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❆♥❛❧②s✐s ♦❢ ◗❯❆❉

❖✇❡♥ ✭❈❤✐❛✲❍s✐♥✮ ❈❤❡♥✱ ◆❛t✐♦♥❛❧ ❚❛✐✇❛♥ ❯♥✐✈❡rs✐t② ▼❛r❝❤ ✷✼✱ ❋❙❊ ✷✵✵✼✱ ▲✉①❡♠❜♦✉r❣ ❲♦r❦ ❛t ❆❝❛❞❡♠✐❛ ❙✐♥✐❝❛ s✉♣❡r✈✐s❡❞ ❜② ❉r✳ ❇♦✲❨✐♥ ❨❛♥❣ ❏♦✐♥t❧② ✇✐t❤ ❉rs✳ ❉❛♥ ❇❡r♥st❡✐♥ ❛♥❞ ❏✐✉♥✲▼✐♥❣ ❈❤❡♥

◗❯❆❉(q, n, r)✱ ❛ ❋❛♠✐❧② ♦❢ ❙tr❡❛♠ ❈✐♣❤❡rs

❙t❛t❡✿ n✲t✉♣❧❡ x = (x1, x2, . . . , xn) ∈ Kn, K = GF(q) ❯♣❞❛t❡✿ x ← (Q1(x), Q2(x), . . . , Qn(x))✳ ❍❡r❡ ❡❛❝❤ Qj ✐s ❛ r❛♥❞♦♠❧② ❝❤♦s❡♥✱ ♣✉❜❧✐❝ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ❖✉t♣✉t✿ r✲t✉♣❧❡ (P1(x), P2(x), . . . , Pr(x)) ❜❡❢♦r❡ ✉♣❞❛t✐♥❣ ✭❛❣❛✐♥✱ ❡❛❝❤ Pj ✐s ❛ r❛♥❞♦♠✱ ♣✉❜❧✐❝ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧✮ ❆t ❊✉r♦❝r②♣t ✷✵✵✻✱ ❇❡r❜❛✐♥✲●✐❧❜❡rt✲P❛t❛r✐♥ r❡♣♦rt❡❞ s♣❡❡❞s ❢♦r ◗❯❆❉(2, 160, 160), ◗❯❆❉(16, 40, 40)✱ ❛♥❞ ◗❯❆❉(256, 20, 20)✳

✶

❆ ❣r❛♣❤✐❝❛❧ ❉❡♣✐❝t✐♦♥

x0

• x1 = Q(x0)
• x2 = Q(x1)
• x3 = Q(x2)
• · · ·

y0 = P(x0) y1 = P(x1) y2 = P(x2) y3 = P(x3) · · ·

❚②♣✐❝❛❧❧② q ✐s ❛ ♣♦✇❡r ♦❢ 2✱ ❛❧❧♦✇✐♥❣ ❡❛❝❤ ♦✉t♣✉t ✈❡❝t♦r yi ∈ GF(q)r t♦ ❡♥❝r②♣t t❤❡ ♥❡①t r lg q ❜✐ts ♦❢ ♣❧❛✐♥t❡①t ✐♥ ❛ str❛✐❣❤t❢♦r✇❛r❞ ✇❛②✳

✷

◗❯❆❉✱ ✏Pr♦✈❛❜❧② ❙❡❝✉r❡✑❄

• ❙❡❝✉r✐t② ❚❤❡♦r❡♠✿ ❇r❡❛❦✐♥❣ ◗❯❆❉ ✐♠♣❧✐❡s t❤❡ ❝❛♣❛❜✐❧✐t② t♦ s♦❧✈❡

n + r r❛♥❞♦♠ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s ✐♥ n ✈❛r✐❛❜❧❡s✳

• ●❡♥❡r✐❝ MQ ✭▼✉❧t✐✈❛r✐❛t❡ ◗✉❛❞r❛t✐❝s✮ ✐s ❛♥ ◆P✲❤❛r❞ ♣r♦❜❧❡♠✳
• ❆❧❧ ❦♥♦✇♥ ❛❧❣♦r✐t❤♠s t♦ s♦❧✈❡ s✉❝❤ ❛ ❣❡♥❡r✐❝ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧

s②st❡♠ ❤❛✈❡ ❛✈❡r❛❣❡ t✐♠❡ ❝♦♠♣❧❡①✐t② 2an+o(n) ✇❤❡♥ r/n = ❝♦♥st❛♥t❀ ♠♦st ❛❧s♦ r❡q✉✐r❡ ❡①♣♦♥❡♥t✐❛❧ s♣❛❝❡✳

✸

❉✐✣❝✉❧t ●❡♥❡r✐❝❛❧❧②✱ ❇✉t ✳ ✳ ✳

❋♦❧❧♦✇✐♥❣ t❤❡ ♣♦s✐t✐♦♥ ♣❛♣❡r ♦❢ ❑♦❜❧✐t③✲▼❡♥❡③❡s ✭✏❆♥♦t❤❡r ❧♦♦❦ ❛t Pr♦✈❛❜❧❡ ❙❡❝✉r✐t②✑ ❏✳ ♦❢ ❈r②♣t♦✳✮ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❞✐s❝✉ss t❤❡ ✐♠♣❧✐❝❛t✐♦♥s ♦❢ t❤❡ s❡❝✉r✐t② ♣r♦♦❢✳

• ❍♦✇ t✐❣❤t ✐s t❤❡ s❡❝✉r✐t② r❡❞✉❝t✐♦♥❄
• ❍♦✇ ❞✐✣❝✉❧t ✐s t❤❡ ✉♥❞❡r❧②✐♥❣ ♣r♦❜❧❡♠❄
• ❲❤❛t ✐s t❤❡ ❜❡st ❛tt❛❝❦ ❦♥♦✇♥ t♦❞❛②❄
• ■s t❤❡ s❡❝✉r✐t② r❡❞✉❝t✐♦♥ ❝♦♠♣❧❡t❡❄

