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❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✶

❚♦❞❛②✬s ❡①❡r❝✐s❡s

• ✷✳✶✶✿ ❉❡r❛♥❞♦♠✐③✐♥❣ ❚❤❡♦r❡♠ ✷✳✹
• ✷✳✶✸✿ ❖♥❡ ♦❢ ❋❡✇ ❆❧✇❛②s ❲♦r❦s
• ✷✳✶✺✿ ❆❧❧ ✐♥ ✸✕❛t ▼♦st ✼ ✐♥ ✶✵
• ✷✯✳✶✿ ❘❛♥❞♦♠ ❙❛♠♣❧✐♥❣ ❉♦❡s♥✬t ❲♦r❦
• ■♥ ❈❧❛ss✿ ✸✳✶✿ ❉❡❝✐s✐♦♥ ✈s✳ ❈♦♥str✉❝t✐♦♥
• ■♥ ❈❧❛ss✿ ✸✳✸✿ ❙❛t✐s✜❛❜❧❡ ✇✐t❤ ♠❛♥② ❈❧❛✉s❡s
• ❆❞❞✐t✐♦♥❛❧✿ Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳✼

P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✷

❈♦♥s✐❞❡r s✉❜♠✐tt✐♥❣ ❡①❡r❝✐s❡ s♦❧✉t✐♦♥s ♦♥ ♣❛♣❡r✦

= ⇒ ❊①❝❡❧❧❡♥t ♣r❛❝t✐❝❡ ❢♦r ❡♥❞ ♦❢ t❡r♠ ❡①❛♠

P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✸

✷✳✶✶✳ ❉❡r❛♥❞♦♠✐③✐♥❣ ❚❤❡♦r❡♠ ✷✳✹

❚❤✐s ✇♦r❦s ❡①❛❝t❧② ❧✐❦❡ t❤❡ ❞❡r❛♥❞♦♠✐③❛t✐♦♥ ♦❢ ❚❤❡♦r❡♠ ✷✳✷✳

• s✇✐t❝❤✐♥❣ s✐❣♥s ✐❢ ✉♥✐t ❝❧❛✉s❡s ❛♣♣❡❛r ✐s ❛❧r❡❛❞② ❞❡t❡r♠✐♥✐st✐❝✳
• ❢♦r ❡✈❡r② ❢♦r♠✉❧❛✱ ✇❡ ❝❛♥ ❝❛❧❝✉❧❛t❡ t❤❡ ❡①♣❡❝t❡❞ s❛t✐s✜❡❞ ✇❡✐❣❤t

✐❢ ❡❛❝❤ ✈❛r✐❛❜❧❡ ✐s ♦♥❡ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② Φ

• ❛ss✐❣♥ t❤❡ ✈❛r✐❛❜❧❡s ♦♥❡ ❜② ♦♥❡✱ ❡❛❝❤ t✐♠❡ ❛ss✐❣♥ ❛ ✈❛❧✉❡ s♦ ❛s t♦

♠❛①✐♠✐③❡ t❤❡ ❡①♣❡❝t❛t✐♦♥

P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✹

✷✳✶✸✳ ❖♥❡ ♦❢ ❋❡✇ ❆❧✇❛②s ❲♦r❦s

❆ ❍❛❞❛♠❛r❞✲▼❛tr✐① H ∈ {−1, 1}n×n ✐s ❛ ♠❛tr✐① ✐♥ ✇❤✐❝❤ ❛❧❧ r♦✇s ❛r❡ ♣❛✐r✇✐s❡ ♦rt❤♦❣♦♥❛❧ ❛♥❞ ❛❧❧ ❝♦❧✉♠♥s ❛r❡ ♣❛✐r✇✐s❡ ♦rt❤♦❣♦♥❛❧✳ ❋♦r n ❛ ♣♦✇❡r ♦❢ t✇♦✱ t❤❡s❡ ❝❛♥ ❜❡ ❡❛s✐❧② ❝♦♥str✉❝t❡❞ ❜② st❛rt✐♥❣ ✇✐t❤ H1 := (1) ❛♥❞ ❜② r❡❝✉rs✐✈❡❧② ❧❡tt✐♥❣ H2n :=

• Hn

Hn Hn −Hn

• .

