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Distribution of the Boolean Functions of N Variables According to Their Algebraic Degrees
Distribution of the Boolean Functions of N Variables According to Their Algebraic Degrees
Valentin Bakoev2019, Serdica Journal of Computing
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The probabilistic degree of a Boolean function f : {0, 1}n → {0, 1} is defined to be the smallest d such that there is a random polynomial P of degree at most d that agrees with f at each point with high probability. Introduced by Razborov (1987), upper and lower bounds on probabilistic degrees of Boolean functions — specifically symmetric Boolean functions — have been used to prove explicit lower bounds, design pseudorandom generators, and devise algorithms for combinatorial problems. In this paper, we characterize the probabilistic degrees of all symmetric Boolean functions up to polylogarithmic factors over all fields of fixed characteristic (positive or zero).
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An Upper Bound on the Number of Iterations for Transforming a Boolean Function of Degree Greater or Equal Than 4 to a Function of Degree 32001 •
The difficulty involved in characterizing the weight distribution of all Boolean functions of degree ≥ 3 is well-known [2, p. 446]. In [1] the author introduces a transformation on Boolean functions which changes their weights in a way that is easy to follow, and which, when iterated, reduces the degree of the function to 2 or 3. He concludes that it is just as difficult to characterize the weight of any function of degree 3 as it is for any other degree. The application of this transformation on a Boolean function defined on \(V_m = GF(2)^m ,m \in \mathbb{N}\) , increases the number of its variables by two. On the other hand, in order to reduce the degree of a function to 2 or 3 it is necessary to apply the tranformation a number of times that grows exponentially with respect to m. In this paper, a factorization method on Boolean functions that allows the establishment of an upper bound for the number of applications of the transformation is presented. It shows that, in general, it is possible to significantly decrease the number of iterations in this process of degree reduction.
2006 •
