦s❡s t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z♥

q ✭▲❡❡ ♠❡tr✐❝✮✱

❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞(①, ②) = ♠✐♥{|① − ②|, q − |① − ②|} ❢♦r ①, ② ∈ Zq ❋♦r ❛♥② ♥✱ ✐❢ ① = (①✶, . . . , ①♥) ∈ Z♥

q ❡ ② = (②✶, . . . , ②♥) ∈ Z♥ q✿

❞(①, ②) = ♥

✐=✶ ❞(①✐, ②✐)

✭q = ✷, ✸ ⇒ ▲❡❡❂❍❛♠♠✐♥❣✮✳ ❊①❛♠♣❧❡✿ ■♥ Z✷

✾ ✇❡ ❤❛✈❡ ❞((✷, ✶), (✼, ✻)) =

✹ ✹ ✽✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✻ ✴ ✺✾

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SLIDE 12

Codes in the Lee metric

❈♦♥s✐❞❡r ❛ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥✿ ❈ ⊆ Z♥

q✳

■♥ ✶✾✺✽ ❈✳ ❨✳ ▲❡❡ ♣r♦♣♦s❡s t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z♥

q ✭▲❡❡ ♠❡tr✐❝✮✱

❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞(①, ②) = ♠✐♥{|① − ②|, q − |① − ②|} ❢♦r ①, ② ∈ Zq ❋♦r ❛♥② ♥✱ ✐❢ ① = (①✶, . . . , ①♥) ∈ Z♥

q ❡ ② = (②✶, . . . , ②♥) ∈ Z♥ q✿

❞(①, ②) = ♥

✐=✶ ❞(①✐, ②✐)

✭q = ✷, ✸ ⇒ ▲❡❡❂❍❛♠♠✐♥❣✮✳ ❊①❛♠♣❧❡✿ ■♥ Z✷

✾ ✇❡ ❤❛✈❡ ❞((✷, ✶), (✼, ✻)) = ✹ +

✹ ✽✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✻ ✴ ✺✾

slide-13
SLIDE 13

Codes in the Lee metric

❈♦♥s✐❞❡r ❛ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥✿ ❈ ⊆ Z♥

q✳

■♥ ✶✾✺✽ ❈✳ ❨✳ ▲❡❡ ♣r♦♣♦s❡s t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z♥

q ✭▲❡❡ ♠❡tr✐❝✮✱

❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞(①, ②) = ♠✐♥{|① − ②|, q − |① − ②|} ❢♦r ①, ② ∈ Zq ❋♦r ❛♥② ♥✱ ✐❢ ① = (①✶, . . . , ①♥) ∈ Z♥

q ❡ ② = (②✶, . . . , ②♥) ∈ Z♥ q✿

❞(①, ②) = ♥

✐=✶ ❞(①✐, ②✐)

✭q = ✷, ✸ ⇒ ▲❡❡❂❍❛♠♠✐♥❣✮✳ ❊①❛♠♣❧❡✿ ■♥ Z✷

✾ ✇❡ ❤❛✈❡ ❞((✷, ✶), (✼, ✻)) = ✹ + ✹

✽✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✻ ✴ ✺✾

slide-14
SLIDE 14

Codes in the Lee metric

❈♦♥s✐❞❡r ❛ q✲❛r② ❝♦❞❡ ♦❢ ❧❡♥❣t❤ ♥✿ ❈ ⊆ Z♥

q✳

■♥ ✶✾✺✽ ❈✳ ❨✳ ▲❡❡ ♣r♦♣♦s❡s t❤❡ ✉s❡ ♦❢ ❛ ♠❡tr✐❝ ✐♥ Z♥

q ✭▲❡❡ ♠❡tr✐❝✮✱

❛♣♣r♦♣r✐❛t❡ t♦ ❝♦rr❡❝t ❡rr♦rs ✐♥ ❝❡rt❛✐♥ t②♣❡s ♦❢ ❝❤❛♥♥❡❧s✳ ❋♦r ♥ = ✶✿ ❞(①, ②) = ♠✐♥{|① − ②|, q − |① − ②|} ❢♦r ①, ② ∈ Zq ❋♦r ❛♥② ♥✱ ✐❢ ① = (①✶, . . . , ①♥) ∈ Z♥

q ❡ ② = (②✶, . . . , ②♥) ∈ Z♥ q✿

❞(①, ②) = ♥

✐=✶ ❞(①✐, ②✐)

✭q = ✷, ✸ ⇒ ▲❡❡❂❍❛♠♠✐♥❣✮✳ ❊①❛♠♣❧❡✿ ■♥ Z✷

✾ ✇❡ ❤❛✈❡ ❞((✷, ✶), (✼, ✻)) = ✹ + ✹ = ✽✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✻ ✴ ✺✾

slide-15
SLIDE 15

Codes in the Lee metric

▲❡❡ ♠❡tr✐❝ ❛s t❤❡ ❞✐st❛♥❝❡ ✐♥ t❤❡ ❣r❛♣❤ ✭t♦r✉s✮

❊①❛♠♣❧❡✿ ■♥ Z✷

✾ ✇❡ ❤❛✈❡ ❞((✷, ✶), (✼, ✻)) = ✹ + ✹ = ✽✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✼ ✴ ✺✾

slide-16
SLIDE 16

Codes in the Lee metric

▲❡❡ ♠❡tr✐❝ ❛s t❤❡ ❞✐st❛♥❝❡ ✐♥ t❤❡ ❣r❛♣❤ ✭t♦r✉s✮

❊①❛♠♣❧❡✿ ■♥ Z✷

✾ ✇❡ ❤❛✈❡ ❞((✷, ✶), (✼, ✻)) = ✹ +✹ = ✽✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✽ ✴ ✺✾

slide-17
SLIDE 17

Codes in the Lee metric

▲❡❡ ♠❡tr✐❝ ❛s t❤❡ ❞✐st❛♥❝❡ ✐♥ t❤❡ ❣r❛♣❤ ✭t♦r✉s✮

❊①❛♠♣❧❡✿ ■♥ Z✷

✾ ✇❡ ❤❛✈❡ ❞((✷, ✶), (✼, ✻)) = ✹ + ✹ = ✽✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✾ ✴ ✺✾

slide-18
SLIDE 18

Codes in the Lee metric

❚❤❡ r❡s✉r❣❡♥❝❡ ♦❢ ▲❡❡ ❈♦❞❡s

❊♥❣✐♥❡❡r✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s ❈♦♥str❛✐♥❡❞ ❛♥❞ ♣❛rt✐❛❧✲r❡s♣♦♥s❡ ❝❤❛♥♥❡❧s✳

❘✳ ▼✳ ❘♦t❤ ❛♥❞ P✳ ❍✳ ❙✐❡❣❡❧✳ ▲❡❡✲♠❡tr✐❝ ❇❈❍ ❝♦❞❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥ t♦ ❝♦♥str❛✐♥❡❞ ❛♥❞ ♣❛rt✐❛❧✲r❡s♣♦♥s❡ ❝❤❛♥♥❡❧s✳ ■❊❊❊ ❚r❛♥s✳ ♦♥ ■♥❢♦r♠✳ ❚❤❡♦r②✱ ✈♦❧✳ ■❚✲✹✵✱ ♣♣✳✶✵✽✸✲✶✵✾✻✱ ❏✉❧② ✶✾✾✹✳

■♥t❡r❧❡❛✈✐♥❣ s❝❤❡♠❡s✳

▼✳ ❇❧❛✉♠✱ ❏✳ ❇r✉❝❦ ❛♥❞ ❆✳ ❱❛r❞②✳ ■♥t❡r❧❡❛✈✐♥❣ s❝❤❡♠❡s ❢♦r ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❝❧✉st❡r ❡rr♦rs✳ ■❊❊❊ ❚r❛♥s✳ ■♥❢♦r♠✳ ❚❤❡♦r②✱ ✈♦❧✳ ■❚✲✹✹✱ ♣♣✳ ✼✸✵✲✼✹✸✱ ▼❛r❝❤ ✶✾✾✽✳

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❜✉rst✲❡rr♦r✲❝♦rr❡❝t✐♦♥✳

❚✳ ❊t③✐♦♥ ❛♥❞ ❊✳ ❨❛❛❦♦❜✐✳ ❊rr♦r✲❝♦rr❡❝t✐♦♥ ♦❢ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❜✉rsts✳ ■❊❊❊ ❚r❛♥s✳ ♦♥ ■♥❢♦r♠✳ ❚❤❡♦r②✱ ✈♦❧✳ ■❚✲✺✺✱ ♣♣✳ ✾✻✶✲✾✼✻✱ ▼❛r❝❤ ✷✵✵✾✳

❊rr♦r✲❝♦rr❡❝t✐♦♥ ❢♦r ✢❛s❤ ♠❡♠♦r✐❡s✳

❆✳ ❇❛r❣ ❛♥❞ ❆✳ ▼❛③✉♠❞❛r✳ ❈♦❞❡s ✐♥ ♣❡r♠✉t❛t✐♦♥s ❛♥❞ ❡rr♦r ❝♦rr❡❝t✐♦♥ ❢♦r r❛♥❦ ♠♦❞✉❧❛t✐♦♥✳ ■❊❊❊ ❚r❛♥s✳ ■♥❢✳ ❚❤❡♦r②✱ ✈♦❧✳ ✺✻✱ ♥♦✳ ✼✱ ♣♣✳✸✶✺✽✲✸✶✻✺✱ ❏✉❧✳ ✷✵✶✵✳ ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✵ ✴ ✺✾

slide-19
SLIDE 19

Codes in the Lee metric

❚❤❡ r❡s✉r❣❡♥❝❡ ♦❢ ▲❡❡ ❈♦❞❡s

❚❤❡♦r❡t✐❝❛❧ r❡s❡❛r❝❤ ❊♥✉♠❡r❛t✐♥❣ ❛♥❞ ❞❡❝♦❞✐♥❣ ♣❡r❢❡❝t ❧✐♥❡❛r ▲❡❡ ❝♦❞❡s✳

❇✳ ❆❧❇❞❛✐✇✐✱ P✳ ❍♦r❛❦✱ ▲✳ ▼✐❧❛③③♦✳ ❊♥✉♠❡r❛t✐♥❣ ❛♥❞ ❞❡❝♦❞✐♥❣ ♣❡r❢❡❝t ❧✐♥❡❛r ▲❡❡ ❝♦❞❡s✳ ❉❡s✳ ❈♦❞❡s✳ ❈r②♣t✳✱ ✈♦❧✳ ✺✷ ♥♦✳ ✷✱ ♣♣✳ ✶✺✺✲✶✻✷✱ ✷✵✵✾✳

❉❡♥s❡ ▲❡❡ ❈♦❞❡s✳

❚✳ ❊t③✐♦♥✱ ❆✳ ❱❛r❞②✱ ❊✳ ❨❛❛❦♦❜✐✳ ❉❡♥s❡ ❡rr♦r✲❝♦rr❡❝t✐♥❣ ❝♦❞❡s ✐♥ t❤❡ ▲❡❡ ♠❡tr✐❝✳ ■♥❢♦r♠❛t✐♦♥ ❚❤❡♦r② ❲♦r❦s❤♦♣ ✭■❚❲✮✱ ✷✵✶✵ ■❊❊❊✳

❙♣❡❝✐❛❧ ❝♦♥str✉❝t✐♦♥s ❢♦r ♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s✳

❚✳ ❊t③✐♦♥✳ Pr♦❞✉❝t ❝♦♥str✉❝t✐♦♥s ❢♦r ♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s✳ ■❊❊❊ ❚r❛♥s✳ ■♥❢♦r♠✳ ❚❤✳✺✼✭✷✵✶✶✮✱ ♥♦✳✶✶✱ ✼✹✼✸✲✼✹✽✶✳