◆♦t ❢♦r ❛❧❧ n ❦♥♦✇♥✱ ❡✳❣✳ n = 668✳ ❊✈❡r② ❍❛❞❛♠❛r❞ ♠❛tr✐① ❤❛s ♦r❞❡r ✶✱✷✱ ♦r ❛ ♠✉❧t✐♣❧❡ ♦❢ ✹✳

P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✺

▲❡♠♠❛✳ ■❢ Hn ∈ {−1, 1}n, n ≥ 4, ✐s ❛ ❍❛❞❛♠❛r❞ ♠❛tr✐① ✇✐t❤ t❤❡ ✜rst ❝♦❧✉♠♥ ❝♦♥t❛✐♥✐♥❣ ♦♥❧② ♦♥❡s✱ t❤❡♥ ❢♦r ❛♥② 1 < i, j ≤ n✱ t❤❡r❡ ✐s ❛♥ ❡q✉❛❧ ♥✉♠❜❡r ♦❢ r♦✇s k s✉❝❤ t❤❛t ({Hn}ki, {Hn}kj) ❡q✉❛❧s (1, 1)✱ (1, −1)✱ (−1, 1) ❛♥❞ (−1, −1)✳ Pr♦♦❢✳ ❋✐① i ❛♥❞ j✳ ■❢ n++, n+−, n−+, n−− ❛r❡ t❤❡ ♥✉♠❜❡rs ♦❢ r♦✇s ✇❤❡r❡ ({Hn}ki, {Hn}kj) ❡q✉❛❧s (1, 1)✱ (1, −1)✱ (−1, 1) ❛♥❞ (−1, −1)✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡♥ ✇❡ ❤❛✈❡ n++ + n+− = n/2 ✭❜❡❝❛✉s❡ ❝♦❧✉♠♥s ✶ ❛♥❞ i ❛r❡ ♦rt❤♦❣♦♥❛❧✮✱ n++ + n−+ = n/2 ✭❜❡❝❛✉s❡ ❝♦❧✉♠♥s ✶ ❛♥❞ j ❛r❡ ♦rt❤♦❣♦♥❛❧✮ ❛♥❞ n++ + n−− = n/2 ✭❜❡❝❛✉s❡ ❝♦❧✉♠♥s i ❛♥❞ j ❛r❡ ♦rt❤♦❣♦♥❛❧✮✳ ❋✐♥❛❧❧②✱ n++ + n+− + n−+ + n−− = n✳ ❖♥❧② n++ = n+− = n−+ = n−− = n/4 s❛t✐s✜❡s ❛❧❧ t❤❡s❡ ❝♦♥str❛✐♥ts✳ ✷

P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✻

✷✳✶✸✳ ❖♥❡ ♦❢ ❋❡✇ ❆❧✇❛②s ❲♦r❦s ✭✷✮

❈❤♦♦s❡ t t♦ ❜❡ t❤❡ s♠❛❧❧❡st ♣♦✇❡r ♦❢ t✇♦ s✉❝❤ t❤❛t t ≥ n + 1✳ ◆♦t❡ t❤❛t t ❝❛♥ ❜❡ ❝❤♦s❡♥ t♦ ❜❡ ❛t ♠♦st 2n✳ ❈♦♥str✉❝t t❤❡ ❍❛❞❛♠❛r❞ ♠❛tr✐① Ht ❛♥❞ ❝❤♦♦s❡ ♦♥❡ ♦❢ t❤❡ t r♦✇s ✉♥✐❢♦r♠❧② ❛t r❛♥❞♦♠✳ ■♥t❡r♣r❡t t❤❡ ✈❛❧✉❡s ❢r♦♠ t❤✐s r♦✇ ✐♥ t❤❡ ❝♦❧✉♠♥s 2 t❤r♦✉❣❤ n + 1 ❛s ❛ss✐❣♥♠❡♥ts t♦ t❤❡ n ✈❛r✐❛❜❧❡s x1 t❤r♦✉❣❤ xn r❡s♣❡❝t✐✈❡❧②✳ ❇② t❤❡ ♣r❡✈✐♦✉s ▲❡♠♠❛✱ ❡❛❝❤ 2✲❝❧❛✉s❡ C ∈ F ❤❛s ❛ ♣r♦❜❛❜✐❧✐t② ♦❢ 3

4

♦❢ ❜❡✐♥❣ s❛t✐s✜❡❞✳ ❚❤❡r❡❢♦r❡ ♦♥❡ ♦❢ t❤❡ t ❛ss✐❣♥♠❡♥ts ✭❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ r♦✇s ♦❢ Ht✮ ♠✉st ❛❧✇❛②s s❛t✐s❢② ❛t ❧❡❛st 3