Algebraic attacks have received a lot of attention in studying security of symmetric ciphers. The function used in a symmetric cipher should have high algebraic immunity ( ${\cal AI}$ ) to resist algebraic attacks. In this paper we are interested in finding ${\cal AI}$ of Boolean power functions. We give an upper bound on the ${\cal AI}$ of any Boolean power function and a formula to find its corresponding low degree multiples. We prove that the upper bound on the ${\cal AI}$ for Boolean power functions with Inverse, Kasami and Niho exponents are $\lfloor \sqrt{n}\rfloor + \lceil \frac{n}{\lfloor \sqrt{n} \rfloor}\rceil -2$ , $\lfloor \sqrt{n} \rfloor + \lceil \frac{n}{\lfloor \sqrt{n} \rfloor}\rceil$ and $\lfloor \sqrt{n} \rfloor + \lceil \frac{n}{\lfloor \sqrt{n} \rfloor}\rceil$ respectively. We also generalize this idea to Boolean polynomial functions. All existing algorithms to determine ${\cal AI}$ and corresponding low degree multiples become too complex if the function has more than 25 variables. In our approach no algorithm is required. The ${\cal AI}$ and low degree multiples can be obtained directly from the given formula.
❙❡r❞✐❝❛
❙❡r❞✐❝❛ ❏✳ ❈♦♠♣✉t✐♥❣ ✶✸ ✭✷✵✶✾✮✱ ◆♦ ✶✕✷✱ ✶✼✕✷✻
❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t✐♥❣
❇✉❧❣❛r✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s
■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ■♥❢♦r♠❛t✐❝s
❉■❙❚❘■❇❯❚■❖◆ ❖❋ ❚❍❊ ❇❖❖▲❊❆◆ ❋❯◆❈❚■❖◆❙
❖❋ n ❱❆❘■❆❇▲❊❙ ❆❈❈❖❘❉■◆● ❚❖ ❚❍❊■❘ ❆▲●❊❇❘❆■❈
❉❊●❘❊❊❙
∗
❱❛❧❡♥t✐♥ ❇❛❦♦❡✈
❑♥♦✇❧❡❞❣❡ ❛❜♦✉t t❤❡ ❡♥✉♠❡r❛t✐♦♥ ❛♥❞ ❞✐str✐❜✉t✐♦♥ ♦❢ ❇♦♦❧❡❛♥
❢✉♥❝t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s ✐s ✐♠♣♦rt❛♥t ❢♦r t❤❡ t❤❡♦r②
❛s ✇❡❧❧ ❛s ❢♦r ✐ts ❛♣♣❧✐❝❛t✐♦♥s✳ ❆s ♦❢ ♥♦✇✱ t❤✐s ❦♥♦✇❧❡❞❣❡ ✐s ♥♦t ❝♦♠♣❧❡t❡✿ ❢♦r
❡①❛♠♣❧❡✱ ✐t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t ❤❛❧❢ ♦❢ ❛❧❧ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❤❛✈❡ ❛ ♠❛①✐♠❛❧
❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡✳ ■♥ t❤❡ ♣r❡s❡♥t ♣❛♣❡r✱ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ❛❧❧
❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♦❢ n ✈❛r✐❛❜❧❡s ❛♥❞ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡ = k ✐s ❞❡r✐✈❡❞✳ ❆ ❞✐r❡❝t
❝♦♥s❡q✉❡♥❝❡ ❢r♦♠ ✐t ✐s t❤❡ ❛ss❡rt✐♦♥ ✭❢♦r♠✉❧❛t❡❞ ❛❧r❡❛❞② ❜② ❈❧❛✉❞❡ ❈❛r❧❡t✮
t❤❛t ✇❤❡♥ n → ∞✱ ❛❧♠♦st ❛ ❤❛❧❢ ♦❢ ❛❧❧ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♦❢ n ✈❛r✐❛❜❧❡s ❤❛✈❡
❛♥ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡ = n − 1✳ ❚❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ❜② t❤✐s ❢♦r♠✉❧❛ ✇❡r❡ ✉s❡❞
✐♥ ❝r❡❛t✐♥❣ t❤❡ s❡q✉❡♥❝❡ ❆✸✶✾✺✶✶ ✐♥ t❤❡ ❖❊■❙✳ ❚❤❡ ❞✐s❝r❡t❡ ♣r♦❜❛❜✐❧✐t② ❛