✹

■♥st❛♥❝❡s ❛♥❞ Pr♦✈❛❜✐❧✐t②

❲❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ♣r♦♣♦s❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ✐♥st❛♥❝❡s ♦❢ ❢❛♠✐❧✐❡s ♦❢ ❝r②♣t♦s②st❡♠s ❝♦✈❡r❡❞ ❜② s❡❝✉r✐t② r❡❞✉❝t✐♦♥s✿ ❇r♦❦❡♥✿ ❲❡ ❝❛♥ ❛tt❛❝❦ ❛♥❞ ❜r❡❛❦ t❤❡ ✐♥st❛♥❝❡✳ ❯♥♣r♦✈❛❜❧❡✿ ❲❡ ❝❛♥ s♦❧✈❡ t❤❡ ✉♥❞❡r❧②✐♥❣ ❤❛r❞ ♣r♦❜❧❡♠✳ ❯♥♣r♦✈❡♥✿ ❆ ♣✉t❛t✐✈❡ ❢❡❛s✐❜❧❡ ❛tt❛❝❦ ♦♥ t❤❡ ✐♥st❛♥❝❡ ♥❡❡❞ ♥♦t ❧❡❛❞ t♦ ❛♥ ✐♠♣r♦✈❡♠❡♥t ♦♥ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ❤❛r❞ ♣r♦❜❧❡♠ ❞✉❡ t♦ t❤❡ ❧♦♦s❡♥❡ss ❢❛❝t♦r ✐♥ t❤❡ s❡❝✉r✐t② r❡❞✉❝t✐♦♥✳ Pr♦✈❡❞✿ ❙❡❝✉r✐t② ♣r♦♦❢ ✇♦r❦s ❛s ❛❞✈❡rt✐s❡❞ ❢♦r t❤✐s ✐♥st❛♥❝❡✳

✺

❚♦❞❛②✬s ❙②st❡♠✲❙♦❧✈✐♥❣

❙t❛t❡✲♦❢✲t❤❡✲❛rt ❛❧❣♦r✐t❤♠s t♦ s♦❧✈❡ m ❣❡♥❡r✐❝ ♣♦❧②♥♦♠✐❛❧ ❡q✉❛t✐♦♥s ✐♥ n GF(q)✲✈❛r✐❛❜❧❡s ❛r❡ ❛❧❧ r❡❧❛t❡❞ ✐♥ s♦♠❡ ✇❛② t♦ ❇✉❝❤❜❡r❣❡r✬s ❛❧❣♦r✐t❤♠ ❢♦r ❝♦♠♣✉t✐♥❣ ●rö❜♥❡r ❇❛s❡s✳

• ❳▲✱ ✜rst ♣r♦♣♦s❡❞ ❜② ▲❛③❛r❞ ❛♥❞ r❡❞✐s❝♦✈❡r❡❞ ❜② ❈♦✉rt♦✐s ❡t ❛❧✳

❊ss❡♥❝❡✿ ❛♥ ❡❧✐♠✐♥❛t✐♦♥ ♦♥ ❛ ▼❛❝❛✉❧❛② ▼❛tr✐①✳ ❆❧s♦ t❤❡ ❛❞❥✉♥❝ts ✕ ❋❳▲ ✭❵❋✬ ❢♦r ✏✜①✑✮ ✐♥tr♦❞✉❝❡s ❣✉❡ss✐♥❣ ✈❛r✐❛❜❧❡s✳ ✕ ❳▲✷✱ r✉♥♥✐♥❣ t❤❡ ❡❧✐♠✐♥❛t✐♦♥ ♦♥ t❤❡ ❤✐❣❤❡st ♠♦♥♦♠✐❛❧s ♦♥❧② ❛♥❞ t❤❡♥ r❡♣❡❛t❡❞❧② ♠✉❧t✐♣❧② ❜② ✈❛r✐❛❜❧❡s t♦ r❛✐s❡ ❞❡❣r❡❡s✳

• F4 ✭♥♦✇ ✐♥ ▼❆●▼❆✮ ❛♥❞ F5✱ ♦❢ ✇❤✐❝❤ ❳▲✷ ✐s ❛♥ ✐♥❢❡r✐♦r ❢♦r♠✳

✻

❋❛❝ts ♦❢ ▲✐❢❡ ❢♦r ❳▲

★ ♠♦♥♦♠✐❛❧s✿ T = [tD]

• (1 − tq)n(1 − t)−(n+1)

; ✭✶✮ ★ ❢r❡❡ ♠♦♥♦♠s✿ T − I ≥ [tD]

• (1 − tq)n

(1 − t)n+1

m

• i=1

1 − tdi 1 − tqdi

• . ✭✷✮

❍❡r❡ deg pi := di✱ [u]s := ❝♦❡✣❝✐❡♥t ♦❢ u ✐♥ ❡①♣❛♥s✐♦♥ ♦❢ s✳ ❲❡ ❡①♣❡❝t ❛ s♦❧✉t✐♦♥ ❛t DXL = min{D : ❘❍❙ ♦❢ ❊q✳ ✷ ≤ 0}✳ ■❢ t❤❡ (pi) ✐s q✲s❡♠✐✲r❡❣✉❧❛r ✭tr✉❡ ❛❧♠♦st ❛❧✇❛②s✮✱ ❊q✳ ✷ ✐s = ❛s ❧♦♥❣ ❛s ✐ts ❘❍❙ r❡♠❛✐♥s ♣♦s✐t✐✈❡✳ T = n+D

D

• ,

T − I = [tD]

• (1 − t)m−n−1 (1 + t)m

✐s t❤❡ r❡❞✉❝❡❞ ❝❛s❡ ❢♦r ❧❛r❣❡ ✜❡❧❞s ✭q > D✮✳ CXL ≈ 3kT 2(c0 + c1 lg T) ✉s✐♥❣ ❛ ♠♦❞✐✜❡❞ ❲✐❡❞❡♠❛♥♥ ❛❧❣♦r✐t❤♠ ✭k ✐s ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ t❡r♠s ♣❡r ❡q✉❛t✐♦♥✮✳