❉✐❛♠❡t❡r ♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s✳

P✳ ❍♦r❛❦✱ ❇✳❋✳ ❆❧❇❞❛✐✇✐✳ ❉✐❛♠❡t❡r ♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s✳ ■❊❊❊ ❚r❛♥s✳ ■♥❢♦r♠✳ ❚❤✳✺✽✭✷✵✶✷✮✱ ♥♦✳✽✱ ✺✹✾✵✲✺✹✾✾✳ ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✶ ✴ ✺✾

slide-20
SLIDE 20

Codes in the Lee metric

▲❡t ❈ ⊆ Z♥

q ❜❡ ❛ q✲❛r② ❝♦❞❡✳

❉❡✜♥✐t✐♦♥s

❆s ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❍❛♠♠✐♥❣ ♠❡tr✐❝✱ ❈ ✐s ❛ ♣❡r❢❡❝t ▲❡❡ ❝♦❞❡ ✇❤❡♥ Z♥

q = ❝∈❈ ❇(❝, ❡)✱ ✇❤❡r❡ ❡ ✐s t❤❡ ♣❛❝❦✐♥❣ r❛❞✐✉s ❛♥❞ t❤❡ ❜❛❧❧s ❛r❡

▲❡❡✲❜❛❧❧s✳ ❲❡ ❞❡♥♦t❡ ❜②

P▲ ♥ ❡ q ❈

♥ q

❈ ✐s ❡✲♣❡r❢❡❝t ▲P▲ ♥ ❡ q ❈ P▲ ♥ ❡ q ❈ ✐s ❧✐♥❡❛r P▲ ♥ ❡ ❈

♥

❈ ✐s ❡✲♣❡r❢❡❝t ❞ ① ②

♥ ✐ ✶ ①✐

②✐ ▲P▲ ♥ ❡ ❈ P▲ ♥ ❡ ❈ ✐s ❧✐♥❡❛r

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✷ ✴ ✺✾

slide-21
SLIDE 21

Codes in the Lee metric

▲❡t ❈ ⊆ Z♥

q ❜❡ ❛ q✲❛r② ❝♦❞❡✳

❉❡✜♥✐t✐♦♥s

❆s ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❍❛♠♠✐♥❣ ♠❡tr✐❝✱ ❈ ✐s ❛ ♣❡r❢❡❝t ▲❡❡ ❝♦❞❡ ✇❤❡♥ Z♥

q = ❝∈❈ ❇(❝, ❡)✱ ✇❤❡r❡ ❡ ✐s t❤❡ ♣❛❝❦✐♥❣ r❛❞✐✉s ❛♥❞ t❤❡ ❜❛❧❧s ❛r❡

▲❡❡✲❜❛❧❧s✳ ❲❡ ❞❡♥♦t❡ ❜②

P▲(♥, ❡, q) = {❈ ⊆ Z♥

q : ❈ ✐s ❡✲♣❡r❢❡❝t}

▲P▲(♥, ❡, q) = {❈ ∈ P▲(♥, ❡, q) : ❈ ✐s ❧✐♥❡❛r} P▲ ♥ ❡ ❈

♥

❈ ✐s ❡✲♣❡r❢❡❝t ❞ ① ②

♥ ✐ ✶ ①✐

②✐ ▲P▲ ♥ ❡ ❈ P▲ ♥ ❡ ❈ ✐s ❧✐♥❡❛r

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✷ ✴ ✺✾

slide-22
SLIDE 22

Codes in the Lee metric

▲❡t ❈ ⊆ Z♥

q ❜❡ ❛ q✲❛r② ❝♦❞❡✳

❉❡✜♥✐t✐♦♥s

❆s ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❍❛♠♠✐♥❣ ♠❡tr✐❝✱ ❈ ✐s ❛ ♣❡r❢❡❝t ▲❡❡ ❝♦❞❡ ✇❤❡♥ Z♥

q = ❝∈❈ ❇(❝, ❡)✱ ✇❤❡r❡ ❡ ✐s t❤❡ ♣❛❝❦✐♥❣ r❛❞✐✉s ❛♥❞ t❤❡ ❜❛❧❧s ❛r❡

▲❡❡✲❜❛❧❧s✳ ❲❡ ❞❡♥♦t❡ ❜②

P▲(♥, ❡, q) = {❈ ⊆ Z♥

q : ❈ ✐s ❡✲♣❡r❢❡❝t}

▲P▲(♥, ❡, q) = {❈ ∈ P▲(♥, ❡, q) : ❈ ✐s ❧✐♥❡❛r} P▲(♥, ❡) = {❈ ⊆ Z♥ : ❈ ✐s ❡✲♣❡r❢❡❝t},

  • ❞(①, ②) = ♥

✐=✶ |①✐ − ②✐|

  • ▲P▲(♥, ❡) = {❈ ∈ P▲(♥, ❡) : ❈ ✐s ❧✐♥❡❛r}

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✷ ✴ ✺✾

slide-23
SLIDE 23

Existence of Perfect Lee Codes

▼❛✐♥ ♣r♦❜❧❡♠

❈❤❛r❛❝t❡r✐③❡ t❤❡ tr✐♣❧❡ts (♥, ❡, q) ❢♦r ✇❤✐❝❤ P▲(♥, ❡, q) = ∅✳

  • ♦❧♦♠❜✲❲❡❧❝❤ ✭✶✾✼✵✮

❙✳ ❲✳ ●♦❧♦♠❜✱ ▲✳ ❘✳ ❲❡❧❝❤✳ P❡r❢❡❝t ❈♦❞❡s ✐♥ t❤❡ ▲❡❡ ♠❡tr✐❝ ❛♥❞ t❤❡ ♣❛❝❦✐♥❣ ♦❢ ♣♦❧②♥♦♠✐♥♦❡s✱ ❙■❆▼ ❏♦✉r♥❛❧ ❆♣♣❧✐❡❞ ▼❛t❤✳✱ ✈♦❧✳ ✶✽✱ ♣♣✳ ✸✵✷✲✸✶✼✳ ✶✾✼✵✳

❋♦r ❡ ✶ ✇❡ ❤❛✈❡ P▲ ♥ ✶ ❢♦r ❛❧❧ ♥✳ ❋♦r ♥ ✷ ✇❡ ❤❛✈❡ P▲ ✷ ❡ ❢♦r ❛❧❧ ❡✳ ❋♦r ❡❛❝❤ ♥ t❤❡r❡ ❡①✐sts ❡♥ s✳t✳ P▲ ♥ ❡ ❢♦r ❛❧❧ ❡ ❡♥✳ ❈♦♥❥❡❝t✉r❡✿ ❋♦r ♥ ✷ ❛♥❞ ❡ ✶ ✇❡ ❤❛✈❡ P▲ ♥ ❡ ✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✸ ✴ ✺✾

slide-24
SLIDE 24

Existence of Perfect Lee Codes

▼❛✐♥ ♣r♦❜❧❡♠

❈❤❛r❛❝t❡r✐③❡ t❤❡ tr✐♣❧❡ts (♥, ❡, q) ❢♦r ✇❤✐❝❤ P▲(♥, ❡, q) = ∅✳

  • ♦❧♦♠❜✲❲❡❧❝❤ ✭✶✾✼✵✮

❙✳ ❲✳ ●♦❧♦♠❜✱ ▲✳ ❘✳ ❲❡❧❝❤✳ P❡r❢❡❝t ❈♦❞❡s ✐♥ t❤❡ ▲❡❡ ♠❡tr✐❝ ❛♥❞ t❤❡ ♣❛❝❦✐♥❣ ♦❢ ♣♦❧②♥♦♠✐♥♦❡s✱ ❙■❆▼ ❏♦✉r♥❛❧ ❆♣♣❧✐❡❞ ▼❛t❤✳✱ ✈♦❧✳ ✶✽✱ ♣♣✳ ✸✵✷✲✸✶✼✳ ✶✾✼✵✳

❋♦r ❡ = ✶ ✇❡ ❤❛✈❡ P▲(♥, ✶) = ∅ ❢♦r ❛❧❧ ♥✳ ❋♦r ♥ = ✷ ✇❡ ❤❛✈❡ P▲(✷, ❡) = ∅ ❢♦r ❛❧❧ ❡✳ ❋♦r ❡❛❝❤ ♥ t❤❡r❡ ❡①✐sts ❡♥ s✳t✳ P▲(♥, ❡) = ∅ ❢♦r ❛❧❧ ❡ ≥ ❡♥✳ ❈♦♥❥❡❝t✉r❡✿ ❋♦r ♥ ✷ ❛♥❞ ❡ ✶ ✇❡ ❤❛✈❡ P▲ ♥ ❡ ✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✸ ✴ ✺✾

slide-25
SLIDE 25

Existence of Perfect Lee Codes

▼❛✐♥ ♣r♦❜❧❡♠

❈❤❛r❛❝t❡r✐③❡ t❤❡ tr✐♣❧❡ts (♥, ❡, q) ❢♦r ✇❤✐❝❤ P▲(♥, ❡, q) = ∅✳

  • ♦❧♦♠❜✲❲❡❧❝❤ ✭✶✾✼✵✮

❙✳ ❲✳ ●♦❧♦♠❜✱ ▲✳ ❘✳ ❲❡❧❝❤✳ P❡r❢❡❝t ❈♦❞❡s ✐♥ t❤❡ ▲❡❡ ♠❡tr✐❝ ❛♥❞ t❤❡ ♣❛❝❦✐♥❣ ♦❢ ♣♦❧②♥♦♠✐♥♦❡s✱ ❙■❆▼ ❏♦✉r♥❛❧ ❆♣♣❧✐❡❞ ▼❛t❤✳✱ ✈♦❧✳ ✶✽✱ ♣♣✳ ✸✵✷✲✸✶✼✳ ✶✾✼✵✳

❋♦r ❡ = ✶ ✇❡ ❤❛✈❡ P▲(♥, ✶) = ∅ ❢♦r ❛❧❧ ♥✳ ❋♦r ♥ = ✷ ✇❡ ❤❛✈❡ P▲(✷, ❡) = ∅ ❢♦r ❛❧❧ ❡✳ ❋♦r ❡❛❝❤ ♥ t❤❡r❡ ❡①✐sts ❡♥ s✳t✳ P▲(♥, ❡) = ∅ ❢♦r ❛❧❧ ❡ ≥ ❡♥✳ ❈♦♥❥❡❝t✉r❡✿ ❋♦r ♥ > ✷ ❛♥❞ ❡ > ✶ ✇❡ ❤❛✈❡ P▲(♥, ❡) = ∅✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✸ ✴ ✺✾

slide-26
SLIDE 26

Existence of Perfect Lee Codes

▼❛✐♥ ♣r♦❜❧❡♠

❈❤❛r❛❝t❡r✐③❡ t❤❡ tr✐♣❧❡ts (♥, ❡, q) ❢♦r ✇❤✐❝❤ P▲(♥, ❡, q) = ∅✳

  • ♦❧♦♠❜✲❲❡❧❝❤ ✭✶✾✼✵✮

❙✳ ❲✳ ●♦❧♦♠❜✱ ▲✳ ❘✳ ❲❡❧❝❤✳ P❡r❢❡❝t ❈♦❞❡s ✐♥ t❤❡ ▲❡❡ ♠❡tr✐❝ ❛♥❞ t❤❡ ♣❛❝❦✐♥❣ ♦❢ ♣♦❧②♥♦♠✐♥♦❡s✱ ❙■❆▼ ❏♦✉r♥❛❧ ❆♣♣❧✐❡❞ ▼❛t❤✳✱ ✈♦❧✳ ✶✽✱ ♣♣✳ ✸✵✷✲✸✶✼✳ ✶✾✼✵✳

❋♦r ❡ = ✶ ✇❡ ❤❛✈❡ P▲(♥, ✶) = ∅ ❢♦r ❛❧❧ ♥✳ ❋♦r ♥ = ✷ ✇❡ ❤❛✈❡ P▲(✷, ❡) = ∅ ❢♦r ❛❧❧ ❡✳ ❋♦r ❡❛❝❤ ♥ t❤❡r❡ ❡①✐sts ❡♥ s✳t✳ P▲(♥, ❡) = ∅ ❢♦r ❛❧❧ ❡ ≥ ❡♥✳ ❈♦♥❥❡❝t✉r❡✿ ❋♦r ♥ > ✷ ❛♥❞ ❡ > ✶ ✇❡ ❤❛✈❡ P▲(♥, ❡) = ∅✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✹ ✴ ✺✾

slide-27
SLIDE 27

Existence of Perfect Lee Codes

Pr♦♦❢ ♦❢ ●♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ❢♦r t❤❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✿

  • ♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ♣r❡s❡♥t t❤❡ ❝♦❞❡s ❉❡ = (❡, ❡ + ✶) ⊂ Z✷

q ❢♦r

q = ✷❡✷ + ✷❡ + ✶ ❛♥❞ ♣r♦✈❡ t❤❛t t❤❡s❡ ❝♦❞❡s ❛r❡ ♣❡r❢❡❝t✱ t❤❡♥ P▲(✷, ❡, q) = ∅ ❢♦r q = ✷❡✷ + ✷❡ + ✶ ✭⇒ P▲(✷, ❡) = ∅✮✳

❊①❛♠♣❧❡✿ ❉✷ ✷ ✸

✷ ✶✸

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✺ ✴ ✺✾

slide-28
SLIDE 28

Existence of Perfect Lee Codes

Pr♦♦❢ ♦❢ ●♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ❢♦r t❤❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✿

  • ♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ♣r❡s❡♥t t❤❡ ❝♦❞❡s ❉❡ = (❡, ❡ + ✶) ⊂ Z✷

q ❢♦r

q = ✷❡✷ + ✷❡ + ✶ ❛♥❞ ♣r♦✈❡ t❤❛t t❤❡s❡ ❝♦❞❡s ❛r❡ ♣❡r❢❡❝t✱ t❤❡♥ P▲(✷, ❡, q) = ∅ ❢♦r q = ✷❡✷ + ✷❡ + ✶ ✭⇒ P▲(✷, ❡) = ∅✮✳

❊①❛♠♣❧❡✿ ❉✷ = (✷, ✸) ⊆ Z✷

✶✸

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✺ ✴ ✺✾

slide-29
SLIDE 29

Existence of Perfect Lee Codes

Pr♦♦❢ ♦❢ ●♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ❢♦r t❤❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✿

  • ♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ♣r❡s❡♥t t❤❡ ❝♦❞❡s ❉❡ = (❡, ❡ + ✶) ⊂ Z✷

q ❢♦r

q = ✷❡✷ + ✷❡ + ✶ ❛♥❞ ♣r♦✈❡ t❤❛t t❤❡s❡ ❝♦❞❡s ❛r❡ ♣❡r❢❡❝t✱ t❤❡♥ P▲(✷, ❡, q) = ∅ ❢♦r q = ✷❡✷ + ✷❡ + ✶ ✭⇒ P▲(✷, ❡) = ∅✮✳

❊①❛♠♣❧❡✿ ❉✷ = (✷, ✸) = {(✵, ✵),

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✻ ✴ ✺✾

slide-30
SLIDE 30

Existence of Perfect Lee Codes

Pr♦♦❢ ♦❢ ●♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ❢♦r t❤❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✿

  • ♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ♣r❡s❡♥t t❤❡ ❝♦❞❡s ❉❡ = (❡, ❡ + ✶) ⊂ Z✷

q ❢♦r

q = ✷❡✷ + ✷❡ + ✶ ❛♥❞ ♣r♦✈❡ t❤❛t t❤❡s❡ ❝♦❞❡s ❛r❡ ♣❡r❢❡❝t✱ t❤❡♥ P▲(✷, ❡, q) = ∅ ❢♦r q = ✷❡✷ + ✷❡ + ✶ ✭⇒ P▲(✷, ❡) = ∅✮✳

❊①❛♠♣❧❡✿ ❉✷ = (✷, ✸) = {(✵, ✵), (✷, ✸),

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✼ ✴ ✺✾

slide-31
SLIDE 31

Existence of Perfect Lee Codes

Pr♦♦❢ ♦❢ ●♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ❢♦r t❤❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✿

  • ♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ♣r❡s❡♥t t❤❡ ❝♦❞❡s ❉❡ = (❡, ❡ + ✶) ⊂ Z✷

q ❢♦r

q = ✷❡✷ + ✷❡ + ✶ ❛♥❞ ♣r♦✈❡ t❤❛t t❤❡s❡ ❝♦❞❡s ❛r❡ ♣❡r❢❡❝t✱ t❤❡♥ P▲(✷, ❡, q) = ∅ ❢♦r q = ✷❡✷ + ✷❡ + ✶ ✭⇒ P▲(✷, ❡) = ∅✮✳

❊①❛♠♣❧❡✿ ❉✷ = (✷, ✸) = {(✵, ✵), (✷, ✸), (✹, ✻),

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✽ ✴ ✺✾

slide-32
SLIDE 32

Existence of Perfect Lee Codes

Pr♦♦❢ ♦❢ ●♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ❢♦r t❤❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✿

  • ♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ♣r❡s❡♥t t❤❡ ❝♦❞❡s ❉❡ = (❡, ❡ + ✶) ⊂ Z✷

q ❢♦r

q = ✷❡✷ + ✷❡ + ✶ ❛♥❞ ♣r♦✈❡ t❤❛t t❤❡s❡ ❝♦❞❡s ❛r❡ ♣❡r❢❡❝t✱ t❤❡♥ P▲(✷, ❡, q) = ∅ ❢♦r q = ✷❡✷ + ✷❡ + ✶ ✭⇒ P▲(✷, ❡) = ∅✮✳

❊①❛♠♣❧❡✿ ❉✷ = (✷, ✸) = {(✵, ✵), (✷, ✸), (✹, ✻), (✻, ✾)

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✶✾ ✴ ✺✾

slide-33
SLIDE 33

Existence of Perfect Lee Codes

Pr♦♦❢ ♦❢ ●♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ❢♦r t❤❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✿

  • ♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ♣r❡s❡♥t t❤❡ ❝♦❞❡s ❉❡ = (❡, ❡ + ✶) ⊂ Z✷

q ❢♦r

q = ✷❡✷ + ✷❡ + ✶ ❛♥❞ ♣r♦✈❡ t❤❛t t❤❡s❡ ❝♦❞❡s ❛r❡ ♣❡r❢❡❝t✱ t❤❡♥ P▲(✷, ❡, q) = ∅ ❢♦r q = ✷❡✷ + ✷❡ + ✶ ✭⇒ P▲(✷, ❡) = ∅✮✳

❊①❛♠♣❧❡✿ ❉✷ = (✷, ✸) = {(✵, ✵), (✷, ✸), (✹, ✻), (✻, ✾), . . . , (✶✶, ✶✵)

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷✵ ✴ ✺✾

slide-34
SLIDE 34

Existence of Perfect Lee Codes

Pr♦♦❢ ♦❢ ●♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ❢♦r t❤❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝❛s❡✿

  • ♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤ ♣r❡s❡♥t t❤❡ ❝♦❞❡s ❉❡ = (❡, ❡ + ✶) ⊂ Z✷

q ❢♦r

q = ✷❡✷ + ✷❡ + ✶ ❛♥❞ ♣r♦✈❡ t❤❛t t❤❡s❡ ❝♦❞❡s ❛r❡ ♣❡r❢❡❝t✱ t❤❡♥ P▲(✷, ❡, q) = ∅ ❢♦r q = ✷❡✷ + ✷❡ + ✶ ✭⇒ P▲(✷, ❡) = ∅✮✳

❊①❛♠♣❧❡✿ ❉✷ = (✷, ✸) = {(✵, ✵), (✷, ✸), (✹, ✻), (✻, ✾), . . . , (✶✶, ✶✵)

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷✶ ✴ ✺✾

slide-35
SLIDE 35

❘❡❧❛t❡❞ q✉❡st✐♦♥s✳

❋♦r ✇❤✐❝❤ (❡, q) ✇❡ ❤❛✈❡ P▲(✷, ❡, q) = ∅❄ ■♥ t❤❛t ❝❛s❡✱ ✐s ✐t ♣♦ss✐❜❧❡ t♦ ❞❡s❝r✐❜❡ ❛❧❧ t❤❡s❡ ❝♦❞❡s❄ ✭❘❡♠❛r❦✿ ✇❡ ❛r❡ ❝♦♥s✐❞❡r✐♥❣ ❧✐♥❡❛r ❛♥❞ ♥♦♥✲❧✐♥❡❛r ❝♦❞❡s✳✮ ❲❤❛t ❛r❡ t❤❡ ♣♦ss✐❜❧❡ str✉❝t✉r❡s ❛s ❛❜❡❧✐❛♥ ❣r♦✉♣s ♦❢ t❤❡s❡ ❝♦❞❡s❄

❲❡ ❝❛♥ ✉s❡ t❤❡ ❣❡♦♠❡tr② ♦❢ ♣♦❧②♦♠✐♥♦❡s ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛r❣✉♠❡♥ts✳

❙✳❈♦st❛ ❛♥❞ ❈✳◗✉r❡s❤✐✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝♦❞❡s ✐♥ t❤❡ ▲❡❡ ♠❡tr✐❝✳ ❳❳❳■ ❇r❛③✐❧✐❛♥ ❙②♠♣♦s✐✉♠ ♦❢ ❚❡❧❡❝♦♠♠✉♥✐❝❛t✐♦♥s✱ ✷✵✶✸✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷✷ ✴ ✺✾

slide-36
SLIDE 36

❘❡❧❛t❡❞ q✉❡st✐♦♥s✳

❋♦r ✇❤✐❝❤ (❡, q) ✇❡ ❤❛✈❡ P▲(✷, ❡, q) = ∅❄ ■♥ t❤❛t ❝❛s❡✱ ✐s ✐t ♣♦ss✐❜❧❡ t♦ ❞❡s❝r✐❜❡ ❛❧❧ t❤❡s❡ ❝♦❞❡s❄ ✭❘❡♠❛r❦✿ ✇❡ ❛r❡ ❝♦♥s✐❞❡r✐♥❣ ❧✐♥❡❛r ❛♥❞ ♥♦♥✲❧✐♥❡❛r ❝♦❞❡s✳✮ ❲❤❛t ❛r❡ t❤❡ ♣♦ss✐❜❧❡ str✉❝t✉r❡s ❛s ❛❜❡❧✐❛♥ ❣r♦✉♣s ♦❢ t❤❡s❡ ❝♦❞❡s❄

❲❡ ❝❛♥ ✉s❡ t❤❡ ❣❡♦♠❡tr② ♦❢ ♣♦❧②♦♠✐♥♦❡s ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛r❣✉♠❡♥ts✳

❙✳❈♦st❛ ❛♥❞ ❈✳◗✉r❡s❤✐✳ ❈❧❛ss✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❜✐❞✐♠❡♥s✐♦♥❛❧ ❝♦❞❡s ✐♥ t❤❡ ▲❡❡ ♠❡tr✐❝✳ ❳❳❳■ ❇r❛③✐❧✐❛♥ ❙②♠♣♦s✐✉♠ ♦❢ ❚❡❧❡❝♦♠♠✉♥✐❝❛t✐♦♥s✱ ✷✵✶✸✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷✷ ✴ ✺✾

slide-37
SLIDE 37

Theorem

❋♦r ❡, q ∈ Z+ ✇❡ ❞❡✜♥❡✿ q❡ = ❡✷ + (❡ + ✶)✷, ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡), η✶ = (✶, −(✷❡ + ✶)), η✷ = (✵, q❡) ∈ Z✷

q,

❉❡ = ν✶Z + ν✷Z ❛♥❞ ✈ = (−①, ②) ✐s t❤❡ ❝♦♥❥✉❣❛t❡ ♦❢ ✈ = (①, ②)✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷✸ ✴ ✺✾

slide-38
SLIDE 38

Theorem

❋♦r ❡, q ∈ Z+ ✇❡ ❞❡✜♥❡✿ q❡ = ❡✷ + (❡ + ✶)✷, ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡), η✶ = (✶, −(✷❡ + ✶)), η✷ = (✵, q❡) ∈ Z✷

q, ❉❡ = ν✶Z + ν✷Z ❛♥❞ ✈ = (−①, ②) ✐s t❤❡ ❝♦♥❥✳ ♦❢ ✈ = (①, ②)✳ ✶

✭❊①✐st❡♥❝❡✮ P▲(✷, ❡, q) = ∅ ⇔ q ≡ ✵ (♠♦❞ q❡)✳

✷

✭❈❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❈ P▲ ✷ ❡ q ❈ ❝ ❉❡ ♦r ❈ ❝ ❉❡ ❢♦r ❛♥② ❝ ❈ ✭✐♥ ♣❛rt✐❝✉❧❛r ❈ ❝ ✐s ❛ ❣r♦✉♣✮✳