r❛♥❞♦♠ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ t♦ ❤❛✈❡ ❛ ❝❡rt❛✐♥ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡ ✐s ❞❡✜♥❡❞ ❛♥❞ t❤❡
❝♦rr❡s♣♦♥❞✐♥❣ ❞✐str✐❜✉t✐♦♥ ✐s ❝♦♠♣✉t❡❞✱ ❢♦r 3 ≤ n ≤ 10✳ ❋♦✉r ❛♣♣❧✐❝❛t✐♦♥s
❛r❡ ❝♦♥s✐❞❡r❡❞✿ ❛t t❤❡ ❞❡s✐❣♥ ❛♥❞ ❛♥❛❧②s✐s ♦❢ ❡✣❝✐❡♥t ❛❧❣♦r✐t❤♠s ❢♦r ❡①❛❝t
❆❜str❛❝t✳
❆❈▼ ❈♦♠♣✉t✐♥❣ ❈❧❛ss✐✜❝❛t✐♦♥ ❙②st❡♠ (1998): ●✳✷✳✶✱ ●✳✸✱ ❊✳✸✳
❑❡② ✇♦r❞s✿ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥✱ ❝r②♣t♦❣r❛♣❤②✱ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡✱ ❡♥✉♠❡r❛t✐♦♥✱ ❞✐str✐❜✉t✐♦♥✳
✯
❚❤✐s ✇♦r❦ ✇❛s ♣❛rt✐❛❧❧② s✉♣♣♦rt❡❞ ❜② t❤❡ ❘❡s❡❛r❝❤ ❋✉♥❞ ♦❢ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❱❡❧✐❦♦ ❚❛r♥♦✈♦
✭❇✉❧❣❛r✐❛✮ ✉♥❞❡r ❝♦♥tr❛❝t ❋❙❉✲✸✶✲✸✹✵✲✶✹✴✷✻✳✵✸✳✷✵✶✾✳ ❙♦♠❡ ♦❢ t❤❡ r❡s✉❧ts ✇❡r❡ ❛♥♥♦✉♥❝❡❞ ❛t
t❤❡ ✽t❤ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❆❧❣❡❜r❛✐❝ ■♥❢♦r♠❛t✐❝s ✭❈❆■ ✷✵✶✾✮✱ ◆✐✟s✱ ❙❡r❜✐❛✳
❱❛❧❡♥t✐♥ ❇❛❦♦❡✈
✶✽
♦r ♣r♦❜❛❜✐❧✐st✐❝ ❝♦♠♣✉t✐♥❣ t❤❡ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡ ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s❀ ✇❤❡♥
❝❤❡❝❦✐♥❣ ❢♦✉r t❡st ✜❧❡s ❢♦r r❡♣r❡s❡♥t❛t✐✈❡♥❡ss❀ ✇❤❡♥ ❝r❡❛t✐♥❣ ❜❡♥❝❤♠❛r❦ ✜❧❡s
♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s✳
✶✳ ■♥tr♦❞✉❝t✐♦♥✳ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♣❧❛② ❛♥ ✐♠♣♦rt❛♥t r♦❧❡ ✐♥ ❝♦❞✐♥❣
t❤❡♦r②✱ ♠♦❞❡r♥ ❝r②♣t♦❣r❛♣❤②✱ ❞✐❣✐t❛❧ ❝✐r❝✉✐t t❤❡♦r②✱ ❡t❝✳ ❬✽✱ ✹✱ ✺✱ ✸❪✳ ❉✐✛❡r❡♥t t②♣❡s
♦❢ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥s ❛r❡ ✉s❡❞ ✐♥ t❤❡s❡ ❛r❡❛s✱ ❢♦r ❡①❛♠♣❧❡ ❜②✿ t❤❡ ✈❡❝t♦rs ♦❢
t❤❡✐r ❢✉♥❝t✐♦♥❛❧ ✈❛❧✉❡s ✭❝❛❧❧❡❞ ❚r✉t❤ ❚❛❜❧❡ ✭❚❚✮ ✈❡❝t♦rs✮✱ ❛❧❣❡❜r❛✐❝ ♥♦r♠❛❧ ❢♦r♠s
✭❆◆❋s✮✱ ♥✉♠❡r✐❝❛❧ ♥♦r♠❛❧ ❢♦r♠s✱ ❡t❝✳ ❚❤❡ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡ ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ✐s
❞❡✜♥❡❞ ❜② ✐ts ❆◆❋ ❛♥❞ ✐t ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ❝r②♣t♦❣r❛♣❤✐❝ ♣❛r❛♠❡t❡rs✳
❲❤❡♥ ✐t ✐s ❤✐❣❤❡r✱ t❤❡ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥✭s✮ ✐♥✈♦❧✈❡❞ ✐♥ t❤❡ ❞❡s✐❣♥ ♦❢ ❜❧♦❝❦ ❝✐♣❤❡rs✱
♣s❡✉❞♦✲r❛♥❞♦♠ ♥✉♠❜❡rs ❣❡♥❡r❛t♦rs ✐♥ str❡❛♠ ❝✐♣❤❡rs✱ ❤❛s❤ ❢✉♥❝t✐♦♥s✱ ❡t❝✳✱ ❛r❡
♠♦r❡ r❡s✐st❛♥t ❛❣❛✐♥st ❝r②♣t♦❣r❛♣❤✐❝ ❛tt❛❝❦s✳
❊♥✉♠❡r❛t✐♦♥ ❛♥❞ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s s❛t✐s❢②✐♥❣ ❞❡s✐r❡❞
❝r②♣t♦❣r❛♣❤✐❝ ♣❛r❛♠❡t❡rs t❛❦❡ ❛ ❣r❡❛t ♣❛rt ♦❢ t❤❡✐r r❡s❡❛r❝❤ ❬✼❪✳ ❑♥♦✇❧❡❞❣❡ ❛❜♦✉t
t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s ✐s ✐♠✲
♣♦rt❛♥t ❢♦r t❤❡ t❤❡♦r②✱ ❛s ✇❡❧❧ ❛s ❢♦r t❤❡ ❞❡s✐❣♥ ❛♥❞ ❛♥❛❧②s✐s ♦❢ ❡✣❝✐❡♥t ❛❧❣♦r✐t❤♠s
❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡ ✭❛s t❤♦s❡ ♣r♦♣♦s❡❞ ✐♥ ❬✻❪✮✱ ❢♦r ❣❡♥❡r❛t✐♥❣ t❡st