✼

❳▲ ✇✐t❤ ❍♦♠♦❣❡♥♦✉s ❲✐❡❞❡♠❛♥♥

✶✳ ❈r❡❛t❡ t❤❡ ❡①t❡♥❞❡❞ ▼❛❝❛✉❧❛② ♠❛tr✐① ♦❢ t❤❡ s②st❡♠ t♦ ❛ ❝❡rt❛✐♥ ❞❡❣r❡❡ DXL✿ ▼✉❧t✐♣❧② ❡❛❝❤ ❡q✉❛t✐♦♥ ♦❢ ❞❡❣r❡❡ di ❜② ❛❧❧ ♠♦♥♦♠✐❛❧s ✉♣ t♦ ❞❡❣r❡❡ DXL − di ❛♥❞ t❛❦❡ t❤❡ ♠❛tr✐① ♦❢ ❝♦❡✣❝✐❡♥ts✳ ✷✳ ❘❛♥❞♦♠❧② ❞❡❧❡t❡ s♦♠❡ r♦✇s t❤❡♥ ❛❞❞ s♦♠❡ ❝♦❧✉♠♥s t♦ ❢♦r♠ ❛ sq✉❛r❡ s②st❡♠✱ Ax = 0 ✇❤❡r❡ dim A = βT + (1 − β)R✳ ❯s✉❛❧❧② β = 1 ✇♦r❦s✳ ❑❡❡♣ t❤❡ s❛♠❡ ❞❡♥s✐t② ♦❢ t❡r♠s✳ ✸✳ ❆♣♣❧② t❤❡ ❤♦♠♦❣❡♥❡♦✉s ✈❡rs✐♦♥ ♦❢ ❲✐❡❞❡♠❛♥♥✬s ♠❡t❤♦❞ t♦ s♦❧✈❡ ❢♦r x✿ ✭❛✮ ❙❡t k = 0 ❛♥❞ g0(z) = 1✱ ❛♥❞ t❛❦❡ ❛ r❛♥❞♦♠ b✳ ✭❜✮ ❈❤♦♦s❡ ❛ r❛♥❞♦♠ uk+1 ❬✉s✉❛❧❧② t❤❡ (k + 1)✲st ✉♥✐t ✈❡❝t♦r❪✳ ✭❝✮ ❋✐♥❞ t❤❡ s❡q✉❡♥❝❡ uk+1Aib st❛rt✐♥❣ ❢r♦♠ i = 0 ❛♥❞ ❣♦✐♥❣ ✉♣ t♦ 2N − 1✳ ✭❞✮ ❆♣♣❧② gk ❛s ❛ ❞✐✛❡r❡♥❝❡ ♦♣❡r❛t♦r t♦ t❤✐s s❡q✉❡♥❝❡✱ ❛♥❞ r✉♥ t❤❡ ❇❡r❧❡❦❛♠♣✲▼❛ss❡② ❛❧❣♦r✐t❤♠ ♦✈❡r GF(q) ♦♥ t❤❡ r❡s✉❧t t♦ ✜♥❞ t❤❡ ♠✐♥✐♠❛❧ ♣♦❧②♥♦♠✐❛❧ fk+1✳ ✭❡✮ ❙❡t gk+1 := fk+1gk ❛♥❞ k := k + 1✳ ■❢ deg(gk) < N ❛♥❞ k < n✱ ❣♦ t♦ ✭❜✮✳ ✹✳ ❈♦♠♣✉t❡ t❤❡ s♦❧✉t✐♦♥ x ✉s✐♥❣ t❤❡ ♠✐♥♣♦❧② f(z) = gk(z) = cmzm +cm−1zm−1 +· · ·+cℓzℓ✿ ❚❛❦❡ ❛♥♦t❤❡r r❛♥❞♦♠ b✳ ❙t❛rt ❢r♦♠ x = (cmAm−ℓ+cm−1Am−ℓ−1+· · ·+cℓ1)b✱ ❝♦♥t✐♥✉✐♥❣ t♦ ♠✉❧t✐♣❧② ❜② A ✉♥t✐❧ ✇❡ ✜♥❞ ❛ s♦❧✉t✐♦♥ t♦ Ax = 0✳ ✺✳ ■❢ t❤❡ ♥✉❧❧✐t② ℓ > 1 r❡♣❡❛t t❤❡ ❝❤❡❝❦ ❜❡❧♦✇ ❛t ❡✈❡r② ♣♦✐♥t ♦❢ ❛♥ ❛✣♥❡ s✉❜s♣❛❝❡ ✭q ♣♦✐♥ts ✐❢ ℓ = 2✮✳ ✻✳ ❖❜t❛✐♥ t❤❡ s♦❧✉t✐♦♥ ❢r♦♠ t❤❡ ❧❛st ❢❡✇ ❡❧❡♠❡♥ts ♦❢ x ❛♥❞ ❝❤❡❝❦ ✐ts ❝♦rr❡❝t♥❡ss✳

✽

◗❯❆❉(256, 20, 20) ❯♥♣r♦✈❛❜❧❡ ❢r♦♠ MQ

• ■s ✷✵ GF(256) ✈❛r✐❛❜❧❡s ✐♥ ✹✵ ❡q✉❛t✐♦♥s ❤❛r❞ t♦ s♦❧✈❡❄
• ❲❡ s❛② ♥♦✦ ●❡♥❡r✐❝ ❳▲ s♦❧✈❡s t❤✐s ✐♥ 245 ❝②❝❧❡s✱ ♦♥❧② ❛ ❢❡✇ ❤♦✉rs

♦♥ ❛ ❞❡❝❡♥t ❝♦♠♣✉t❡r✳

• ❚❤❡ t❡❝❤♥✐❝❛❧ ❞❡t❛✐❧s ❛r❡✿ ❝②❝❧❡s ♣❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♦♥ ❛ P✹ ≈ 12

✭✸ ▲✶ ❝❛❝❤❡ ❧♦❛❞s✮❀ DXL = 5 ❛♥❞ T = 53130✳ ▼❛① ♥✉♠❜❡r ♦❢ t❡r♠s ♣❡r ❡q✉❛t✐♦♥ ✐s k 231✱ s♦ CXL ≈ 9 × 1012 245✳