✸

✭❙tr✉❝t✉r❡✮ ▲❡t ❈ P▲ ✷ ❡ q ❛♥❞ ●❈ ❈ ❝ t❤❡ ❣r♦✉♣ ❛ss♦❝✳ ✇✐t❤ ❈✳ ✐✮ ●❈ ✐s ❝②❝❧✐❝ ✐✛ q q❡✳ ■♥ t❤✐s ❝❛s❡ ●❈

q ✇✐t❤ ❣❡♥❡r❛t♦r ✶

❡ ❡ ✶ ✐❢ ●❈ ❉❡ ♦r

✶ ✐❢ ●❈

❉❡✳ ✐✐✮ ■❢ q ❤q❡ ❝♦♠ ❤ ✶ t❤❡♥ ●❈

q ❤✳

▼♦r❡♦✈❡r✱ ●❈

✶ ✷

✐❢ ●❈ ❉❡ ♦r ●❈

✶ ✷

✐❢ ●❈ ❉❡✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷✹ ✴ ✺✾

slide-39
SLIDE 39

Theorem

❋♦r ❡, q ∈ Z+ ✇❡ ❞❡✜♥❡✿ q❡ = ❡✷ + (❡ + ✶)✷, ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡), η✶ = (✶, −(✷❡ + ✶)), η✷ = (✵, q❡) ∈ Z✷

q, ❉❡ = ν✶Z + ν✷Z ❛♥❞ ✈ = (−①, ②) ✐s t❤❡ ❝♦♥❥✳ ♦❢ ✈ = (①, ②)✳ ✶

✭❊①✐st❡♥❝❡✮ P▲(✷, ❡, q) = ∅ ⇔ q ≡ ✵ (♠♦❞ q❡)✳

✷

✭❈❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❈ ∈ P▲(✷, ❡, q) ⇔ ❈ = ❝ + ❉❡ ♦r ❈ = ❝ + ❉❡ ❢♦r ❛♥② ❝ ∈ ❈ ✭✐♥ ♣❛rt✐❝✉❧❛r ❈ − ❝ ✐s ❛ ❣r♦✉♣✮✳

✸

✭❙tr✉❝t✉r❡✮ ▲❡t ❈ P▲ ✷ ❡ q ❛♥❞ ●❈ ❈ ❝ t❤❡ ❣r♦✉♣ ❛ss♦❝✳ ✇✐t❤ ❈✳ ✐✮ ●❈ ✐s ❝②❝❧✐❝ ✐✛ q q❡✳ ■♥ t❤✐s ❝❛s❡ ●❈

q ✇✐t❤ ❣❡♥❡r❛t♦r ✶

❡ ❡ ✶ ✐❢ ●❈ ❉❡ ♦r

✶ ✐❢ ●❈

❉❡✳ ✐✐✮ ■❢ q ❤q❡ ❝♦♠ ❤ ✶ t❤❡♥ ●❈

q ❤✳

▼♦r❡♦✈❡r✱ ●❈

✶ ✷

✐❢ ●❈ ❉❡ ♦r ●❈

✶ ✷

✐❢ ●❈ ❉❡✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷✹ ✴ ✺✾

slide-40
SLIDE 40

Theorem

❋♦r ❡, q ∈ Z+ ✇❡ ❞❡✜♥❡✿ q❡ = ❡✷ + (❡ + ✶)✷, ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡), η✶ = (✶, −(✷❡ + ✶)), η✷ = (✵, q❡) ∈ Z✷

q, ❉❡ = ν✶Z + ν✷Z ❛♥❞ ✈ = (−①, ②) ✐s t❤❡ ❝♦♥❥✳ ♦❢ ✈ = (①, ②)✳ ✶

✭❊①✐st❡♥❝❡✮ P▲(✷, ❡, q) = ∅ ⇔ q ≡ ✵ (♠♦❞ q❡)✳

✷

✭❈❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❈ ∈ P▲(✷, ❡, q) ⇔ ❈ = ❝ + ❉❡ ♦r ❈ = ❝ + ❉❡ ❢♦r ❛♥② ❝ ∈ ❈ ✭✐♥ ♣❛rt✐❝✉❧❛r ❈ − ❝ ✐s ❛ ❣r♦✉♣✮✳

✸

✭❙tr✉❝t✉r❡✮ ▲❡t ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ●❈ = ❈ − ❝ t❤❡ ❣r♦✉♣ ❛ss♦❝✳ ✇✐t❤ ❈✳ ✐✮ ●❈ ✐s ❝②❝❧✐❝ ✐✛ q = q❡✳ ■♥ t❤✐s ❝❛s❡ ●❈ ≃ Zq ✇✐t❤ ❣❡♥❡r❛t♦r ν✶ = (❡, ❡ + ✶) ✐❢ ●❈ = ❉❡ ♦r ν✶ ✐❢ ●❈ = ❉❡✳ ✐✐✮ ■❢ q = ❤q❡ ❝♦♠ ❤ > ✶ t❤❡♥ ●❈ ≃ Zq × Z❤✳ ▼♦r❡♦✈❡r✱ ●❈ = η✶Z ⊕ η✷Z ✐❢ ●❈ = ❉❡ ♦r ●❈ = η✶Z ⊕ η✷Z ✐❢ ●❈ = ❉❡✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷✹ ✴ ✺✾

slide-41
SLIDE 41

Sketch of the proof

■♠♣♦ss✐❜❧❡ ❝♦♥✜❣✉r❛t✐♦♥

▲❡t ■ = {(−✶, −✶), (−✶, ✵), (−✶, ✶), (−✶, ✷), (✵, −✶), (✵, ✷)} ⊆ Z✷

q✳

■❢ ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ❝, ❝′ ∈ ❈ ⇒ ∄ ① ∈ Z✷

q s✉❝❤ t❤❛t

① ∈ ❇(❝, ❡) ∪ ❇(❝′, ❡)✳ ① + ❈ ⊆ ❇(❝, ❡) ∪ ❇(❝′, ❡)

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷✺ ✴ ✺✾

slide-42
SLIDE 42

Sketch of the proof

❉❡❝♦❞✐♥❣ ♦❢ s♣❡❝✐❛❧ ♣♦✐♥ts

■❢ ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ❝ ∈ ❈✱ t❤❡ ♣♦✐♥t ❝ + (✵, ❡ + ✶) ❝❛♥ ♦♥❧② ❜❡ ❞❡❝♦❞❡❞ ✐♥ t✇♦ ✇❛②s✳ ❚❤❡s❡ ♣♦ss✐❜✐❧✐t✐❡s ❛r❡ ❝ + (−❡, ❡ + ✶) ❛♥❞ ❝ + (❡, ❡ + ✶)✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷✻ ✴ ✺✾

slide-43
SLIDE 43

Sketch of the proof

❉❡❝♦❞✐♥❣ ♦❢ s♣❡❝✐❛❧ ♣♦✐♥ts

■❢ ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ❝ ∈ ❈✱ t❤❡ ♣♦✐♥t ❝ + (✵, ❡ + ✶) ❝❛♥ ♦♥❧② ❜❡ ❞❡❝♦❞❡❞ ✐♥ t✇♦ ✇❛②s✳ ❚❤❡s❡ ♣♦ss✐❜✐❧✐t✐❡s ❛r❡ ❝ + (−❡, ❡ + ✶) ❛♥❞ ❝ + (❡, ❡ + ✶)✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷✼ ✴ ✺✾

slide-44
SLIDE 44

Sketch of the proof

❉❡❝♦❞✐♥❣ ♦❢ s♣❡❝✐❛❧ ♣♦✐♥ts

■❢ ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ❝ ∈ ❈✱ t❤❡ ♣♦✐♥t ❝ + (✵, ❡ + ✶) ❝❛♥ ♦♥❧② ❜❡ ❞❡❝♦❞❡❞ ✐♥ t✇♦ ✇❛②s✳ ❚❤❡s❡ ♣♦ss✐❜✐❧✐t✐❡s ❛r❡ ❝ + (−❡, ❡ + ✶) ❛♥❞ ❝ + (❡, ❡ + ✶)✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷✽ ✴ ✺✾

slide-45
SLIDE 45

Sketch of the proof

❉❡❝♦❞✐♥❣ ♦❢ s♣❡❝✐❛❧ ♣♦✐♥ts

■❢ ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ❝ ∈ ❈✱ t❤❡ ♣♦✐♥t ❝ + (✵, ❡ + ✶) ❝❛♥ ♦♥❧② ❜❡ ❞❡❝♦❞❡❞ ✐♥ t✇♦ ✇❛②s✳ ❚❤❡s❡ ♣♦ss✐❜✐❧✐t✐❡s ❛r❡ ❝ + (−❡, ❡ + ✶) ❛♥❞ ❝ + (❡, ❡ + ✶)✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✷✾ ✴ ✺✾

slide-46
SLIDE 46

Sketch of the proof

❉❡❝♦❞✐♥❣ ♦❢ s♣❡❝✐❛❧ ♣♦✐♥ts

■❢ ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ❝ ∈ ❈✱ t❤❡ ♣♦✐♥t ❝ + (✵, ❡ + ✶) ❝❛♥ ♦♥❧② ❜❡ ❞❡❝♦❞❡❞ ✐♥ t✇♦ ✇❛②s✳ ❚❤❡s❡ ♣♦ss✐❜✐❧✐t✐❡s ❛r❡ ❝ + (−❡, ❡ + ✶) ❛♥❞ ❝ + (❡, ❡ + ✶)✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸✵ ✴ ✺✾

slide-47
SLIDE 47

Sketch of the proof

❉❡❝♦❞✐♥❣ ♦❢ s♣❡❝✐❛❧ ♣♦✐♥ts

■❢ ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ❝ ∈ ❈✱ t❤❡ ♣♦✐♥t ❝ + (✵, ❡ + ✶) ❝❛♥ ♦♥❧② ❜❡ ❞❡❝♦❞❡❞ ✐♥ t✇♦ ✇❛②s✳ ❚❤❡s❡ ♣♦ss✐❜✐❧✐t✐❡s ❛r❡ ❝ + (−❡, ❡ + ✶) ❛♥❞ ❝ + (❡, ❡ + ✶)✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸✶ ✴ ✺✾

slide-48
SLIDE 48

Sketch of the proof

❉❡✜♥✐t✐♦♥

❋♦r ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ❝ ∈ ❈ ✇❡ ❞❡✜♥❡ t❤❡ s❡t ω(❝) = {✈✶, . . . , ✈τ} ✇❤❡r❡ t❤❡ ❛❞❥❛❝❡♥t ❜❛❧❧s ♦❢ ❇(❝, ❡) ❛r❡ ❡①❛❝t❧② ❇(❝ + ✈✐, ❡) ❢♦r ✶ ≤ ✐ ≤ τ✳ ❉✷ ✷ ✸

✷ ✶✸

❝ ✸ ✶✶

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸✷ ✴ ✺✾

slide-49
SLIDE 49

Sketch of the proof

❉❡✜♥✐t✐♦♥

❋♦r ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ❝ ∈ ❈ ✇❡ ❞❡✜♥❡ t❤❡ s❡t ω(❝) = {✈✶, . . . , ✈τ} ✇❤❡r❡ t❤❡ ❛❞❥❛❝❡♥t ❜❛❧❧s ♦❢ ❇(❝, ❡) ❛r❡ ❡①❛❝t❧② ❇(❝ + ✈✐, ❡) ❢♦r ✶ ≤ ✐ ≤ τ✳ ❉✷ = (✷, ✸) ⊆ Z✷

