❆♥❛❧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 = ( x 1 , x 2 , . . . , x n ) ∈ K n , K = GF( q ) ❯♣❞❛t❡✿ x ← ( Q 1 ( x ) , Q 2 ( x ) , . . . , Q n ( x )) ✳ ❍❡r❡ ❡❛❝❤ Q j ✐s ❛ r❛♥❞♦♠❧② ❝❤♦s❡♥✱ ♣✉❜❧✐❝ q✉❛❞r❛t✐❝ ♣♦❧②♥♦♠✐❛❧ ❖✉t♣✉t✿ r ✲t✉♣❧❡ ( P 1 ( x ) , P 2 ( x ) , . . . , P r ( x )) ❜❡❢♦r❡ ✉♣❞❛t✐♥❣ ✭❛❣❛✐♥✱ ❡❛❝❤ P j ✐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✐♦♥ x 0 · · · x 1 = Q ( x 0 ) x 2 = Q ( x 1 ) x 3 = Q ( x 2 ) · · · y 0 = P ( x 0 ) y 1 = P ( x 1 ) y 2 = P ( x 2 ) y 3 = P ( x 3 ) ❚②♣✐❝❛❧❧② q ✐s ❛ ♣♦✇❡r ♦❢ 2 ✱ ❛❧❧♦✇✐♥❣ ❡❛❝❤ ♦✉t♣✉t ✈❡❝t♦r y i ∈ 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 2 an + 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✳ • F 4 ✭♥♦✇ ✐♥ ▼❆●▼❆✮ ❛♥❞ F 5 ✱ ♦❢ ✇❤✐❝❤ ❳▲✷ ✐s ❛♥ ✐♥❢❡r✐♦r ❢♦r♠✳ ✻
❋❛❝ts ♦❢ ▲✐❢❡ ❢♦r ❳▲ � (1 − t q ) n (1 − t ) − ( n +1) � [ t D ] ★ ♠♦♥♦♠✐❛❧s✿ T = ; ✭✶✮ � 1 − t d i � m �� (1 − t q ) n � [ t D ] ★ ❢r❡❡ ♠♦♥♦♠s✿ T − I ≥ . ✭✷✮ (1 − t ) n +1 1 − t qd i i =1 ❍❡r❡ deg p i := d i ✱ [ u ] s := ❝♦❡✣❝✐❡♥t ♦❢ u ✐♥ ❡①♣❛♥s✐♦♥ ♦❢ s ✳ ❲❡ ❡①♣❡❝t ❛ s♦❧✉t✐♦♥ ❛t D XL = min { D : ❘❍❙ ♦❢ ❊q✳ ✷ ≤ 0 } ✳ ■❢ t❤❡ ( p i ) ✐s q ✲s❡♠✐✲r❡❣✉❧❛r ✭tr✉❡ ❛❧♠♦st ❛❧✇❛②s✮✱ ❊q✳ ✷ ✐s = ❛s ❧♦♥❣ ❛s ✐ts ❘❍❙ r❡♠❛✐♥s ♣♦s✐t✐✈❡✳ (1 − t ) m − n − 1 (1 + t ) m � � n + D T − I = [ t D ] � � T = , D ✐s t❤❡ r❡❞✉❝❡❞ ❝❛s❡ ❢♦r ❧❛r❣❡ ✜❡❧❞s ✭ q > D ✮✳ C XL ≈ 3 kT 2 ( c 0 + c 1 lg T ) ✉s✐♥❣ ❛ ♠♦❞✐✜❡❞ ❲✐❡❞❡♠❛♥♥ ❛❧❣♦r✐t❤♠ ✭ k ✐s ❛✈❡r❛❣❡ ♥✉♠❜❡r ♦❢ t❡r♠s ♣❡r ❡q✉❛t✐♦♥✮✳ ✼
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