• ❍❡♥❝❡ ♥♦ s❡❝✉r✐t② ✐s ♣r♦✈❛❜❧❡ ❬♥♦r ❝❧❛✐♠❡❞ ❜② ♦r✐❣✳ ◗❯❆❉ ♣❛♣❡r❪

❢r♦♠ MQ ✭20 ✈❛rs✱ 40 ❡qs✮ ♦✈❡r GF(256)✳

✾

❉✐r❡❝t ❆tt❛❝❦

• ❈❛♥ ◗❯❆❉(256, 20, 20) ❜❡ ❛ ❝✐♣❤❡r t❤❛t ✐s ❛❝❝❡♣t❛❜❧② s❡❝✉r❡

✇✐t❤♦✉t ❜❡✐♥❣ ♣r♦✈❛❜❧❡❄ ❲❡ s❛② ♥♦✱ ❛♥❞ ❡st✐♠❛t❡ 263 ❝②❝❧❡s ❢♦r ❛ ❞✐r❡❝t ❛tt❛❝❦ t❤❛t ❜r❡❛❦s ◗❯❆❉(256, 20, 20)✳

• ❖❢t❡♥ ✇❡ ❝❛♥ ❛❝q✉✐r❡ s♦♠❡ ❝✐♣❤❡r str❡❛♠ ✈✐❛ ❦♥♦✇♥ ♣❧❛✐♥t❡①t✳

❚❤✐s ❛tt❛❝❦ ♦♥❧② ✉s❡s t✇♦ ❜❧♦❝❦s ✭29 ❜✐ts✮ ♦❢ ♦✉t♣✉t✳

• ▲❡t t❤❡ ✐♥st❛♥❝❡ ❜❡ xj+1 = Q(xj), yj = P(xj) ✇✐t❤ P, Q :

GF(q)n → GF(q)n✳ ❲✐t❤ ✭❲▲❖●✮ y0 ❛♥❞ y1✱ ✇❡ s♦❧✈❡ ❢♦r x0 ✈✐❛ P(x0) = y0, P(Q(x0)) = y1.

✶✵

✷✵ q✉❛❞r❛t✐❝s✱ ✷✵ q✉❛rt✐❝s ♦✈❡r GF(256)

• 263 ♠✉❧ts ✉♣♣❡r ❜♦✉♥❞✱ r❡❛❧ ✈❛❧✉❡ s❤♦✉❧❞ ❜❡ ♠♦r❡ ❧✐❦❡ 260✳
• ❙✐❣♥✐✜❝❛♥t ♣❛r❛♠❡t❡rs ❛r❡✿

✕ ❞❡❣r❡❡ DXL = 10✱ ✕ ★♠♦♥♦♠✐❛❧s T = 30

10

• = 30045015✱

✕ ★✐♥✐t✐❛❧ ❡q✉❛t✐♦♥s ✐s R = 20× 28

8

• +20×

26

6

• = 66766700✱

✕ t♦t❛❧ ★ t❡r♠s ✐♥ t❤♦s❡ ❡q✉❛t✐♦♥s ✐s τ := kR = 20 28

8

22

2

• + 20

26

6

24

4

• = 63287924700.

❙❤♦✉❧❞ ❜❡ ❞♦❛❜❧❡ ♦♥ ❛ ♠❛❝❤✐♥❡ ♦r ❝❧✉st❡r ✇✐t❤ ✸✽✹●❇ ♦❢ ♠❡♠♦r②✳

✶✶

❚❡st✐♥❣ ❆tt❛❝❦ ✈s✳ ◗❯❆❉(256, n, n)

n ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ❉ ✼ ✼ ✼ ✽ ✽ ✽ ✽ CXL 2.29 · 102 7.55 · 102 2.30 · 103 5.12 · 104 1.54 · 105 4.39 · 105 1.17 · 106 ❧❣CXL ✼✳✽✹ ✾✳✺✻ 1.12 · 10 1.56 · 10 1.72 · 10 1.87 · 10 2.02 · 10 ❚ 1.14 · 104 1.94 · 104 3.28 · 104 1.26 · 105 2.03 · 105 3.20 · 105 4.90 · 105 ❛❚♠ ✶✷✵ ✶✹✼ ✶✼✼ ✷✹✺ ✷✽✽ ✸✸✺ ✸✽✺ ❝❧❦s ✶✹✳✻ ✶✸✳✻ ✶✷✳✶ ✶✸✳✶ ✶✷✳✾ ✶✷✳✽ ✶✷✳✼ ▼❙ ❈✰✰ ✼❀ P✲❉ ✸✳✵●❍③✱ ✷●❇ ❉❉❘✷✲✺✸✸✱ ❚✿ ★♠♦♥♦♠✐❛❧s✱ ❛❚♠✿ ❛✈❡r❛❣❡ t❡r♠s ✐♥ ❛ r♦✇✱ ❝❧❦s✿ ♥✉♠❜❡r ♦❢ ❝❧♦❝❦s ♣❡r ♠✉❧t✐♣❧✐❝❛t✐♦♥✳

• ❙❡r✐❛❧ ❈♦❞❡ ♦♥ i386 r❡q✉✐r❡s t❤r❡❡ ❞❡♣❡♥❞❡♥t ▲✶ ❛❝❝❡ss❡s ♣❡r

♠✉❧t✐♣❧✐❝❛t✐♦♥ ✭✸ ❝②❝❧❡s ❑✽✴❈♦r❡✱ ✹ ❝②❝❧❡s P✹✮ ♣❧✉s ❝❤❛♥❣❡✳

• ❯♥r♦❧❧✐♥❣ ❧♦♦♣s ❢♦r ①✽✻✲✻✹ s❛✈❡s ✷✵✪✕✷✺✪ ❝②❝❧❡s ❛ ♠✉❧t✐♣❧✐❝❛t✐♦♥✳
• 256✲s❡♠✐✲r❡❣✉❧❛r✐t② ❛ss✉♠♣t✐♦♥ ✜ts ❡♠♣✐r✐❝❛❧ ❞❛t❛ ✉♣ t♦ n = 15✳