❡①❛♠♣❧❡s ❢♦r s✉❝❤ ❛❧❣♦r✐t❤♠s✱ ❡t❝✳
❆s ♦❢ ♥♦✇✱ t❤✐s ❦♥♦✇❧❡❞❣❡ ✐s ♣❛rt✐❛❧✖t❤❡r❡
❛r❡ s♦♠❡ t❤❡♦r❡t✐❝❛❧ r❡s✉❧ts ❛❜♦✉t t❤❡ ❡♥✉♠❡r❛t✐♦♥ ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♦❢
n
✈❛r✐❛❜❧❡s ❤❛✈✐♥❣ ❝❡rt❛✐♥ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s✳ ❆ ❧♦t ♦❢ t❤❡♠ ❛r❡ ❞❡r✐✈❡❞ ❜② ❡st❛❜✲
❧✐s❤✐♥❣ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ ❚❚ ✈❡❝t♦r✬s ✇❡✐❣❤t ❛♥❞ t❤❡ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡ ♦❢ t❤❡
❝♦rr❡s♣♦♥❞✐♥❣ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥✳ ■♥ ❣❡♥❡r❛❧✱ ✐t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t ❬✹✱ ✸✱ ✻❪✿
•
❆ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ♦❢
n ✈❛r✐❛❜❧❡s
❤❛s ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡
= n ✐❢ ❛♥❞
♦♥❧② ✐❢
✐ts ❚❚ ✈❡❝t♦r ❤❛s ❛♥ ♦❞❞ ✇❡✐❣❤t✳ ❍❡♥❝❡ ❤❛❧❢ ♦❢ ❛❧❧ s✉❝❤ ❢✉♥❝t✐♦♥s ❤❛✈❡ ❛♥
❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡
•
= n✳
❚❤❡ ♥✉♠❜❡r ♦❢ ❛✣♥❡ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♦❢
❞❡❣r❡❡ ❛t ♠♦st ✶✮ ✐s
2n+1 ✳
❝♦♥st❛♥t ❢✉♥❝t✐♦♥s✮ ❤❛✈❡ ✇❡✐❣❤t
2n−1 ✳
❈♦✉♥t✐♥❣ ♦❢ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡s ❛t ♠♦st
r
❘❡❡❞✲▼✉❧❧❡r ❝♦❞❡s ✐s ♦✉t❧✐♥❡❞ ✐♥ ❬✹✱ ♣✳ ✸✽❪✳
♥♦t❡s✿ ✏❲❤❡♥
n
n ✈❛r✐❛❜❧❡s ✭✐✳ ❡✳✱ ❤❛✈✐♥❣ ❛❧❣❡❜r❛✐❝
❚❤❡ ❚❚ ✈❡❝t♦rs ♦❢ ❛❧❧ s✉❝❤ ❢✉♥❝t✐♦♥s ✭❡①❝❡♣t ❜♦t❤
❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦❞❡✇♦r❞s ❢♦r
❋✉rt❤❡r✱ ♦♥ ♣✳ ✹✾✱ ❈❧❛✉❞❡ ❈❛r❧❡t
t❡♥❞s t♦ ✐♥✜♥✐t②✱ r❛♥❞♦♠ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❤❛✈❡ ❛❧♠♦st s✉r❡❧②
n − 1 s✐♥❝❡
t❤❡ ♥✉♠❜❡r ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♦❢ ❛❧❣❡❜r❛✐❝
Pn−2 n
2n −n−1
(
i=0
i) = 2
❛♥❞ ✐s ♥❡❣❧✐❣✐❜❧❡ ✇✐t❤ r❡s♣❡❝t
n − 2 ❡q✉❛❧s 2
❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s ❛t ❧❡❛st
❞❡❣r❡❡s ❛t ♠♦st
t♦ t❤❡ ♥✉♠❜❡r
22
n
♦❢ ❛❧❧ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s✳ ❇✉t ✇❡ s❤❛❧❧ s❡❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥s
♦❢ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s
n−1
♦r
n
❞♦ ♥♦t ❛❧❧♦✇ ❛❝❤✐❡✈✐♥❣ s♦♠❡ ♦t❤❡r ❝❤❛r❛❝t❡r✐st✐❝s
❉✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❇♦♦❧❡❛♥ ❋✉♥❝t✐♦♥s ♦❢
✭❜❛❧❛♥❝❡❞♥❡ss✱ r❡s✐❧✐❡♥❝②✱ ✳ ✳ ✳ ✮✑✳
n
✶✾
❱❛r✐❛❜❧❡s ✳ ✳ ✳
❚❤✐s ✐s ♦♥❡ ♠♦r❡ r❡❛s♦♥ t♦ ❡①♣❧♦r❡ t❤❡ ✇❤♦❧❡
❞✐str✐❜✉t✐♦♥ ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s✳
❚❤❡ ♦✉t❧✐♥❡ ♦❢ t❤❡ ♣❛♣❡r ✐s ❛s ❢♦❧❧♦✇s✳
❚❤❡ ❜❛s✐❝ ♥♦t✐♦♥s ❛r❡ ❣✐✈❡♥ ✐♥
❙❡❝t✐♦♥ ✷✳ ❙❡❝t✐♦♥ ✸ st❛rts ✇✐t❤ ❚❤❡♦r❡♠ ✶ ✇❤✐❝❤ ❣✐✈❡s ❛ ❢♦r♠✉❧❛ ❢♦r ❡♥✉♠❡r❛t✐♦♥
♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♦❢