✶✸

❝ = (✸, ✶✶)

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸✷ ✴ ✺✾

slide-50
SLIDE 50

Sketch of the proof

❉❡✜♥✐t✐♦♥

❋♦r ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ❝ ∈ ❈ ✇❡ ❞❡✜♥❡ t❤❡ s❡t ω(❝) = {✈✶, . . . , ✈τ} ✇❤❡r❡ t❤❡ ❛❞❥❛❝❡♥t ❜❛❧❧s ♦❢ ❇(❝, ❡) ❛r❡ ❡①❛❝t❧② ❇(❝ + ✈✐, ❡) ❢♦r ✶ ≤ ✐ ≤ τ✳

ω(✸, ✶✶) = {(✷, ✸),

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸✸ ✴ ✺✾

slide-51
SLIDE 51

Sketch of the proof

❉❡✜♥✐t✐♦♥

❋♦r ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ❝ ∈ ❈ ✇❡ ❞❡✜♥❡ t❤❡ s❡t ω(❝) = {✈✶, . . . , ✈τ} ✇❤❡r❡ t❤❡ ❛❞❥❛❝❡♥t ❜❛❧❧s ♦❢ ❇(❝, ❡) ❛r❡ ❡①❛❝t❧② ❇(❝ + ✈✐, ❡) ❢♦r ✶ ≤ ✐ ≤ τ✳

ω(✸, ✶✶) = {(✷, ✸), (−✸, ✷),

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸✹ ✴ ✺✾

slide-52
SLIDE 52

Sketch of the proof

❉❡✜♥✐t✐♦♥

❋♦r ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ❝ ∈ ❈ ✇❡ ❞❡✜♥❡ t❤❡ s❡t ω(❝) = {✈✶, . . . , ✈τ} ✇❤❡r❡ t❤❡ ❛❞❥❛❝❡♥t ❜❛❧❧s ♦❢ ❇(❝, ❡) ❛r❡ ❡①❛❝t❧② ❇(❝ + ✈✐, ❡) ❢♦r ✶ ≤ ✐ ≤ τ✳

ω(✸, ✶✶) = {(✷, ✸), (−✸, ✷), (✷, −✸),

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸✺ ✴ ✺✾

slide-53
SLIDE 53

Sketch of the proof

❉❡✜♥✐t✐♦♥

❋♦r ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ❝ ∈ ❈ ✇❡ ❞❡✜♥❡ t❤❡ s❡t ω(❝) = {✈✶, . . . , ✈τ} ✇❤❡r❡ t❤❡ ❛❞❥❛❝❡♥t ❜❛❧❧s ♦❢ ❇(❝, ❡) ❛r❡ ❡①❛❝t❧② ❇(❝ + ✈✐, ❡) ❢♦r ✶ ≤ ✐ ≤ τ✳

ω(✸, ✶✶) = {(✷, ✸), (−✸, ✷), (✷, −✸), (−✸, −✷)}

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸✻ ✴ ✺✾

slide-54
SLIDE 54

Sketch of the proof

❑✐ss✐♥❣ ▲❡♠♠❛

■❢ ❈ ∈ P▲(✷, ❡, q) t❤❡ s❡t ω(❝) ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ❝✳ ▼♦r❡♦✈❡r ✇❡ ❤❛✈❡ ♦♥❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿ ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✶✮ ♦r ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✷✮✱ ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡)✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸✼ ✴ ✺✾

slide-55
SLIDE 55

Sketch of the proof

❑✐ss✐♥❣ ▲❡♠♠❛

■❢ ❈ ∈ P▲(✷, ❡, q) t❤❡ s❡t ω(❝) ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ❝✳ ▼♦r❡♦✈❡r ✇❡ ❤❛✈❡ ♦♥❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿ ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✶✮ ♦r ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✷✮✱ ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡)✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸✼ ✴ ✺✾

slide-56
SLIDE 56

Sketch of the proof

❑✐ss✐♥❣ ▲❡♠♠❛

■❢ ❈ ∈ P▲(✷, ❡, q) t❤❡ s❡t ω(❝) ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ❝✳ ▼♦r❡♦✈❡r ✇❡ ❤❛✈❡ ♦♥❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿ ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✶✮ ♦r ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✷✮✱ ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡)✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸✽ ✴ ✺✾

slide-57
SLIDE 57

Sketch of the proof

❑✐ss✐♥❣ ▲❡♠♠❛

■❢ ❈ ∈ P▲(✷, ❡, q) t❤❡ s❡t ω(❝) ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ❝✳ ▼♦r❡♦✈❡r ✇❡ ❤❛✈❡ ♦♥❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿ ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✶✮ ♦r ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✷✮✱ ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡)✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✸✾ ✴ ✺✾

slide-58
SLIDE 58

Sketch of the proof

❑✐ss✐♥❣ ▲❡♠♠❛

■❢ ❈ ∈ P▲(✷, ❡, q) t❤❡ s❡t ω(❝) ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ❝✳ ▼♦r❡♦✈❡r ✇❡ ❤❛✈❡ ♦♥❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿ ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✶✮ ♦r ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✷✮✱ ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡)✳

ω(❝) = {ν✶,

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✵ ✴ ✺✾

slide-59
SLIDE 59

Sketch of the proof

❑✐ss✐♥❣ ▲❡♠♠❛

■❢ ❈ ∈ P▲(✷, ❡, q) t❤❡ s❡t ω(❝) ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ❝✳ ▼♦r❡♦✈❡r ✇❡ ❤❛✈❡ ♦♥❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿ ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✶✮ ♦r ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✷✮✱ ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡)✳

ω(❝) = {ν✶, ν✷,

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✶ ✴ ✺✾

slide-60
SLIDE 60

Sketch of the proof

❑✐ss✐♥❣ ▲❡♠♠❛

■❢ ❈ ∈ P▲(✷, ❡, q) t❤❡ s❡t ω(❝) ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ❝✳ ▼♦r❡♦✈❡r ✇❡ ❤❛✈❡ ♦♥❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿ ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✶✮ ♦r ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✷✮✱ ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡)✳

ω(❝) = {±ν✶, ν✷,

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✷ ✴ ✺✾

slide-61
SLIDE 61

Sketch of the proof

❑✐ss✐♥❣ ▲❡♠♠❛

■❢ ❈ ∈ P▲(✷, ❡, q) t❤❡ s❡t ω(❝) ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ❝✳ ▼♦r❡♦✈❡r ✇❡ ❤❛✈❡ ♦♥❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿ ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✶✮ ♦r ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✷✮✱ ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡)✳

ω(❝) = {±ν✶, ±ν✷,

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✸ ✴ ✺✾

slide-62
SLIDE 62

Sketch of the proof

❑✐ss✐♥❣ ▲❡♠♠❛

■❢ ❈ ∈ P▲(✷, ❡, q) t❤❡ s❡t ω(❝) ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ❝✳ ▼♦r❡♦✈❡r ✇❡ ❤❛✈❡ ♦♥❧② t✇♦ ♣♦ss✐❜✐❧✐t✐❡s✿ ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✶✮ ♦r ω(❝) = {±ν✶, ±ν✷} ✭t②♣❡ ✷✮✱ ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡)✳

ω(❝) = {±ν✶, ±ν✷}

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✹ ✴ ✺✾

slide-63
SLIDE 63

Sketch of the proof

❋♦r ❡, q ∈ Z+ ✇❡ ❞❡✜♥❡✿ q❡ = ❡✷ + (❡ + ✶)✷, ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡), η✶ = (✶, −(✷❡ + ✶)), η✷ = (✵, q❡) ∈ Z✷

q, ❉❡ = ν✶Z + ν✷Z ❛♥❞ ✈ = (−①, ②) ✐s t❤❡ ❝♦♥❥✳ ♦❢ ✈ = (①, ②)✳ ✶

✭❊①✐st❡♥❝❡✮ P▲(✷, ❡, q) = ∅ ⇔ q ≡ ✵ (♠♦❞ q❡)✳

✷

✭❈❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❈ ∈ P▲(✷, ❡, q) ⇔ ❈ = ❝ + ❉❡ ♦r ❈ = ❝ + ❉❡ ✇❤❡r❡ ❝ ∈ ❈ ❛♥② ✭✐♥ ♣❛rt✐❝✉❧❛r ❈ − ❝ ✐s ❛ ❣r♦✉♣✮✳

✸

✭❙tr✉❝t✉r❡✮ ▲❡t ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ●❈ = ❈ − ❝ t❤❡ ❣r♦✉♣ ❛ss♦❝✳ ✇✐t❤ ❈✳ ✐✮ ●❈ ✐s ❝②❝❧✐❝ ✐✛ q = q❡✳ ■♥ t❤✐s ❝❛s❡ ●❈ ≃ Zq ✇✐t❤ ❣❡♥❡r❛t♦r ν✶ = (❡, ❡ + ✶) ✐❢ ●❈ = ❉❡ ♦r ν✶ ✐❢ ●❈ = ❉❡✳ ✐✐✮ ■❢ q = ❤q❡ ❝♦♠ ❤ > ✶ t❤❡♥ ●❈ ≃ Zq × Z❤✳ ▼♦r❡♦✈❡r✱ ●❈ = η✶Z ⊕ η✷Z ✐❢ ●❈ = ❉❡ ♦r ●❈ = η✶Z ⊕ η✷Z ✐❢ ●❈ = ❉❡✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✺ ✴ ✺✾

slide-64
SLIDE 64

Sketch of the proof

❋♦r ❡, q ∈ Z+ ✇❡ ❞❡✜♥❡✿ q❡ = ❡✷ + (❡ + ✶)✷, ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡), η✶ = (✶, −(✷❡ + ✶)), η✷ = (✵, q❡) ∈ Z✷

q, ❉❡ = ν✶Z + ν✷Z ❛♥❞ ✈ = (−①, ②) ✐s t❤❡ ❝♦♥❥✳ ♦❢ ✈ = (①, ②)✳ ✶

✭❊①✐st❡♥❝❡✮ P▲(✷, ❡, q) = ∅ ⇔ q ≡ ✵ (♠♦❞ q❡)✳

✷

✭❈❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❈ ∈ P▲(✷, ❡, q) ⇔ ❈ = ❝ + ❉❡ ♦r ❈ = ❝ + ❉❡ ✇❤❡r❡ ❝ ∈ ❈ ❛♥② ✭✐♥ ♣❛rt✐❝✉❧❛r ❈ − ❝ ✐s ❛ ❣r♦✉♣✮✳