4 ♠❛♥② ❝❧❛✉s❡s ♦❢ F✳ P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✼

✷✳✶✺✿ ❆❧❧ ✐♥ ✸✕❛t ▼♦st ✼ ✐♥ ✶✵

F := {{a}, {b}, {c}, {d}

• G21

, {a, x}, {b, x}, {c, d, x}

• G22
• G2

, {c, x}, {d, x}, {a, b, x}

• G1

} P❛rt✐t✐♦♥ t❤❡ ❝❧❛✉s❡s ♦❢ F ✐♥t♦ t✇♦ ❣r♦✉♣s✿ G1 ❛♥❞ G2✱ ✇❤❡r❡ G1 ❝♦♥t❛✐♥s ❛❧❧ ❝❧❛✉s❡s ✇✐t❤ ♦♥❧② ♥❡❣❛t✐✈❡ ❧✐t❡r❛❧s ❛♥❞ G2 ❝♦♥t❛✐♥s ❛❧❧ ❝❧❛✉s❡s ✇✐t❤ ❛t ❧❡❛st ♦♥❡ ♣♦s✐t✐✈❡ ❧✐t❡r❛❧✳ ❈❧❡❛r❧②✱ ❛♥② tr✐♣❧❡ ♦❢ ❝❧❛✉s❡s ❢r♦♠ G1 ♦r ❢r♦♠ G2 ❛r❡ s❛t✐s✜❛❜❧❡✳ ❆ tr✐♣❧❡ ❝♦♥t❛✐♥✐♥❣ 2 ❝❧❛✉s❡s ❢r♦♠ G1 ❛♥❞ 1 ❢r♦♠ G2 ❝❛♥ ❜❡ s❛t✐s✜❡❞ ❜② s❡tt✐♥❣ x → 0 ❛♥❞ ✉s✐♥❣ t❤❡ ❛❞❞✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡ ❢r♦♠ t❤❡ ❝❧❛✉s❡ ❢r♦♠ G2 t♦ s❛t✐s❢② t❤❡ ❝❧❛✉s❡ ❢r♦♠ G2✳

P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✽

◆♦✇ ✇❡ ❤❛✈❡ ❛ tr✐♣❧❡ ♦❢ ❝❧❛✉s❡s ❝♦♥t❛✐♥✐♥❣ 1 ❢r♦♠ G1 ❛♥❞ 2 ❢r♦♠ G2✳ ❲❡ s♣❧✐t G2 ✐♥t♦ t✇♦ ♣❛rts✿ G21 ❛♥❞ G22 ✇❤❡r❡ G21 := {{a}, {b}, {c}, {d}} ❛♥❞ G22 := {{a, x}, {b, x}, {c, x}}✱ ❛♥❞ ❝♦♥s✐❞❡r s✉❜❝❛s❡s✿

• 2 ❢r♦♠ G21✿ ❙❡t x → 0 ❛♥❞ s❛t✐s❢② t❤❡ t✇♦ ❝❧❛✉s❡s ❢r♦♠ G21 ✉s✐♥❣

t❤❡ ✈❛r✐❛❜❧❡s a, b, c, d✳

• 2 ❢r♦♠ G22✿ ❙❡t x → 1 ❛♥❞ s❛t✐s❢② t❤❡ ❝❧❛✉s❡ ❢r♦♠ G1 ✉s✐♥❣ ♦♥❡

♦❢ t❤❡ ✈❛r✐❛❜❧❡s a, b, c, d✳

• 1 ❢r♦♠ ❡❛❝❤ ♦❢ G21 ❛♥❞ G22✿ ■❢ ✇❡ ♣✐❝❦❡❞ {c} ♦r {d} ❢r♦♠ G21

t❤❡♥ ✇❡ s❡t x → 0 ♦t❤❡r✇✐s❡ ✇❡ s❡t x → 1✱ ❛♥❞ t❤✐s ❝❛♥ ❜❡ s❡❡♥ t♦ ❧❡❛❞ t♦ ❛ s❛t✐s❢②✐♥❣ ❛ss✐❣♥♠❡♥t ✭❱❡r✐❢②✦✮✳

P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✾

✷✳✶✺✿ ❆❧❧ ✐♥ ✸✕❛t ▼♦st ✼ ✐♥ ✶✵✿ P❛rt ■■

❚♦ ♣r♦✈❡ t❤❛t ❛♥② ❛ss✐❣♥♠❡♥t s❛t✐s✜❡s ♦♥❧② ❛t ♠♦st 7 ❝❧❛✉s❡s ✐♥ F✱ ❝♦♥s✐❞❡r t❤❡ t✇♦ ❢♦r♠✉❧❛❡ F [x→1] ❛♥❞ F [x→0]✳ ❋♦r ❡①❛♠♣❧❡✱ F [x→1] = {{a}, {b}, {c}, {d}, {¯ c}, {¯ d}, {¯ a,¯ b}}. ■♥ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ❣r♦✉♣s ♦❢ ❝❧❛✉s❡s ✐♥ F [x→1]✿