✶✷

◗❯❆❉(16, 40, 40) ❯♥♣r♦✈❛❜❧❡✱ ❜✉t ♥♦t ❇r♦❦❡♥

• ✽✵ ❡qs✳ ✐♥ ✹✵ GF(16) ✈❛rs✳ ❡st✐♠❛t❡❞ t♦ < 272 ❝②❝❧❡s ✐♥ ❳▲✳
• ❚❡❝❤♥✐❝❛❧ ❞❛t❛✿ DXL = 8✱ T = 377348994✱ ❛♥❞ k 861✳
• ❙♦ ◗❯❆❉(16, 40, 40) ❝❛♥ ♥❡✈❡r ❜❡ ✏♣r♦✈❛❜❧② s❡❝✉r❡✑ ❢r♦♠ MQ

✭✹✵✱✽✵✮✳ ❇✉t ✇❡ ❞♦♥✬t ❦♥♦✇ ❤♦✇ t♦ ❜r❡❛❦ ✐t ✐♥ 280✳

• ❉✐r❡❝t s♦❧✉t✐♦♥ t❛❦❡s 295 ♠✉❧ts ✭❣✉❡sst✐♠❛t❡❞ ❛t 2100 ❝②❝❧❡s✮

✈✐❛ ❳▲✲❲✐❡❞❡♠❛♥♥ ✭DXL = 14✱ T = 3245372870670✮✳

• ❉❛t❛ ❝♦♠♣❧❡①✐t② ✐s 10000 ❚❇ ✭♦♥❧② ∼ 256 ❜✐ts✮ ❢♦r t❤❡ ♠❛tr✐①✳

✶✸

❲❤② ❖♥❧② ✷ ❇❧♦❝❦s❄

• Pr❛❝t✐❝❛❧ ❛♥s✇❡r✿ ✇❡ t❡st ✇✐t❤ ❞❡❣r❡❡✲✽ ❡q✉❛t✐♦♥s❀ ❞♦❡s♥✬t ❤❡❧♣✳
• ❚❤❡♦r❡t✐❝❛❧ ❛♥s✇❡r✿ t❤❡ ❳▲ ♦♣❡r❛t✐♥❣ ❞❡❣r❡❡ ✐s

DXL = min

• D : [tD]
• (1 − t2)(1 − t4)

n (1 − t)n+1 < 0

• ,

❍❡♥❝❡ w := DXL/n ≈ t❤❡ s♠❛❧❧❡st ♣♦s✐t✐✈❡ ③❡r♦ ♦❢ fn(w) := (1 − z2)n(1 − z4)n (1 − z)n+1zwn+1 dz =

• dz

z(1 − z) (1 + z)(1 − z4) zw n

✶✹

❉✐♠✐♥✐s❤✐♥❣ ❘❡t✉r♥s ✭❢♦r ❧❛r❣❡ q✮

■♥ ❛s②♠♣t♦t✐❝ ❛♥❛❧②s✐s✱ fn(w) =

• dz

z(1−z)

• (1+z)(1−z4)

zw

n ❝❛♥ ♦♥❧② ✈❛♥✐s❤ ✐❢ t❤❡ s❛❞❞❧❡ ♣♦✐♥t ❡q✉❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡❣r❛❧✱ ❧❡tt✐♥❣ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❡①♣r❡ss✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♣❛r❡♥ ❜❡ ③❡r♦✿ (w − 5)z4 + z3 − z2 + z − w = 0 ❤❛s ❞♦✉❜❧❡ r♦♦ts ✭❛ ✏♠♦♥❦❡② s❛❞❞❧❡✑✮✱ ✇❤✐❝❤ ❤❛♣♣❡♥s ✇❤❡♥ w ✐s ✈❡r② ❝❧♦s❡ t♦ 0.2 ✭❛❝t✉❛❧❧② ≈ 0.200157957✮✳ ❙✐♠✐❧❛r ❝♦♠♣✉t❛t✐♦♥s ✐♥❝❧✉❞✐♥❣ ❞❡❣r❡❡✲✽ ❡q✉❛t✐♦♥s ♦♥❧② ♠❛❦❡ ✐t w ≈ 0.1998✳ ❈❧❡❛r❧② ♥♦t ✇♦rt❤ ♦✉r t✐♠❡✳

✶✺

◗❯❆❉(2, 160, 160)✿ ❆♥ ❯♥♣r♦✈❡♥ ❈❛s❡

• ◗❯❆❉(2, 160, 160) t❛❦❡s ≈ 2180 ♠✉❧t✐♣❧✐❝❛t✐♦♥s t♦ ❛tt❛❝❦ ❞✐r❡❝t❧②✿

❥✉st s♦❧✈❡ 160 ❡q✉❛t✐♦♥s ✐♥ 160 ✈❛r✐❛❜❧❡s ✉s✐♥❣ ❳▲✳

• ❋♦r n < 200✱ t❤❡ ❡✛❡❝t ♦❢ ✉s✐♥❣ q✉❛rt✐❝ ❛♥❞ ❞❡❣r❡❡✲✽ ❡q✉❛t✐♦♥s

✭✷♥❞✱ ✸r❞ ♦✉t♣✉t ❜❧♦❝❦s ❛♥❞ ❜❡②♦♥❞✮ ✐s ♥♦t ❞✐s❝❡r♥✐❜❧❡✳

• ❙✐♠✐❧❛r ❛s②♠♣t♦t✐❝s ❛s ❛❜♦✈❡ s❤♦✇s t❤❛t ❢♦r ❧❛r❣❡ n t❤❡②

✭❡✈❡♥t✉❛❧❧②✮ ♠❛❦❡ ❛ ❜✐❣ ❞✐✛❡r❡♥❝❡✳

• ❚❤❡ ✉♥❞❡r❧②✐♥❣ MQ ♣r♦❜❧❡♠ ♦❢ ✶✻✵ ✈❛rs ❛♥❞ ✸✷✵ ❡q✉❛t✐♦♥s t❛❦❡s

2140 ♠✉❧t✐♣❧✐❝❛t✐♦♥s✱ ✇❤✐❝❤ s❡❡♠s ❤✐❣❤ ❡♥♦✉❣❤✱ ❜✉t ✳ ✳ ✳

✶✻

❚✐❣❤t♥❡ss ♦❢ ❘❡❞✉❝t✐♦♥

• ◗❯❆❉ ❛tt❛❝❦ ✐♠♣❧✐❡s ❛♥ MQ ❛tt❛❝❦ ✇✐t❤ ❛ ❧♦ss ♦❢ ❡✣❝✐❡♥❝②✳
• ❙♣❡❝✐✜❝❛❧❧②✱ ✐❢ λr ❜✐ts ♦❢ ♦✉t♣✉t ❢r♦♠ ◗❯❆❉(2, n, r) ❝❛♥ ❜❡