n
✈❛r✐❛❜❧❡s ✐♥ ❛❝❝♦r❞❛♥❝❡ ✇✐t❤ t❤❡✐r ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s✳
❚❤✐s ❢♦r♠✉❧❛ ✇❛s ✉s❡❞ ✐♥ ❝r❡❛t✐♥❣ t❤❡ s❡q✉❡♥❝❡ ❆✸✶✾✺✶✶ ❬✾❪✳ ❚❤❡ ❝✐t❡❞ ❛ss❡rt✐♦♥
♦❢ ❈❛r❧❡t ❢♦❧❧♦✇s ❛s ❛ ❞✐r❡❝t ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ t❤❡♦r❡♠✳ ■t ✐s ❞❡r✐✈❡❞ ✐♥ ❛ ❞✐✛❡r❡♥t
✇❛② ✐♥ ❈♦r♦❧❧❛r② ✶✳ ■♥ ❙❡❝t✐♦♥ ✹✱ ❢♦✉r ❛♣♣❧✐❝❛t✐♦♥s ❛r❡ ❞✐s❝✉ss❡❞✿ ❛t t❤❡ ❞❡s✐❣♥ ❛♥❞
❛♥❛❧②s✐s ♦❢ ❡✣❝✐❡♥t ❛❧❣♦r✐t❤♠s ❢♦r ❡①❛❝t ♦r ♣r♦❜❛❜✐❧✐st✐❝ ❝♦♠♣✉t✐♥❣ t❤❡ ❛❧❣❡❜r❛✐❝
❞❡❣r❡❡ ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s❀ ✇❤❡♥ ❝❤❡❝❦✐♥❣ ❢♦✉r t❡st ✜❧❡s ✭✇✐t❤ s❛♠♣❧❡s ♦❢ ❇♦♦❧❡❛♥
❢✉♥❝t✐♦♥s✮ ❢♦r r❡♣r❡s❡♥t❛t✐✈❡♥❡ss❀ ✇❤❡♥ ❝r❡❛t✐♥❣ ♦❢ ❜❡♥❝❤♠❛r❦ ✜❧❡s ♦❢ ❇♦♦❧❡❛♥
❢✉♥❝t✐♦♥s✳
F2 = {0, 1} ❜❡ t❤❡ ✜❡❧❞ ♦❢ t✇♦ ❡❧❡♠❡♥ts ✇✐t❤ ❜♦t❤
x ⊕ y ✭s✉♠ ♠♦❞✉❧♦ ✷✱ ❳❖❘✮ ❛♥❞ x.y ✭♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❆◆❉✱ ❞❡♥♦t❡❞
n
s✐♠♣❧② ❜② xy ✮✱ ❢♦r x, y ∈ F2 ✳ F2 ✐s t❤❡ n✲❞✐♠❡♥s✐♦♥❛❧ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r F2 ✱
n
X
n
n
❝♦♥t❛✐♥✐♥❣ ❛❧❧ 2 ❜✐♥❛r② ✈❡❝t♦rs✳ ■❢ a = (a1 , a2 , . . . , an ) ∈ F2 ✱ t❤❡♥ ā =
ai .2n−i
✷✳ ❇❛s✐❝ ♥♦t✐♦♥s✳ ▲❡t
♦♣❡r❛t✐♦♥s✿
i=1
❞❡♥♦t❡s t❤❡ ♥❛t✉r❛❧ ♥✉♠❜❡r ❝♦rr❡s♣♦♥❞✐♥❣ t♦
a✳ ❆ ✭❍❛♠♠✐♥❣✮ ✇❡✐❣❤t ♦❢
n
X
wt(a) =
ai ✱ ✐✳ ❡✳✱ ✐t ✐s t❤❡ ♥✉♠❜❡r
t❤❡ ✈❡❝t♦r
❢✉♥❝t✐♦♥
i=1
♦❢
n
❞❡♥♦t❡❞ ❜②
❛♥❞
ā
✐s ❝❛❧❧❡❞ ❛
t❤❡ s❛♠❡ ✈❡❝t♦r
a
s❡r✐❛❧ ♥✉♠❜❡r
♦❢
✐s t❤❡ ♥❛t✉r❛❧ ♥✉♠❜❡r
♦❢ ♥♦♥✲③❡r♦ ❝♦♦r❞✐♥❛t❡s ♦❢
a✳
❆
❇♦♦❧❡❛♥
f : Fn2 → F2 ✳ ❙♦✱ ✐❢ x1 , x2 , . . . , xn ❞❡♥♦t❡
n
❜✐♥❛r② ✐♥♣✉t x = (x1 , x2 , . . . , xn ) ∈ F2 t♦ ❛ s✐♥❣❧❡
❚❤❡ s❡t ♦❢ ❛❧❧ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♦❢ n ✈❛r✐❛❜❧❡s ✐s
✈❛r✐❛❜❧❡s ✐s ❛ ♠❛♣♣✐♥❣
f ✱ ✐t ♠❛♣s ❛♥②
y = f (x) ∈ F2 ✳
t❤❡ ✈❛r✐❛❜❧❡s ♦❢
❜✐♥❛r② ♦✉t♣✉t
a
Bn ✳
❆♥② ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥
f ∈ Bn
❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ ❛♥ ✉♥✐q✉❡ ✇❛② ❜②
t❤❡ ✈❡❝t♦r ♦❢ ✐ts ❢✉♥❝t✐♦♥❛❧ ✈❛❧✉❡s✱ ❝❛❧❧❡❞ ❛
❚r✉t❤ ❚❛❜❧❡
✈❡❝t♦r✳ ■t ✐s ❞❡♥♦t❡❞ ❜②
T T (f ) = (f0 , f1 , . . . f
)✱ ✇❤❡r❡ fi = f (ai ) ❛♥❞ ai ✐s t❤❡ i✲t❤ ❧❡①✐❝♦❣r❛♣❤✐❝ ✈❡❝t♦r
n
n
n
♦❢ F2 ✱ ❢♦r i = 0, 1, . . . , 2 − 1✳ ❙✐♥❝❡ t❤❡r❡ ❛r❡ 2 ❜✐♥❛r② ♦✉t♣✉ts ❝♦rr❡s♣♦♥❞✐♥❣ t♦
n
2n
t❤❡ ✈❡❝t♦rs ♦❢ F2 ✱ ✐t ❢♦❧❧♦✇s t❤❛t |Bn | = 2 ✳
2n −1
❚❤❡
f ∈ Bn ✳
❆❧❣❡❜r❛✐❝ ◆♦r♠❛❧ ❋♦r♠
✭❆◆❋✮ ✐s ♦t❤❡r ✉♥✐q✉❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛♥②
■t ✐s ❛ ♠✉❧t✐✈❛r✐❛t❡ ♣♦❧②♥♦♠✐❛❧ ♦❢ t❤❡ ❢♦r♠
f (x1 , x2 , . . . , xn ) =
✭✶✮
M
aū xu .
u∈Fn
2
❍❡r❡
u = (u1 , u2 , . . . , un ) ∈ Fn2 ✱ aū ∈ {0, 1}✱
❛♥❞
xu
♠❡❛♥s t❤❡ ♠♦♥♦♠✐❛❧
✷✵
❱❛❧❡♥t✐♥ ❇❛❦♦❡✈
n
Y
✱ ✇❤❡r❡ x0i = 1 ❛♥❞ x1i = xi✱ ❢♦r i = 1, 2, . . . , n✳ ❚❤❡