• {{c}, {¯

c}}✱

• {{d}, {¯

d}}✱ ❛♥❞

P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✶✵

• {{a}, {b}, {¯

a,¯ b}} ❛♥② ❛ss✐❣♥♠❡♥t ✇✐❧❧ ✉♥s❛t✐s❢② ❛t ❧❡❛st ♦♥❡ ❝❧❛✉s❡ ✐♥ ❡❛❝❤ ❣r♦✉♣✳ ❚❤❡ ❛r❣✉♠❡♥t ❢♦r t❤❡ ❢♦r♠✉❧❛ F [x→0] ✐s s✐♠✐❧❛r✳ ❚❤❡r❡❢♦r❡ ❛♥② ❛ss✐❣♥♠❡♥t t♦ t❤❡ ✈❛r✐❛❜❧❡s ✐♥ t❤❡ ❢♦r♠✉❧❛ F ✇✐❧❧ ✉♥✲ s❛t✐s❢② ❛t ❧❡❛st 3 ❝❧❛✉s❡s✳

P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✶✶

✷✯✳✶✿ ❘❛♥❞♦♠ ❙❛♠♣❧✐♥❣ ❉♦❡s♥✬t ❲♦r❦

❚r✐✈✐❛❧ s♦❧✉t✐♦♥✿ m ❞✐s❥♦✐♥t ❝❧❛✉s❡s ✐♥ ❛ k✲❈◆❋❀ s❛t✐s✜❡❞ ❜② ❛ r❛♥❞♦♠ ❛ss✐❣♠❡♥t ✇✐t❤ ♣r♦❜❛❜✐❧✐t② (1 − 2−k)m✳

P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✶✷

✸✳✶✿ ❉❡❝✐s✐♦♥ ✈s✳ ❈♦♥str✉❝t✐♦♥

❙✉♣♣♦s❡ F ✐s s❛t✐s✜❛❜❧❡✳ ❲❡ ❜✉✐❧❞ ❛ s❛t✐s❢②✐♥❣ ❛ss✐❣♥♠❡♥t α ❛s ❢♦❧❧♦✇s✿

• ❈❤❡❝❦ ✐❢ F [x→0] ❛♥❞ ✐❢ F [x→1] ✐s s❛t✐s✜❛❜❧❡ ✭❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠

♠✉st ❜❡✮✳

• ❆ss✐❣♥ x s✳t✳ t❤❡ r❡s✉❧t✐♥❣ ❢♦r♠✉❧❛ ✐s s❛t✐s✜❛❜❧❡✱ t❤❡♥ r❡♣❡❛t ✇✐t❤

F ← F [x→α(x)]✳ ◆♦t❡ t❤❛t t❤✐s ❞♦❡s ♥♦t ✇♦r❦ ✐❢ ✇❡ ✇❛♥t t♦ ❝♦✉♥t t❤❡ ♥✉♠❜❡r ♦❢ s❛t✐s❢②✐♥❣ ❛ss✐❣♥♠❡♥ts✳ ✭✷✲❙❆❚ ❝❛♥ ❜❡ ❞❡❝✐❞❡❞ ✐♥ ♣♦❧②✲t✐♠❡✱ ❜✉t ❝♦✉♥t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ❛ss✐❣♥♠❡♥ts ✐s ◆P✲❤❛r❞✮

P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐

❙❆❚ ❡①❡r❝✐s❡s ▼❛r❝❤ ✽✱ ✷✵✶✻ s❧✐❞❡ ✶✸

✸✳✸✿ ❙❛t✐s✜❛❜❧❡ ✇✐t❤ ♠❛♥② ❈❧❛✉s❡s

❆ s❛t✐s✜❛❜❧❡ (≤ 2)✲❈◆❋ ❢♦r♠✉❧❛ F ♠✉st ❤❛✈❡ ❛t ❧❡❛st ♦♥❡ s❛t✐s❢②✐♥❣ ❛ss✐❣♥♠❡♥t✱ s❛② α := (x1 → t1, . . . , xn → tn)✳ ❚❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ 1 ❝❧❛✉s❡s ✐♥ F ✐s ❛t ♠♦st n✳ ❚❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ 2 ❝❧❛✉s❡s ✐♥ F ✐s ❛t ♠♦st

n

2

• · 3✳

❚❤✉s✱ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝❧❛✉s❡s ✐s ❛t ♠♦st n + 3

n

2

• ✭❝❧❡❛r❧② t❤✐s ✐s

❛❧s♦ t✐❣❤t✮✳

P❛rt✐❛❧ s❛t✐s❢❛❝t✐♦♥✱ ❆❧❣♦r✐t❤♠s ❢♦r ✷✲❙❆❚ ❈❤✐❞❛♠❜❛r❛♠ ❆♥♥❛♠❛❧❛✐