❞✐st✐♥❣✉✐s❤❡❞ ❢r♦♠ ✉♥✐❢♦r♠ ✇✐t❤ ❛❞✈❛♥t❛❣❡ ǫ ✐♥ t✐♠❡ T✱ t❤❡♥ ❛ r❛♥❞♦♠ MQ s②st❡♠ ♦❢ n + r ❡q✉❛t✐♦♥s ✐♥ n ✈❛r✐❛❜❧❡s ♦✈❡r GF(2) ❝❛♥ ❜❡ s♦❧✈❡❞ ✇✐t❤ ♣r♦❜❛❜✐❧✐t② 2−3ǫ/λ ✐♥ t✐♠❡ T ′ ≤ 27n2λ2 ǫ2

• T + (λ + 2)TS + log

27nλ2 ǫ2

• + 2
• +27nλ2

ǫ2 TS ✇❤❡r❡ TS := t✐♠❡ t♦ r✉♥ ♦♥❡ ❜❧♦❝❦ ♦❢ ◗❯❆❉(2, n, r)✳

✶✼

Pr♦✈❡♥ ❛♥❞ ❯♥♣r♦✈❡❞ ❈❛s❡s ❢♦r q = 2

❚❤❡ ❧♦♦s❡♥❡ss ❢❛❝t♦r ✐s ❛❜♦✉t 210n2λ3/ǫ3✳ ■❢ ǫ = 0.01✱ n = r✱ ❛♥❞ L = λn = 240✱ t❤✐s ❢❛❝t♦r ✐s t❤❡♥ 2150/n✳ ❚❤❡ t❤❡♦r❡♠ ❝❛♥♥♦t ❝♦♥❝❧✉❞❡ T ≥ 280 ✇✐t❤♦✉t ❛ss✉♠✐♥❣ t❤❛t T ′ ≥ 2230/n✳

• n = 160 ✐s ❤❡♥❝❡ ❯♥♣r♦✈❡♥ ✭♦r✐❣✐♥❛❧ ◗❯❆❉ ♣❛♣❡r st❛t❡s t❤✐s✮✳
• n = 256✿

Pr♦✈❡♥ ❢♦r L = 222, ǫ = 0.01✱ T ′ ≈ 2205 ✭♠✉❧t✐♣❧✐❝❛t✐♦♥s✮✳ ■♥ ❢❛❝t ✇❡ ♦♥❧② ♥❡❡❞ T ′ ≥ 2168✳

• n = 350✿

Pr♦✈❡♥ ❢♦r L = 240, ǫ = 0.01✱ T ′ ≈ 2263 ✭♠✉❧t✐♣❧✐❝❛t✐♦♥s✮✳ ❲❡ ♦♥❧② ♥❡❡❞❡❞ T ′ ≥ 2221✳

✶✽

❆ ◆♦t❡ ♦♥ T 2.376

• ❖❢t❡♥ T 2.376 ✐s ✉s❡❞ ❛s t❤❡ ❝♦st ♦❢ ❡❧✐♠✐♥❛t✐♦♥s✳
• ❚❤✐s ❞✐s❝♦✉♥ts t❤❡ ❤✉❣❡ ❝♦♥st❛♥t t❤❛t ✐s ❡①♣❡❝t❡❞ ❢r♦♠ t❤❡

❈♦♣♣❡rs♠✐t❤✲❲✐♥♦❣r❛❞ ♣❛♣❡r✳

• ❲❡ ✐♠♣r♦✈❡ T 2.376 t♦ T 2✱ ✉s✐♥❣ ❛ s♣❛rs❡ ♠❛tr✐① ❛❧❣♦r✐t❤♠✱ ❜✉t

t❤❡r❡ ❛r❡ st✐❧❧ ❢❛❝t♦rs ✐♥ ❢r♦♥t ♦❢ T 2✳

• ❚❤✐s ❡①♣❧❛✐♥s t❤❡ ❣❛♣ ✐♥ t❤❡ ❛♥❛❧②s✐s ❢♦r ◗❯❆❉(2, 350, 350)✳

✶✾

❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❚❖❉❖s

• ●❡♥❡r✐❝❛❧❧② MQ ✐s ❜❡❧✐❡✈❡❞ t♦ ❜❡ ❡①♣♦♥❡♥t✐❛❧ ✐♥ n✳ ❈♦♠♣❧❡①✐t②

♦❢ ❜r❡❛❦✐♥❣ ◗❯❆❉ ✇♦✉❧❞ t❤❡♥ ❛❧s♦ ❜❡ ♦❢ t❤❡ ❢♦r♠ 2an+o(n)✳ ❇✉t t❤❡ ❝♦❡✣❝✐❡♥t a ✭= a(q, r/n)✮ ❝❛♥ ❜❡ s✉r♣r✐s✐♥❣❧② s♠❛❧❧✳

• ◗❯❆❉ ✐s ❝❧❡❛r❧② ❛ ✇♦rt❤✇❤✐❧❡ ❛tt❡♠♣t ❛♥❞ ✇♦rt❤ ♦♣t✐♠✐③✐♥❣ ❢✉rt❤❡r✳
• ❲❡ ♥❡❡❞ t✐❣❤t❡r r❡❞✉❝t✐♦♥s✳ ❆t t❤❡ ♠♦♠❡♥t✱ ✇❡ ❛r❡ r❡❞✉❝✐♥❣ ❢r♦♠