✸

✭❙tr✉❝t✉r❡✮ ▲❡t ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ●❈ = ❈ − ❝ t❤❡ ❣r♦✉♣ ❛ss♦❝✳ ✇✐t❤ ❈✳ ✐✮ ●❈ ✐s ❝②❝❧✐❝ ✐✛ q = q❡✳ ■♥ t❤✐s ❝❛s❡ ●❈ ≃ Zq ✇✐t❤ ❣❡♥❡r❛t♦r ν✶ = (❡, ❡ + ✶) ✐❢ ●❈ = ❉❡ ♦r ν✶ ✐❢ ●❈ = ❉❡✳ ✐✐✮ ■❢ q = ❤q❡ ❝♦♠ ❤ > ✶ t❤❡♥ ●❈ ≃ Zq × Z❤✳ ▼♦r❡♦✈❡r✱ ●❈ = η✶Z ⊕ η✷Z ✐❢ ●❈ = ❉❡ ♦r ●❈ = η✶Z ⊕ η✷Z ✐❢ ●❈ = ❉❡✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✻ ✴ ✺✾

slide-65
SLIDE 65

Sketch of the proof

❋♦r t❤❡ s❡❝♦♥❞ ♣❛rt✳✳✳

▲❡t ❈ ∈ P▲(✷, q, ❡) ❛♥❞ ✜① ❛♥② ❝ ∈ ❈✳ ❋♦r t❤❡ ❦✐ss✐♥❣ ❧❡♠♠❛❀ ❝

✶ ✷

❈✳ ❋♦r t❤❡ ♦t❤❡r ✐♥❝❧✉s✐♦♥✱ ✐❢ ❝ ❈ ✐t ✐s s✉✣❝✐❡♥t t♦ ❝♦♥s✐❞❡r ❛ ❝❤❛✐♥ ♦❢ ❛❞❥❛❝❡♥t ❜❛❧❧s ❢r♦♠ ❇ ❝ ❡ t♦ ❇ ❝ ❡ ❛♥❞ ✉s❡ ❦✐ss✐♥❣ ❧❡♠♠❛ ❛❣❛✐♥✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✼ ✴ ✺✾

slide-66
SLIDE 66

Sketch of the proof

❋♦r t❤❡ s❡❝♦♥❞ ♣❛rt✳✳✳

▲❡t ❈ ∈ P▲(✷, q, ❡) ❛♥❞ ✜① ❛♥② ❝ ∈ ❈✳ ❋♦r t❤❡ ❦✐ss✐♥❣ ❧❡♠♠❛❀ ❝ + ν✶Z + ν✷Z ⊆ ❈✳ ❋♦r t❤❡ ♦t❤❡r ✐♥❝❧✉s✐♦♥✱ ✐❢ ❝ ❈ ✐t ✐s s✉✣❝✐❡♥t t♦ ❝♦♥s✐❞❡r ❛ ❝❤❛✐♥ ♦❢ ❛❞❥❛❝❡♥t ❜❛❧❧s ❢r♦♠ ❇ ❝ ❡ t♦ ❇ ❝ ❡ ❛♥❞ ✉s❡ ❦✐ss✐♥❣ ❧❡♠♠❛ ❛❣❛✐♥✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✼ ✴ ✺✾

slide-67
SLIDE 67

Sketch of the proof

❋♦r t❤❡ s❡❝♦♥❞ ♣❛rt✳✳✳

▲❡t ❈ ∈ P▲(✷, q, ❡) ❛♥❞ ✜① ❛♥② ❝ ∈ ❈✳ ❋♦r t❤❡ ❦✐ss✐♥❣ ❧❡♠♠❛❀ ❝ + ν✶Z + ν✷Z ⊆ ❈✳ ❋♦r t❤❡ ♦t❤❡r ✐♥❝❧✉s✐♦♥✱ ✐❢ ❝′ ∈ ❈ ✐t ✐s s✉✣❝✐❡♥t t♦ ❝♦♥s✐❞❡r ❛ ❝❤❛✐♥ ♦❢ ❛❞❥❛❝❡♥t ❜❛❧❧s ❢r♦♠ ❇(❝′, ❡) t♦ ❇(❝, ❡) ❛♥❞ ✉s❡ ❦✐ss✐♥❣ ❧❡♠♠❛ ❛❣❛✐♥✳

  • ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮

❉❡❝❡♠❜❡r ✷✵✶✸ ✹✼ ✴ ✺✾

slide-68
SLIDE 68

Sketch of the proof

❋♦r ❡, q ∈ Z+ ✇❡ ❞❡✜♥❡✿ q❡ = ❡✷ + (❡ + ✶)✷, ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡), η✶ = (✶, −(✷❡ + ✶)), η✷ = (✵, q❡) ∈ Z✷

q, ❉❡ = ν✶Z + ν✷Z ❛♥❞ ✈ = (−①, ②) ✐s t❤❡ ❝♦♥❥✳ ♦❢ ✈ = (①, ②)✳ ✶

✭❊①✐st❡♥❝❡✮ P▲(✷, ❡, q) = ∅ ⇔ q ≡ ✵ (♠♦❞ q❡)✳

✷

✭❈❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❈ ∈ P▲(✷, ❡, q) ⇔ ❈ = ❝ + ❉❡ ♦r ❈ = ❝ + ❉❡ ✇❤❡r❡ ❝ ∈ ❈ ❛♥② ✭✐♥ ♣❛rt✐❝✉❧❛r ❈ − ❝ ✐s ❛ ❣r♦✉♣✮✳

✸

✭❙tr✉❝t✉r❡✮ ▲❡t ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ●❈ = ❈ − ❝ t❤❡ ❣r♦✉♣ ❛ss♦❝✳ ✇✐t❤ ❈✳ ✐✮ ●❈ ✐s ❝②❝❧✐❝ ✐✛ q = q❡✳ ■♥ t❤✐s ❝❛s❡ ●❈ ≃ Zq ✇✐t❤ ❣❡♥❡r❛t♦r ν✶ = (❡, ❡ + ✶) ✐❢ ●❈ = ❉❡ ♦r ν✶ ✐❢ ●❈ = ❉❡✳ ✐✐✮ ■❢ q = ❤q❡ ❝♦♠ ❤ > ✶ t❤❡♥ ●❈ ≃ Zq × Z❤✳ ▼♦r❡♦✈❡r✱ ●❈ = η✶Z ⊕ η✷Z ✐❢ ●❈ = ❉❡ ♦r ●❈ = η✶Z ⊕ η✷Z ✐❢ ●❈ = ❉❡✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✽ ✴ ✺✾

slide-69
SLIDE 69

Sketch of the proof

❋♦r t❤❡ ✜rst ♣❛rt✳✳✳

✭⇒✮▲❡t ❈ ∈ P▲(✷, q, ❡) ❛♥❞ ✜① ❛♥② ❝ ∈ ❈✳ ❲❡ ❝❛♥ s✉♣♣♦s❡ ❈ = ❝ + ν✶Z + ν✷Z ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡) ∈ Z✷

q✳

❆s ❣❝❞ ❡ ❡ ✶ ✶

✶

q✳ ❯s✐♥❣ ▲❛❣r❛♥❣❡ ❚❤❡♦r❡♠ q

✶ ✶ ✷

❈ ❈ q❤ ❢♦r s♦♠❡ ❤ ✳ ❇② t❤❡ s♣❤❡r❡ ♣❛❝❦✐♥❣ ❝♦♥❞✐t✐♦♥✿ ❇ ✵ ❡ ❈ q✷ q❡ q❤ q✷ q q❡❤ ❛♥❞ s♦ q ✵ ♠♦❞ q❡ ✳ ✭ ✮ ●♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✾ ✴ ✺✾

slide-70
SLIDE 70

Sketch of the proof

❋♦r t❤❡ ✜rst ♣❛rt✳✳✳

✭⇒✮▲❡t ❈ ∈ P▲(✷, q, ❡) ❛♥❞ ✜① ❛♥② ❝ ∈ ❈✳ ❲❡ ❝❛♥ s✉♣♣♦s❡ ❈ = ❝ + ν✶Z + ν✷Z ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡) ∈ Z✷

q✳

❆s ❣❝❞(❡, ❡ + ✶) = ✶ ⇒ |ν✶Z| = q✳ ❯s✐♥❣ ▲❛❣r❛♥❣❡ ❚❤❡♦r❡♠ q

✶ ✶ ✷

❈ ❈ q❤ ❢♦r s♦♠❡ ❤ ✳ ❇② t❤❡ s♣❤❡r❡ ♣❛❝❦✐♥❣ ❝♦♥❞✐t✐♦♥✿ ❇ ✵ ❡ ❈ q✷ q❡ q❤ q✷ q q❡❤ ❛♥❞ s♦ q ✵ ♠♦❞ q❡ ✳ ✭ ✮ ●♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✾ ✴ ✺✾

slide-71
SLIDE 71

Sketch of the proof

❋♦r t❤❡ ✜rst ♣❛rt✳✳✳

✭⇒✮▲❡t ❈ ∈ P▲(✷, q, ❡) ❛♥❞ ✜① ❛♥② ❝ ∈ ❈✳ ❲❡ ❝❛♥ s✉♣♣♦s❡ ❈ = ❝ + ν✶Z + ν✷Z ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡) ∈ Z✷

q✳

❆s ❣❝❞(❡, ❡ + ✶) = ✶ ⇒ |ν✶Z| = q✳ ❯s✐♥❣ ▲❛❣r❛♥❣❡ ❚❤❡♦r❡♠ q = |ν✶Z| | |ν✶Z + ν✷Z| = #❈ ⇒ #❈ = q❤ ❢♦r s♦♠❡ ❤ ∈ Z+✳ ❇② t❤❡ s♣❤❡r❡ ♣❛❝❦✐♥❣ ❝♦♥❞✐t✐♦♥✿ ❇ ✵ ❡ ❈ q✷ q❡ q❤ q✷ q q❡❤ ❛♥❞ s♦ q ✵ ♠♦❞ q❡ ✳ ✭ ✮ ●♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✾ ✴ ✺✾

slide-72
SLIDE 72

Sketch of the proof

❋♦r t❤❡ ✜rst ♣❛rt✳✳✳

✭⇒✮▲❡t ❈ ∈ P▲(✷, q, ❡) ❛♥❞ ✜① ❛♥② ❝ ∈ ❈✳ ❲❡ ❝❛♥ s✉♣♣♦s❡ ❈ = ❝ + ν✶Z + ν✷Z ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡) ∈ Z✷

q✳

❆s ❣❝❞(❡, ❡ + ✶) = ✶ ⇒ |ν✶Z| = q✳ ❯s✐♥❣ ▲❛❣r❛♥❣❡ ❚❤❡♦r❡♠ q = |ν✶Z| | |ν✶Z + ν✷Z| = #❈ ⇒ #❈ = q❤ ❢♦r s♦♠❡ ❤ ∈ Z+✳ ❇② t❤❡ s♣❤❡r❡ ♣❛❝❦✐♥❣ ❝♦♥❞✐t✐♦♥✿ #❇(✵, ❡) · #❈ = q✷ ⇔ q❡ · q❤ = q✷ ⇔ q = q❡❤ ❛♥❞ s♦ q ≡ ✵ (♠♦❞ q❡)✳ ✭ ✮ ●♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✹✾ ✴ ✺✾

slide-73
SLIDE 73

Sketch of the proof

❋♦r t❤❡ ✜rst ♣❛rt✳✳✳

✭⇒✮▲❡t ❈ ∈ P▲(✷, q, ❡) ❛♥❞ ✜① ❛♥② ❝ ∈ ❈✳ ❲❡ ❝❛♥ s✉♣♣♦s❡ ❈ = ❝ + ν✶Z + ν✷Z ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡) ∈ Z✷

q✳

❆s ❣❝❞(❡, ❡ + ✶) = ✶ ⇒ |ν✶Z| = q✳ ❯s✐♥❣ ▲❛❣r❛♥❣❡ ❚❤❡♦r❡♠ q = |ν✶Z| | |ν✶Z + ν✷Z| = #❈ ⇒ #❈ = q❤ ❢♦r s♦♠❡ ❤ ∈ Z+✳ ❇② t❤❡ s♣❤❡r❡ ♣❛❝❦✐♥❣ ❝♦♥❞✐t✐♦♥✿ #❇(✵, ❡) · #❈ = q✷ ⇔ q❡ · q❤ = q✷ ⇔ q = q❡❤ ❛♥❞ s♦ q ≡ ✵ (♠♦❞ q❡)✳ ✭⇐✮ ●♦❧♦♠❜ ❛♥❞ ❲❡❧❝❤✳

  • ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮

❉❡❝❡♠❜❡r ✷✵✶✸ ✹✾ ✴ ✺✾

slide-74
SLIDE 74

Sketch of the proof

❋♦r ❡, q ∈ Z+ ✇❡ ❞❡✜♥❡✿ q❡ = ❡✷ + (❡ + ✶)✷, ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡), η✶ = (✶, −(✷❡ + ✶)), η✷ = (✵, q❡) ∈ Z✷

q, ❉❡ = ν✶Z + ν✷Z ❛♥❞ ✈ = (−①, ②) ✐s t❤❡ ❝♦♥❥✳ ♦❢ ✈ = (①, ②)✳ ✶

✭❊①✐st❡♥❝❡✮ P▲(✷, ❡, q) = ∅ ⇔ q ≡ ✵ (♠♦❞ q❡)✳

✷

✭❈❤❛r❛❝t❡r✐③❛t✐♦♥✮ ❈ ∈ P▲(✷, ❡, q) ⇔ ❈ = ❝ + ❉❡ ♦r ❈ = ❝ + ❉❡ ✇❤❡r❡ ❝ ∈ ❈ ❛♥② ✭✐♥ ♣❛rt✐❝✉❧❛r ❈ − ❝ ✐s ❛ ❣r♦✉♣✮✳