i=1
❞❡❣r❡❡ ♦❢ t❤❡ ♠♦♥♦♠✐❛❧ xu ✐s ❡q✉❛❧ t♦ t❤❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ✐♥ t❤❡ ♣r♦❞✉❝t
❛♥❞ ✐t ✐s ❞❡♥♦t❡❞ ❜② deg(xu)✳ ❖❜✈✐♦✉s❧②✱ deg(xu) = wt(u)✳ ❚❤❡ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡
♦❢ t❤❡ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f ∈ Bn ✐s t❤❡ ❤✐❣❤❡st ❞❡❣r❡❡ ❛♠♦♥❣ ❛❧❧ ♠♦♥♦♠✐❛❧s ✐♥
t❤❡ ❆◆❋ ♦❢ f ✳ ■t ✐s ❞❡♥♦t❡❞ ❜② deg(f )✳ ❲❤❡♥ t❤❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥s ③❡r♦ ❛♥❞
♦♥❡ ❛r❡ ❝♦♥s✐❞❡r❡❞ ❛s ❢✉♥❝t✐♦♥s ♦❢ n ✈❛r✐❛❜❧❡s✱ t❤❡② ❛r❡ ❞❡♥♦t❡❞ ❜② 0̃n ❛♥❞ 1̃n✱
❝♦rr❡s♣♦♥❞✐♥❣❧②✳ ■❢ n = 0✱ t❤❡♥ 0̃0 = 0 ❛♥❞ 1̃0 = 1✱ ✐✳ ❡✳✱ t❤❡ ❇♦♦❧❡❛♥ ✈❛❧✉❡s 0
❛♥❞ 1 ❛r❡ ❝♦♥s✐❞❡r❡❞ ❛s ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ ♦❢ 0 ✈❛r✐❛❜❧❡s✳ ❯s✉❛❧❧②✱ t❤❡ ❞❡❣r❡❡ ♦❢ 0̃n
✐s ❞❡✜♥❡❞ ❛s deg(0̃n) = −∞✱ ❜✉t deg(1̃n) = 0✱ ✐♥❞❡♣❡♥❞❡♥t❧② ♦❢ t❤❡ ♥✉♠❜❡r ♦❢
t❤❡ ✈❛r✐❛❜❧❡s✳
❲❤❡♥ f ∈ Bn ✐s ❣✐✈❡♥ ❜② ✐ts T T (f )✱ ✐ts ❆◆❋ ❝❛♥ ❜❡ ❝♦♠✲
♣✉t❡❞ ❡✐t❤❡r ❜② ❛ ✇❡❧❧✲❦♥♦✇♥ tr❛♥s❢♦r♠❛t✐♦♥ ✭❛❧❣♦r✐t❤♠✮ ❞❡✜♥❡❞ ❜② ❛ s♣❡❝✐❛❧
tr❛♥s❢♦r♠❛t✐♦♥ ♠❛tr✐① ❬✶❪✱ ♦r ❜② ❛ ❢✉♥❝t✐♦♥ ❝❛❧❧❡❞ t❤❡ ❜✐♥❛r② ▼☎♦❜✐✉s tr❛♥s❢♦r♠
❬✸❪✱ ♦r ❜② ❛ s✐♠♣❧❡ ❞✐✈✐❞❡✲❛♥❞✲❝♦♥q✉❡r ❜✉tt❡r✢② ❛❧❣♦r✐t❤♠ ❬✹✱ ✼❪✱ ❡t❝✳ ❆❧❧ t❤❡s❡
tr❛♥s❢♦r♠❛t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✱ ❛♥❞ ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❛r❡❛ ♦❢ ❝♦♥s✐❞❡r❛t✐♦♥ t❤❡②
❛r❡ ❦♥♦✇♥ ❛s ❆◆❋ ❚r❛♥s❢♦r♠ ✭❆◆❋❚✮✱ ❢❛st ▼☎♦❜✐✉s ✭♦r ▼♦❡❜✐✉s✮ ❚r❛♥s❢♦r♠✱
❩❤❡❣❛❧❦✐♥ ❚r❛♥s❢♦r♠✱ P♦s✐t✐✈❡ P♦❧❛r✐t② ❘❡❡❞✲▼✉❧❧❡r ❚r❛♥s❢♦r♠✱ ❡t❝✳ ❬✶❪✳ ❋♦r ❛♥②
f ∈ Bn t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ❆◆❋ ♦❢ ✐t✖❛s ❩❤❡❣❛❧❦✐♥✬s ❢❛♠♦✉s t❤❡♦r❡♠ st❛t❡s✱
♦r ❛s ✐s s❤♦✇♥ ✐♥ ❬✸✱ ✹❪✳ ❙♦ t❤❡ ❆◆❋❚ ✐s ❛ ❜✐❥❡❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❢✉♥❝t✐♦♥s ✐♥ Bn
❛♥❞ t❤❡ s❡t ♦❢ t❤❡✐r ❆◆❋s✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ ❆◆❋❚ ❝♦✐♥❝✐❞❡s ✇✐t❤ ✐ts ✐♥✈❡rs❡
tr❛♥s❢♦r♠❛t✐♦♥ ❛♥❞ s♦ ✐t ✐s ❛♥ ✐♥✈♦❧✉t✐♦♥ ❬✶✱ ✸✱ ✹✱ ✼❪✳
xu1 1 xu2 2 . . . xunn =
xui i
❘❡♠❛r❦ ✶✳
✸✳ ❊♥✉♠❡r❛t✐♦♥ ❛♥❞ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s
▲❡t d(n, k) ❞❡♥♦t❡s t❤❡ ♥✉♠❜❡r ♦❢
❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♦❢ f ∈ Bn s✉❝❤ t❤❛t deg(f ) = k✳
❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s✳
❚❤❡♦r❡♠ ✶✳
✭✷✮
❋♦r ❛♥② ✐♥t❡❣❡rs n ≥ 0 ❛♥❞ 0 ≤ k ≤ n✱ t❤❡ ♥✉♠❜❡r
(
1,
✐❢ k = 0;
Pk−1 n
n
d(n, k) =
)
(
)
(
i=0
i ,
✐❢ 1 ≤ k ≤ n.
(2 k − 1).2
P r ♦ ♦ ❢✳
❛✮ ❚❤❡ ♣❛rt✐❝✉❧❛r ❝❛s❡ k = 0 ♠❡❛♥s t❤❛t f = 1̃n = 1✱ ✇❤❡r❡ ✶ ✐s ❝♦♥s✐❞❡r❡❞
❛s ❛ ✉♥✐q✉❡ ♠♦♥♦♠✐❛❧ t❤❛t ❝♦♥t❛✐♥s ♥♦ ✈❛r✐❛❜❧❡s✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ❛ss❡rt✐♦♥ ♦❢