✇❤❛t s❡❡♠s t♦ ❜❡ ❛ ♠♦r❡ ❞✐✣❝✉❧t ♣r♦❜❧❡♠ t♦ ❛♥ ❡❛s✐❡r ♣r♦❜❧❡♠✳

• ❈♦♠♣❛r✐s♦♥s ❜❡t✇❡❡♥ ❝✐♣❤❡rs ✇✳ ♣r♦✈❛❜❧② s❡❝✉r❡ ♣❛r❛♠❡t❡rs❄
• ❚❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t st♦r❛❣❡ ❛❝❝❡ss ❞❡❧❛②s ❛♥❞ ♣❛r❛❧❧❡❧✐s♠❄

✷✵

❚❤❛♥❦s t♦

• ❖✉r ❣r❛❝✐♦✉s ❤♦sts ❛♥❞ ♦r❣❛♥✐③❡rs
• ❆❝❛❞❡♠✐❛ ❙✐♥✐❝❛ ❛♥❞ ❚❲■❙❈ ✭❚❛✐✇❛♥ ■♥❢♦✳ ❙❡❝✉r✐t② ❈❡♥t❡r✮
• ❉r✳ ❇♦✲❨✐♥ ❨❛♥❣✱ Pr♦❢✳ ❉❛♥ ❇❡r♥st❡✐♥✱ ❉r✳ ❏✐✉♥✲▼✐♥❣ ❈❤❡♥✳
• ❊✈❡r②♦♥❡ ❢♦r ❜❡✐♥❣ ❤❡r❡✳

q✉❡st✐♦♥s❄❄

✷✶

❲❤② ❲✐❡❞❡♠❛♥♥ ❛♥❞ ♥♦t ▲❛♥❝③♦s

❚❤❡ t✇♦ s❤♦✉❧❞ ❜❡ ♠♦r❡ ♦r ❧❡ss ❡q✉✐✈❛❧❡♥t ✐♥ ♠♦❞❡r♥ ❢♦r♠s✳ ❲❡ ❝❤♦s❡ ❲✐❡❞❡♠❛♥♥ ♦✈❡r ▲❛♥❝③♦s ❜❡❝❛✉s❡ ✐♥ t❤❡ ✏♥❛✐✈❡✑ ❢♦r♠s

• ❇❡❝❛✉s❡ ✐t ✐s ❡❛s✐❡r t♦ ♣r♦❣r❛♠ ✇❡❧❧✳ ▲❛♥❝③♦s r❡q✉✐r❡s ♠✉❧t✐♣❧②✐♥❣

❜② ❛ s♣❛rs❡ ♠❛tr✐① ✐♥ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥s✳

• ❲❡ ❞♦♥✬t ♥❡❡❞ t♦ ✉s❡ ❛ r❛♥❞♦♠ ❞✐❛❣♦♥❛❧ ✈❡❝t♦r✳
• ❲❡ ❥✉st ❤❛❞ t❤❡ ❝♦❞❡ r❡❛❞② t♦ ✉s❡✳

✷✷

❲❤② ❳▲ ❛♥❞ ♥♦t F5

• ❚❤❡♦r❡t✐❝❛❧✿ ❲♦r❦✐♥❣ ♦♥ t❤❡ t♦♣ ❞❡❣r❡❡ ♠♦♥♦♠✐❛❧s✱ ❢♦r ❧❛r❣❡ ✜❡❧❞s

❳▲✷✴F4✴F5 ♣❧❛② ✇✐t❤ ♦♥❡ ❢❡✇❡r ✈❛r✐❛❜❧❡✳ ❚❤✐s ♠❛② ♥♦t ♦✛s❡t ❞❡♥s❡ ✈s✳ s♣❛rs❡ ♠❛tr✐① ❡q✉❛t✐♦♥ s♦❧✈✐♥❣ ❞✐✛❡r❡♥❝❡ ✐❢ ω > 2✳

• Pr❛❝t✐❝❛❧✿

■❢ t❤❡ ♠❛tr✐❝❡s ♦❢ F4✴F5 ✇✐❧❧ ❡✈❡♥t✉❛❧❧② ❜❡❝♦♠❡ ♠♦❞❡r❛t❡❧② ❞❡♥s❡✱ ✇❡ ✇✐❧❧ r✉♥ ♦✉t ♦❢ ♠❡♠♦r② ❜❡❢♦r❡ t✐♠❡✳

m − n DXL Dreg n = 9 n = 10 n = 11 n = 12 n = 13 2m m ✻✳✵✾✵ ✹✻✳✼✼✵ ✸✺✵✳✺✸✵ ✸✸✷✷✳✻✸✵ s✐❣♠❡♠ 1 m ⌈m+1