✸

✭❙tr✉❝t✉r❡✮ ▲❡t ❈ ∈ P▲(✷, ❡, q) ❛♥❞ ●❈ = ❈ − ❝ t❤❡ ❣r♦✉♣ ❛ss♦❝✳ ✇✐t❤ ❈✳ ✐✮ ●❈ ✐s ❝②❝❧✐❝ ✐✛ q = q❡✳ ■♥ t❤✐s ❝❛s❡ ●❈ ≃ Zq ✇✐t❤ ❣❡♥❡r❛t♦r ν✶ = (❡, ❡ + ✶) ✐❢ ●❈ = ❉❡ ♦r ν✶ ✐❢ ●❈ = ❉❡✳ ✐✐✮ ■❢ q = ❤q❡ ❝♦♠ ❤ > ✶ t❤❡♥ ●❈ ≃ Zq × Z❤✳ ▼♦r❡♦✈❡r✱ ●❈ = η✶Z ⊕ η✷Z ✐❢ ●❈ = ❉❡ ♦r ●❈ = η✶Z ⊕ η✷Z ✐❢ ●❈ = ❉❡✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺✵ ✴ ✺✾

slide-75
SLIDE 75

Sketch of the proof

❋♦r t❤❡ ❧❛st ♣❛rt✳✳✳

❲❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t ❈ ✐s ❧✐♥❡❛r ❛♥❞ t②♣❡ ✶✱ t❤❛t ✐s ❈ = ν✶Z + ν✷Z ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡)✳ ❆s ✶ ✶ ❡ ✶ ❡

✶ ✷ ✶ ✷

❛♥❞ ❞❡t ✶ ✶ ❡ ✶ ❡ ✶✱ t❤❡♥ ❈

✶ ✷

✇❤❡r❡

✶

✶ ✷❡ ✶ ❛♥❞

✷

✵ q❡ ✭✐♥

✷ q✮✳

❈❧❡❛r❧②

✶ ✷

✵ ✭t❤❡r❡❢♦r❡ ❈

✶ ✷

✮ ❛♥❞

✶

q ❡

✷ q q❡

❤ ❢r♦♠ ✇❤❡r❡ ✇❡ ❝❛♥ ❝♦♥❝❧✉❞❡ t❤❛t ❈

✶ ✷ q ❤✳

❆s ❤ q✱ ✐t ✐s ❝❧❡❛r t❤❛t ❈ ✐s ❝②❝❧✐❝ ✐✛ ❤ ✶✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺✶ ✴ ✺✾

slide-76
SLIDE 76

Sketch of the proof

❋♦r t❤❡ ❧❛st ♣❛rt✳✳✳

❲❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t ❈ ✐s ❧✐♥❡❛r ❛♥❞ t②♣❡ ✶✱ t❤❛t ✐s ❈ = ν✶Z + ν✷Z ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡)✳ ❆s   −✶ −✶ ❡ + ✶ ❡     ν✶ ν✷   =   η✶ η✷   ❛♥❞ ❞❡t   −✶ −✶ ❡ + ✶ ❡   = ✶✱ t❤❡♥ ❈ = η✶Z + η✷Z ✇❤❡r❡ η✶ = (✶, −(✷❡ + ✶)) ❛♥❞ η✷ = (✵, q❡) ✭✐♥ Z✷

q✮✳

❈❧❡❛r❧②

✶ ✷

✵ ✭t❤❡r❡❢♦r❡ ❈

✶ ✷

✮ ❛♥❞

✶

q ❡

✷ q q❡

❤ ❢r♦♠ ✇❤❡r❡ ✇❡ ❝❛♥ ❝♦♥❝❧✉❞❡ t❤❛t ❈

✶ ✷ q ❤✳

❆s ❤ q✱ ✐t ✐s ❝❧❡❛r t❤❛t ❈ ✐s ❝②❝❧✐❝ ✐✛ ❤ ✶✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺✶ ✴ ✺✾

slide-77
SLIDE 77

Sketch of the proof

❋♦r t❤❡ ❧❛st ♣❛rt✳✳✳

❲❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t ❈ ✐s ❧✐♥❡❛r ❛♥❞ t②♣❡ ✶✱ t❤❛t ✐s ❈ = ν✶Z + ν✷Z ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡)✳ ❆s   −✶ −✶ ❡ + ✶ ❡     ν✶ ν✷   =   η✶ η✷   ❛♥❞ ❞❡t   −✶ −✶ ❡ + ✶ ❡   = ✶✱ t❤❡♥ ❈ = η✶Z + η✷Z ✇❤❡r❡ η✶ = (✶, −(✷❡ + ✶)) ❛♥❞ η✷ = (✵, q❡) ✭✐♥ Z✷

q✮✳

❈❧❡❛r❧② η✶Z ∩ η✷Z = (✵) ✭t❤❡r❡❢♦r❡ ❈ = η✶Z ⊕ η✷Z)✮ ❛♥❞ |η✶Z| = q ❡ |η✷Z| = q

q❡ = ❤ ❢r♦♠ ✇❤❡r❡ ✇❡ ❝❛♥ ❝♦♥❝❧✉❞❡ t❤❛t

❈ = η✶Z ⊕ η✷Z ≃ Zq × Z❤✳ ❆s ❤ q✱ ✐t ✐s ❝❧❡❛r t❤❛t ❈ ✐s ❝②❝❧✐❝ ✐✛ ❤ ✶✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺✶ ✴ ✺✾

slide-78
SLIDE 78

Sketch of the proof

❋♦r t❤❡ ❧❛st ♣❛rt✳✳✳

❲❡ ❝❛♥ s✉♣♣♦s❡ t❤❛t ❈ ✐s ❧✐♥❡❛r ❛♥❞ t②♣❡ ✶✱ t❤❛t ✐s ❈ = ν✶Z + ν✷Z ✇❤❡r❡ ν✶ = (❡, ❡ + ✶), ν✷ = (−(❡ + ✶), ❡)✳ ❆s   −✶ −✶ ❡ + ✶ ❡     ν✶ ν✷   =   η✶ η✷   ❛♥❞ ❞❡t   −✶ −✶ ❡ + ✶ ❡   = ✶✱ t❤❡♥ ❈ = η✶Z + η✷Z ✇❤❡r❡ η✶ = (✶, −(✷❡ + ✶)) ❛♥❞ η✷ = (✵, q❡) ✭✐♥ Z✷

q✮✳

❈❧❡❛r❧② η✶Z ∩ η✷Z = (✵) ✭t❤❡r❡❢♦r❡ ❈ = η✶Z ⊕ η✷Z)✮ ❛♥❞ |η✶Z| = q ❡ |η✷Z| = q

q❡ = ❤ ❢r♦♠ ✇❤❡r❡ ✇❡ ❝❛♥ ❝♦♥❝❧✉❞❡ t❤❛t

❈ = η✶Z ⊕ η✷Z ≃ Zq × Z❤✳ ❆s ❤|q✱ ✐t ✐s ❝❧❡❛r t❤❛t ❈ ✐s ❝②❝❧✐❝ ✐✛ ❤ = ✶✳

  • ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮

❉❡❝❡♠❜❡r ✷✵✶✸ ✺✶ ✴ ✺✾

slide-79
SLIDE 79

Example

❊①❛♠♣❧❡

❋♦r q = ✶✶✵✺✳ ❚❤❡r❡ ❛r❡ ❡①❛❝t❧② ✺ ❝♦❞❡s ✐♥ Z✶✶✵✺ × Z✶✶✵✺ ✉♣ t♦ tr❛♥s❧❛t✐♦♥s ❛♥❞ ❝♦♥❥✉❣❛t✐♦♥ ✭✶✶✵✺ = ✺ · ✶✸ · ✶✼✮✳ ❖♥❡ ♦❢ t❤❡s❡ ❝♦❞❡s ✐s ❝②❝❧✐❝ ❛♥❞ t❤❡ ♦t❤❡rs ❛r❡ ♥♦♥✲❝②❝❧✐❝✳ ❚❤❡s❡ ❝♦❞❡s ❛r❡ ❣✐✈❡♥ ❜②✿ ❈✶ = (✶, −✸)Z✶✶✵✺ ⊕ (✵, ✺)Z✶✶✵✺ (❡ = ✶) ❈✷ = (✶, −✺)Z✶✶✵✺ ⊕ (✵, ✶✸)Z✶✶✵✺ (❡ = ✷) ❈✸ = (✶, −✶✸)Z✶✶✵✺ ⊕ (✵, ✽✺)Z✶✶✵✺ (❡ = ✻) ❈✹ = (✶, −✷✶)Z✶✶✵✺ ⊕ (✵, ✷✷✶)Z✶✶✵✺ (❡ = ✶✵) ❈✺ = (✷✸, ✷✹)Z✶✶✵✺ (❡ = ✷✸)

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺✷ ✴ ✺✾

slide-80
SLIDE 80

▼♦r❡ r❡❧❡✈❛♥t r❡s✉❧ts r❡❧❛t❡❞ t♦ t❤❡ ●♦❧♦♠❜✲❲❡❧❝❤ ❝♦♥❥❡❝t✉r❡

P▲(✷, ❡, q❡) = ∅✱ P▲(♥, ✶, ✷♥ + ✶) = ∅✱ P▲(✸, ✷) = ∅✱ P▲(♥, ❡) = ∅ ❢♦r ❡ ≥ ❡♥✳

❙✳ ❲✳ ●♦❧♦♠❜✱ ▲✳ ❘✳ ❲❡❧❝❤✳ P❡r❢❡❝t ❈♦❞❡s ✐♥ t❤❡ ▲❡❡ ♠❡tr✐❝ ❛♥❞ t❤❡ ♣❛❝❦✐♥❣ ♦❢ ♣♦❧②♥♦♠✐♥♦❡s✱ ❙■❆▼ ❏♦✉r♥❛❧ ❆♣♣❧✐❡❞ ▼❛t❤✳✱ ✈♦❧✳ ✶✽✱ ♣♣✳ ✸✵✷✲✸✶✼✳ ✶✾✼✵✳

P▲(♥, ❡, q) = ∅ ❢♦r ✸ ≤ ♥ ≤ ✺, ❡ ≥ ♥ − ✶, q ≥ ✷❡ + ✶ ❛♥❞ ❢♦r ♥ ≥ ✻, ❡ ≥ ✷♥−✸

✷ √ ✷ − ✶ ✷ ❑✳❆✳P♦st✳ ◆♦♥❡①✐st❡♥❝❡ t❤❡♦r❡♠ ♦♥ ♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s ♦✈❡r ❧❛r❣❡ ❛❧♣❤❛❜❡ts✳ ■♥❢✳ ❛♥❞ ❝♦♥tr♦❧ ✷✾✱ ✸✻✾✲✸✽✵✳ ✶✾✼✺✳

P▲(♥, ✷, q) = ∅ ❢♦r q = ✶✸✱ q ♥♦t ❞✐✈✐s✐❜❧❡ ❜② ❛ ♣r✐♠❡ ˙ ✹ + ✶✱ ❛♥❞ q = ♣❦ ✇✐t❤ ♣ ♣r✐♠❡✱ ♣ = ✶✸ ❛♥❞ ♣ <

  • ♥✷ + (♥ + ✶)✷✳

❏✳❆st♦❧❛✳ ❖♥ ♣❡r❢❡❝t ❝♦❞❡s ✐♥ t❤❡ ▲❡❡ ♠❡tr✐❝✳ ❆♥♥✳ ❯♥✐✈✳ ❚✉r❦✉ ✭❆✮ ✶✼✻ ✭✶✮✱ ✺✻✳ ✶✾✼✽✳