t❤❡ t❤❡♦r❡♠ ✐s tr✉❡✳
❜✮ ▲❡t 1 ≤ k ≤ n ❛♥❞ X = {x1, x2, . . . , xn} ❜❡ ❛ s❡t ♦❢ ✈❛r✐❛❜❧❡s✳ ❚❤❡ s❡t
♦❢ ♠♦♥♦♠✐❛❧s ✐♥ ❛♥② ❆◆❋ ✭✐✳ ❡✳✱ ❢♦r♠✉❧❛ ♦❢ t❤❡ t②♣❡ ✭✶✮✮ ♦❢ n ✈❛r✐❛❜❧❡s ❝❛♥ ❜❡
❉✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❇♦♦❧❡❛♥ ❋✉♥❝t✐♦♥s ♦❢
n
✷✶
❱❛r✐❛❜❧❡s ✳ ✳ ✳
A ❛♥❞ B ✳ ❆ s✉❜s❡t ♦❢ t❤❡ t②♣❡ A
❝♦♥t❛✐♥s ❛❧❧ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ = k ❛♥❞ s♦ A 6= ∅✳ ❆ s✉❜s❡t ♦❢ t❤❡ t②♣❡ B
❝♦♥t❛✐♥s ❛❧❧ ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡s ❧❡ss t❤❛♥ k ❛♥❞ t❤❡ ❝❛s❡ B = ∅ ✐s ♣♦ss✐❜❧❡✳ ❲❡
s❤❛❧❧ ❡♥✉♠❡r❛t❡ ❛❧❧ ♣♦ss✐❜❧❡ s✉❜s❡ts ♦❢ t❤❡ t②♣❡ A ❛♥❞ B ✱ ❝♦rr❡s♣♦♥❞✐♥❣❧②✿
n
✶✳ ❚❤❡r❡ ❛r❡
✇❛②s t♦ ❝❤♦♦s❡ k ✈❛r✐❛❜❧❡s ❢r♦♠ X ❛♥❞ t♦ ❢♦r♠ ❛ ♠♦♥♦♠✐❛❧
k
♦❢ ❞❡❣r❡❡ = k ✳ ❙♦✱ ✐❢ M ✐s t❤❡ s❡t ♦❢ ❛❧❧ s✉❝❤ ♠♦♥♦♠✐❛❧s✱ t❤❡♥ |M | =
n
( n)
✳ ❚❤❡r❡ ❛r❡ 2 k − 1 ✇❛②s ❢♦r ❛t ❧❡❛st ♦♥❡ ♦❢ t❤❡♠ t♦ ❜❡ ❝❤♦s❡♥ ❛♥❞
k
t♦ ❢♦r♠ t❤❡ s✉❜s❡t A ✐♥ t❤❡ ❆◆❋✱ s✐♥❝❡ s♦ ♠❛♥② ❛r❡ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡
♣♦✇❡r s❡t ♦❢ M ✇✐t❤♦✉t t❤❡ ❡♠♣t② s❡t✳ ■♥ ❝♦♥❝❧✉s✐♦♥✱ t❤❡ ✜rst ♠✉❧t✐♣❧✐❡r ✐♥
♣❛rt✐t✐♦♥❡❞ ✐♥t♦ t✇♦ t②♣❡s s✉❜s❡ts ❞❡♥♦t❡❞ ❜②
❢♦r♠✉❧❛ ✭✷✮ r❡♣r❡s❡♥ts t❤❡ ♥✉♠❜❡r ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✭❛t ❧❡❛st
♦♥❡✮ ♠♦♥♦♠✐❛❧s ❤❛✈✐♥❣ ❞❡❣r❡❡
♣♦ss✐❜❧❡ s✉❜s❡ts ♦❢ t❤❡ t②♣❡
✷✳ ❆♥❛❧♦❣♦✉s❧②✱ ✇❡ ❤❛✈❡
n
i
=k
✐♥ t❤❡ ❆◆❋s✱ ✇❤✐❝❤ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❛❧❧
A✳
♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡
❚❤✉s t❤❡ s❡t ♦❢ ❛❧❧ ♦❢ t❤❡♠ ❝♦♥t❛✐♥s
❤❛s ❛ ❝❛r❞✐♥❛❧✐t② ♦❢
♦❢ t❤❡ t②♣❡
B✳
2
Pk−1
i=0
k−1
X
n
i=0
(ni)
i
<k
❢♦r
i = 0, 1, . . . , k − 1✳
♠♦♥♦♠✐❛❧s✳
B=∅
✇❤✐❝❤ ♠❡❛♥s t❤❛t ♥♦
✐s ❝❤♦s❡♥ ❛♥❞ s♦ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❆◆❋s ❝♦♥t❛✐♥
♦♥❧② ♠♦♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡
= k✳
■♥ ❝♦♥❝❧✉s✐♦♥✱ t❤❡ s❡❝♦♥❞ ♠✉❧t✐♣❧✐❡r ✐♥
❢♦r♠✉❧❛ ✭✷✮ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ s✉❜s❡ts ♦❢ t❤❡ t②♣❡
❋✐♥❛❧❧②✱ ❛♥② s✉❜s❡t ♦❢ t❤❡ t②♣❡
t❤❡ t②♣❡
B
■ts ♣♦✇❡r s❡t
✇❤✐❝❤ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ s✉❜s❡ts
❚❤✐s ♥✉♠❜❡r ✐♥❝❧✉❞❡s t❤❡ ❝❛s❡
♠♦♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡
= i✱
A
B✳
❝❛♥ ❜❡ ❝♦♠❜✐♥❡❞ ✇✐t❤ ❛♥② s✉❜s❡t ♦❢
✐♥ t❤❡ ❢♦r♠✉❧❛s ♦❢ t❤❡ t②♣❡ ✭✶✮✱ ✇❤✐❝❤ ✐s t❤❡ r❡❛s♦♥ ❢♦r ❛♣♣❧②✐♥❣ t❤❡
♠✉❧t✐♣❧✐❝❛t✐♦♥ r✉❧❡ ❜❡t✇❡❡♥ ❜♦t❤ t❡r♠s ✐♥ ❢♦r♠✉❧❛ ✭✷✮✳
❚❛❜❧❡ ✶ r❡♣r❡s❡♥ts t❤❡ ✈❛❧✉❡s ♦❢
0, 1, . . . , 5
❛♥❞
d(n, k)
♦❜t❛✐♥❡❞ ❜② ❢♦r♠✉❧❛ ✭✷✮✱ ❢♦r
n=
0 ≤ k ≤ n✳
n✱ ✇❡ ❝❛♥ ❞❡✜♥❡ t❤❡ r❡❧❛t✐♦♥
Bn ❛s ❢♦❧❧♦✇s✿ ❛r❜✐tr❛r② ❢✉♥❝t✐♦♥s f, g ∈ Bn