2

⌉ ✶✳✷✹✵ ✽✳✾✼✵ ✺✸✳✼✸✵ ✹✶✸✳✼✽✵ ✷✺✸✽✳✽✼✵ 2 ⌈m+1

2

⌉ ⌈m+2−√m+2

2

⌉ ✵✳✸✷✵ ✷✳✷✸✵ ✶✷✳✹✺✵ ✽✽✳✶✽✵ ✹✸✻✳✻✵✵

❚❡st r❡s✉❧ts ❣✐✈❡♥ ♦♥ P✹✲✸✳✷●✱ ✷●❇ ❘❆▼✱ ▼❆●▼❆✲✷✳✶✷ ✇✐t❤ F4✳

• Pr❛❣♠❛t✐❝✿ ✇❡ ❞♦♥✬t ❤❛✈❡ ❛ ❝♦♣② ♦❢ F5 t♦ ♣❧❛② ✇✐t❤✳

✷✸

❇❛s✐❝ ❳▲ ❛t ❉❡❣r❡❡ D

▲❡t T (D):={deg ≤ D ♠♦♥♦♠✐❛❧s}✱ T := |T (D)|✳

• ❡❳t❡♥❞✿ ✜rst ♠✉❧t✐♣❧② ❡❛❝❤ pi ♦❢ ❞❡❣r❡❡ di ❜② ❡✈❡r② ♠♦♥♦♠✐❛❧

xb := xb1

1 · · · xbn n ∈ T (D−di) t♦ ❣❡t ❡q✉❛t✐♦♥s R(D)✳

• ▲✐♥❡❛r✐③❡✿ t❤❡♥ r❡❞✉❝❡ R(D) ❛s ❛ ❧✐♥❡❛r s②st❡♠ ✐♥ ❛❧❧ t❤❡

xb ∈ T (D)✳ ❲❡ ♠❛② ❜❡ ❛❜❧❡ t♦ s♦❧✈❡ t❤❡ s②st❡♠ ♦r t♦ r❡❞✉❝❡ ❞♦✇♥ t♦ ❛ ✉♥✐✈❛r✐❛t❡ ❡q✉❛t✐♦♥ ✭s❛② ✐♥ x1✮✳ R := |R(D)| ❛♥❞ I ❝♦✉♥ts r❡s♣✳ ❡q✉❛t✐♦♥s ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ❡q✉❛t✐♦♥s ❛♠♦♥❣ R(D)✳

✷✹

❚♦② ❳▲ ❡①❛♠♣❧❡ ♦✈❡r GF(7)

p1 : x2+ 4y2+ z2+ 5xy+ 2xz+ 6yz+ 5x+ 3y+ 5z+ 1 = p2 : 3x2+ 2y2+ 3z2+ 4xy+ 6xz+ 2yz+ 6x+ 4y+ 3z+ 2 = p3 : 2x2+ 3y2+ 2z2+ 5xy+ 2yz+ 4x+ y+ z+ 4 = p4 : 6x2+ 3y2+ 3z2+ 5xz+ yz+ 5y+ 2z+ 2 =

❍❡r❡ n = 3✱ m = 4✱ ✇❡ ✇✐❧❧ ✉s❡ D = 3✱ ❛♥❞ ♠✉❧t✐♣❧② ❡✈❡r② ❡q✉❛t✐♦♥ ❜② 1, x, y, z t♦ ❣❡t 4

3

• = 20 ♠♦♥♦♠✐❛❧s ✭✐♥❝❧✉❞✐♥❣ ✶✮

❛♥❞ 4 × 4 = 16 ❡q✉❛t✐♦♥s✳

✷✺

❚❤❡ ❊①t❡♥❞❡❞ ▼❛❝❛✉❧❛② ▼❛tr✐①

x 2 y x 2 z y 2 x x y z z 2 x y 2 z z 2 y xy xz yz x 3 x 2 x y 3 y 2 y z 3 z 2 z 1

5 2 6 1 5 4 3 1 5 1 4 6 2 3 6 2 4 3 3 2 5 2 2 4 3 1 2 1 4 5 1 6 3 5 3 2 2 5 2 4 6 1 3 5 1 5 1 1 5 2 6 1 5 5 4 3 1 1 5 2 4 6 5 3 1 5 1 4 6 2 2 3 4 3 3 6 2 3 4 6 2 3 6 3 2 4 2 3 4 6 2 2 6 4 3 3 2 5 3 2 2 1 1 2 4 4 2 5 2 2 4 1 3 1 4 2 5 3 2 4 1 2 1 4 5 3 1 3 5 2 6 2 6 5 1 3 2 3 5 2 6 5 3 1 5 3 2 2 ✷✻

❚❤❡ ❘❡s✉❧t ♦❢ ❊❧✐♠✐♥❛t✐♦♥

x 2 y x 2 z y 2 x x y z z 2 x y 2 z z 2 y xy xz yz x 3 x 2 x y 3 y 2 y z 3 z 2 z 1

5 2 4 6 1 3 5 1 5 1 1 5 4 6 1 3 6 5 4 6 4 4 3 1 3 6 3 4 1 2 6 5 6 2 5 4 1 2 3 4 5 3 2 1 2 4 2 5 5 5 4 6 5 3 1 3 3 4 6 1 5 1 5 3 2 4 1 4 1 2 1 2 6 6 4 2 5 1 5 6 5 6 1 5 2 2 4 3 1 2 1 4 5 1 6 3 5 3 2 2 2 4 3 2 4 2 6 6 3 1 4 1 6 1 2 1 4 3 1 3 1 2 4 2 1 1 4 6 1 5 6 3 6 1 5 5 5 2 1 6 ✷✼

❖♣❡r❛t✐✈❡ ❈♦♥❞✐t✐♦♥ ❛♥❞ ❈♦st ♦❢ ❳▲

• ❳▲ s♦❧✈❡s ❛ s②st❡♠ ✐❢ T − I ≤ min(D, q − 1)✳
• ❖t❤❡r s✐t✉❛t✐♦♥s ✇❤❡r❡ ❳▲ ❛❧s♦ s✉❝❝❡❡❞s ❛r❡ ❝❛❧❧❡❞ ✏♣❛t❤♦❧♦❣✐❝❛❧

t❡r♠✐♥❛t✐♦♥s✑✳ ❬❖✉r ❡①❛♠♣❧❡ ❛❜♦✈❡ ✐s ♦♥❡✳❪

• ▲❡t E(N, M) := t❤❡ t✐♠❡ ❝♦♠♣❧❡①✐t② ♦❢ ❡❧✐♠✐♥❛t✐♦♥ ♦♥ N

✈❛r✐❛❜❧❡s ❛♥❞ M ❡q✉❛t✐♦♥s✱ t❤❡♥ ❳▲ t❛❦❡s t✐♠❡ C①❧ ≈ E(T, R)✳

• ❆s②♠♣t♦t✐❝❛❧❧② lg E(T, R) ∼ ω lg T✱ ✇❤❡r❡ ω ✐s ✏t❤❡ ♦r❞❡r ♦❢

♠❛tr✐① ♠✉❧t✐♣❧✐❝❛t✐♦♥✑✳ ❆♥ ♦❢t❡♥✲❝✐t❡❞ ♥✉♠❜❡r ✐s 2.376✳

✷✽