P▲(✸, ❡) = ∅ ❢♦r ❡ ≥ ✷✳

❙✳●r❛✈✐❡r✱ ▼✳▼♦❧❧❛r❞✱ ❈✳P❛②❛♥✳ ❖♥ t❤❡ ◆♦♥✲❡①✐st❡♥❝❡ ♦❢ ✸✲❞✐♠❡♥s✐♦♥❛❧ t✐❧✐♥❣ ✐♥ t❤❡ ▲❡❡ ♠❡tr✐❝✳ ❊✉r♦♣✳ ❏✳ ❈♦♠❜✐♥❛t♦r✐❝s ✶✾✳ ♣♣✳✺✻✼✲✺✼✷✳ ✶✾✾✽✳

P▲(✹, ❡) = ∅ ❢♦r ❡ ≥ ✷✳

❙✳❙♣❛❝❛♣❛♥✳ ❆ ❝♦♠♣❧❡t❡ ♣r♦♦❢ ♦❢ t❤❡ ♥♦♥❡①✐st❡♥❝❡ ♦❢ r❡❣✉❧❛r ❢♦✉r ❞✐♠❡♥s✐♦♥❛❧ t✐❧✐♥❣s ✐♥ t❤❡ ▲❡❡ ♠❡tr✐❝✳ Pr❡♣r✐♥t ❙❡r✐❡s✱ ✈♦❧✳✹✷✭✾✾✸✮✳ ✷✵✵✹ ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺✸ ✴ ✺✾

slide-81
SLIDE 81

▼♦r❡ r❡❧❡✈❛♥t r❡s✉❧ts r❡❧❛t❡❞ t♦ t❤❡ ●♦❧♦♠❜✲❲❡❧❝❤ ❝♦♥❥❡❝t✉r❡

P▲(♥, ✷) = ∅ ❢♦r ✺ ≤ ♥ ≤ ✶✷ ❢♦r ❧✐♥❡❛r ❝♦❞❡s

P✳❍♦r❛❦✳ ❖♥ ♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s✳ ❉✐s❝r✳ ▼❛t❤✳ ✸✵✾✳ ✺✺✺✶✲✺✺✻✶✳ ✷✵✵✾✳ P✳❍♦r❛❦✱ ❖✳●r♦s❡❦✳ ❆ ♥❡✇ ❛♣♣r♦❛❝❤ t♦✇❛r❞s t❤❡ ●♦❧♦♠❜✲❲❡❧❝❤ ❝♦♥❥❡❝t✉r❡✳ ♣r❡♣r✐♥t ❛r①✐✈✳♦r❣/♣❞❢/✶✷✵✺✳✹✽✼✺✈✸✳♣❞❢✳ ✷✵✶✸✳ ✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺✹ ✴ ✺✾

slide-82
SLIDE 82

❋✉t✉r❡ ✇♦r❦

❆♣♣r♦❛❝❤ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♥✲❞✐♠❡♥s✐♦♥❛❧ ♣❡r❢❡❝t s✐♥❣❧❡✲❡rr♦r✲❝♦rr❡❝t✐♥❣ ▲❡❡ ❝♦❞❡s ✉s✐♥❣ t❤❡ ❣❡♦♠❡tr② ♦❢ ♣♦❧②♦♠✐♥♦❡s ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛r❣✉♠❡♥ts✳ Pr♦✈❡ t❤❡ ♥♦♥✲❡①✐st❡♥❝❡ ♦❢ ❡✲♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s ✐♥ s♦♠❡ ❞✐♠❡♥s✐♦♥ ♥ > ✸ ✉s✐♥❣ t❤❡s❡ t❡❝❤♥✐q✉❡s✳ ❈♦♥str✉❝t q✉❛s✐✲♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s ❛♥❞ ❞❡♥s❡ ❝♦❞❡s ✉s✐♥❣ t❤✐s ❛♣♣r♦❛❝❤✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺✺ ✴ ✺✾

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SLIDE 83

❋✉t✉r❡ ✇♦r❦

❆♣♣r♦❛❝❤ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♥✲❞✐♠❡♥s✐♦♥❛❧ ♣❡r❢❡❝t s✐♥❣❧❡✲❡rr♦r✲❝♦rr❡❝t✐♥❣ ▲❡❡ ❝♦❞❡s ✉s✐♥❣ t❤❡ ❣❡♦♠❡tr② ♦❢ ♣♦❧②♦♠✐♥♦❡s ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛r❣✉♠❡♥ts✳ Pr♦✈❡ t❤❡ ♥♦♥✲❡①✐st❡♥❝❡ ♦❢ ❡✲♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s ✐♥ s♦♠❡ ❞✐♠❡♥s✐♦♥ ♥ > ✸ ✉s✐♥❣ t❤❡s❡ t❡❝❤♥✐q✉❡s✳ ❙✳ ●r❛✈✐❡r✱ ▼✳ ▼♦❧❧❛r❞✱ ❈✳ P❛②❛♥ s✉❝❝❡❡❞ ❢♦r ♥ = ✸✳

❖♥ t❤❡ ♥♦♥❡①✐st❡♥❝❡ ♦❢ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ t✐❧✐♥❣ ✐♥ t❤❡ ▲❡❡ ♠❡tr✐❝ ■■✳ ❉✐s❝r✳▼❛t❤ ✷✸✺✱ ✶✺✶✲✶✺✼✳ ✷✵✵✶✳

❈♦♥str✉❝t q✉❛s✐✲♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s ❛♥❞ ❞❡♥s❡ ❝♦❞❡s ✉s✐♥❣ t❤✐s ❛♣♣r♦❛❝❤✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺✻ ✴ ✺✾

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SLIDE 84

❋✉t✉r❡ ✇♦r❦

❆♣♣r♦❛❝❤ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♥✲❞✐♠❡♥s✐♦♥❛❧ ♣❡r❢❡❝t s✐♥❣❧❡✲❡rr♦r✲❝♦rr❡❝t✐♥❣ ▲❡❡ ❝♦❞❡s ✉s✐♥❣ t❤❡ ❣❡♦♠❡tr② ♦❢ ♣♦❧②♦♠✐♥♦❡s ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛r❣✉♠❡♥ts✳ Pr♦✈❡ t❤❡ ♥♦♥✲❡①✐st❡♥❝❡ ♦❢ ❡✲♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s ✐♥ s♦♠❡ ❞✐♠❡♥s✐♦♥ ♥ > ✸ ✉s✐♥❣ t❤❡s❡ t❡❝❤♥✐q✉❡s✳ ❈♦♥str✉❝t q✉❛s✐✲♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s ❛♥❞ ❞❡♥s❡ ❝♦❞❡s ✉s✐♥❣ t❤✐s ❛♣♣r♦❛❝❤✳ ■s ✐t ♣♦ss✐❜❧❡ t♦ ✉s❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❞❡s✐❣♥s t♦ ♦❜t❛✐♥ ♥♦♥✲tr✐✈✐❛❧ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ♣❛r❛♠❡t❡rs ♥ ❡ q ♦❢ ♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s✱ ❧✐❦❡ ✐♥ t❤❡ ❍❛♠♠✐♥❣ ❝❛s❡❄

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺✼ ✴ ✺✾

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SLIDE 85

❋✉t✉r❡ ✇♦r❦

❆♣♣r♦❛❝❤ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♥✲❞✐♠❡♥s✐♦♥❛❧ ♣❡r❢❡❝t s✐♥❣❧❡✲❡rr♦r✲❝♦rr❡❝t✐♥❣ ▲❡❡ ❝♦❞❡s ✉s✐♥❣ t❤❡ ❣❡♦♠❡tr② ♦❢ ♣♦❧②♦♠✐♥♦❡s ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛r❣✉♠❡♥ts✳ Pr♦✈❡ t❤❡ ♥♦♥✲❡①✐st❡♥❝❡ ♦❢ ❡✲♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s ✐♥ s♦♠❡ ❞✐♠❡♥s✐♦♥ ♥ > ✸ ✉s✐♥❣ t❤❡s❡ t❡❝❤♥✐q✉❡s✳ ❈♦♥str✉❝t q✉❛s✐✲♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s ❛♥❞ ❞❡♥s❡ ❝♦❞❡s ✉s✐♥❣ t❤✐s ❛♣♣r♦❛❝❤✳ ■s ✐t ♣♦ss✐❜❧❡ t♦ ✉s❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❞❡s✐❣♥s t♦ ♦❜t❛✐♥ ♥♦♥✲tr✐✈✐❛❧ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ♣❛r❛♠❡t❡rs (♥, ❡, q) ♦❢ ♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s✱ ❧✐❦❡ ✐♥ t❤❡ ❍❛♠♠✐♥❣ ❝❛s❡❄

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺✼ ✴ ✺✾

slide-86
SLIDE 86

❋✉t✉r❡ ✇♦r❦

❆♣♣r♦❛❝❤ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ♥✲❞✐♠❡♥s✐♦♥❛❧ ♣❡r❢❡❝t s✐♥❣❧❡✲❡rr♦r✲❝♦rr❡❝t✐♥❣ ▲❡❡ ❝♦❞❡s ✉s✐♥❣ t❤❡ ❣❡♦♠❡tr② ♦❢ ♣♦❧②♦♠✐♥♦❡s ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❛❧ ❛r❣✉♠❡♥ts✳ Pr♦✈❡ t❤❡ ♥♦♥✲❡①✐st❡♥❝❡ ♦❢ ❡✲♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s ✐♥ s♦♠❡ ❞✐♠❡♥s✐♦♥ ♥ > ✸✳ ❈♦♥str✉❝t q✉❛s✐✲♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s ❛♥❞ ❞❡♥s❡ ❝♦❞❡s ✉s✐♥❣ t❤✐s ❛♣♣r♦❛❝❤✳ ■s ✐t ♣♦ss✐❜❧❡ t♦ ✉s❡ ❝♦♠❜✐♥❛t♦r✐❛❧ ❞❡s✐❣♥s t♦ ♦❜t❛✐♥ ♥♦♥✲tr✐✈✐❛❧ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ♣❛r❛♠❡t❡rs (♥, ❡, q) ♦❢ ♣❡r❢❡❝t ▲❡❡ ❝♦❞❡s✱ ❧✐❦❡ ✐♥ t❤❡ ❍❛♠♠✐♥❣ ❝❛s❡ ❄

❙♦♠❡ r❡❢❡r❡♥❝❡s ✐♥ ♣♦❧②♦♠✐♥♦❡s

❏✳❍✳❈♦♥✇❛② ❛♥❞ ❏✳❈✳▲❛❣❛r✐❛s✳ ❚✐❧✐♥❣ ✇✐t❤ ♣♦❧②♦♠✐♥♦❡s ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❛❧ ❣r♦✉♣ t❤❡♦r②✳ ❏♦✉r♥❛❧ ♦❢ ❈♦♠❜✳ ❚❤❡♦r②✱ ❙❡r✐❡s ❆✳✺✸✱ ✶✽✸✲✷✵✽✱ ✶✾✾✵✳ ❙✳❲✳ ●♦❧♦♠❜✳ P♦❧②♦♠✐♥♦❡s✿ P✉③③❧❡s✱ P❛tt❡r♥s✱ Pr♦❜❧❡♠s ❛♥❞ P❛❝❦✐♥❣s✳ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ s❡❝♦♥❞ ❡❞✐t✐♦♥✱ ✶✾✾✻✳ ▼✳❘✳ ❑♦r♥✳ ●❡♦♠❡tr✐❝ ❛♥❞ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ ♣♦❧②♦♠✐♥♦ t✐❧✐♥❣s✳ P❤❉✳ ❚❤❡s✐s✱ ▼❛ss❛❝❤✉s❡tts ■♥st✐t✉t❡ ♦❢ ❚❡❝❤♥♦❧♦❣②✱ ✷✵✵✹✳

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺✽ ✴ ✺✾

slide-87
SLIDE 87

❚❤❛♥❦s ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

✭◗✉r❡s❤✐ ✲ ❈❛♠♣✐♥❛s ❯♥✐✈❡rs✐t②✱ ❇r❛③✐❧✮ ❉❡❝❡♠❜❡r ✷✵✶✸ ✺✾ ✴ ✺✾