❜❡❧♦♥❣ t♦ t❤✐s r❡❧❛t✐♦♥ ✐✛ deg(f ) = deg(g)✳ ■t ✐s ❡❛s② t♦ ✈❡r✐❢② t❤❛t ✐t ✐s ❛♥
❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳ ❚❤❡r❡❢♦r❡ t❤✐s r❡❧❛t✐♦♥ ♣❛rt✐t✐♦♥s t❤❡ s❡t Bn ✐♥t♦ n + 2
❘❡♠❛r❦ ✷✳ ❋♦r ❛♥② ♣♦s✐t✐✈❡ ♥❛t✉r❛❧ ♥✉♠❜❡r
✏ ❡q✉❛❧
❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s ✑
♦✈❡r t❤❡ s❡t
❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s✱ ❡❛❝❤ ♦❢ ✇❤✐❝❤ ❝♦♥t❛✐♥s ❛❧❧ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♦❢ ❡q✉❛❧ ❞❡❣r❡❡s✳
❚❤❡ ✜rst ♦❢ t❤❡♠ ❝♦♥t❛✐♥s ♦♥❧② t❤❡ ❝♦♥st❛♥t ③❡r♦ ❢✉♥❝t✐♦♥✱ ❛♥❞ t❤❡ ❝❛r❞✐♥❛❧✐t✐❡s
♦❢ t❤❡ r❡♠❛✐♥✐♥❣ ❝❧❛ss❡s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② ❚❤❡♦r❡♠ ✶✳
❝❛r❞✐♥❛❧✐t✐❡s ❢♦r
0 ≤ n ≤ 5✳
❚❛❜❧❡ ✶ s❤♦✇s t❤❡s❡
❱❛❧❡♥t✐♥ ❇❛❦♦❡✈
✷✷
❚❛❜❧❡ ✶✳ ❚❤❡ ✈❛❧✉❡s ♦❢
d(n, k)✱
❢♦r
n = 0, 1, . . . , 5
d(n, k)✱
k=3
❚❤❡ ✈❛❧✉❡s ♦❢
k=1
k=2
❛♥❞
0≤k≤n
❢♦r✿
n=
k=0
✵
✶
k=4
✶
✶
✷
✶
✻
✽
✸
✶
✶✹
✶✶✷
✹
✶
✸✵
✷✵✶✻
✸✵✼✷✵
✸✷✼✻✽
✺
✶
✻✷
✻✺✹✼✷
✻✼✵✹✸✸✷✽
✷✵✽✵✸✼✹✼✽✹
k=5
✷
✶✷✽
✷✶✹✼✹✽✸✻✹✽
▼♦r❡ ❝♦♠♠❡♥ts✱ r❡❧❛t✐♦♥s ❛♥❞ r❡s✉❧ts ❛❜♦✉t t❤❡ ♥✉♠❜❡rs d(n, k) ✭❢♦r
n ≤ 10✮ ❝❛♥ ❜❡ s❡❡♥ ✐♥ ❬✾❪✱ s❡q✉❡♥❝❡ ❆✸✶✾✺✶✶✳ ❚❤❡② s✉❣❣❡st t❤❡ ❢♦❧❧♦✇✐♥❣ ❛s✲
s❡rt✐♦♥✳
❈♦r♦❧❧❛r② ✶✳ ❚❤❡ ♥✉♠❜❡r
d(n, n − 1)
t❡♥❞s t♦
1
· |Bn |
2
✇❤❡♥
n → ∞✳
P r ♦ ♦ ❢✳ ❆♣♣❧②✐♥❣ ❢♦r♠✉❧❛ ✭✷✮ ❢♦r k = n − 1✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❧✐♠✐t ♦❢ t❤❡ r❛t✐♦
❜❡t✇❡❡♥ d(n, n − 1) ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❛❧❧ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ✐♥ Bn ❛s ❢♦❧❧♦✇s✿
P
n
n−2 n
n
d(n, n − 1)
(2(n−1) − 1).2 i=0 ( i )
(2n − 1).22 −n−1
lim
= lim
= lim
=
n→∞
n→∞
n→∞
|Bn |
22n
22 n
n
n
1
1
1
22 −1 − 22 −n−1
= lim
− n+1 = .
lim
n
2
n→∞ 2
n→∞
2
2
2
❖❜✈✐♦✉s❧②✱ ✇❤❡♥ n ❣r♦✇s ❛♥❞ k ❜❡❝♦♠❡s ❝❧♦s❡ t♦ n✱ t❤❡ ✈❛❧✉❡s ♦❢ d(n, k)
❣r♦✇ ❡①tr❡♠❡❧② ❢❛st✳ ■t ✐s ❝♦♥✈❡♥✐❡♥t t♦ ❞❡✜♥❡ ❛♥❞ ✉s❡ t❤❡ ❞✐s❝r❡t❡ ♣r♦❜❛❜✐❧✐t②
p(n, k) ❛ r❛♥❞♦♠ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥ f ∈ Bn t♦ ❤❛✈❡ ❛♥ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡ = k ✱
p(n, k) =
d(n, k)
d(n, k)
,
=
|Bn |
22n
❢♦r n ≥ 0 ❛♥❞ 0 ≤ k ≤ n✳ ❚❤❡ ✈❛❧✉❡s ♦❢ p(n, k) ♦❜t❛✐♥❡❞ ❢♦r ❛ ✜①❡❞ n ❣✐✈❡ t❤❡
❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥s ❢r♦♠ Bn ✐♥ ❛❝❝♦r❞❛♥❝❡ ✇✐t❤ t❤❡✐r ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s✳
❚❛❜❧❡ ✷ r❡♣r❡s❡♥ts ♣❛rt✐❛❧❧②✶ t❤✐s ❞✐str✐❜✉t✐♦♥✱ ❢♦r 3 ≤ n ≤ 10 ❛♥❞ n − 3 ≤ k ≤ n✳
❚❤❡ ✈❛❧✉❡s ♦❢ p(n, k) ✐♥ ✐t ❛r❡ r♦✉♥❞❡❞ ✉♣ t♦ ✶✵ ❞✐❣✐ts ❛❢t❡r t❤❡ ❞❡❝✐♠❛❧ ♣♦✐♥t✳
t✐♦♥s✿
✶
✹✳ ❆♣♣❧✐❝❛t✐♦♥s✳
❚❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ❤❡r❡ ❤❛✈❡ s♦♠❡ ✉s❡❢✉❧ ❛♣♣❧✐❝❛✲
❇❡❝❛✉s❡ t❤❡ r❡♠❛✐♥✐♥❣ ✈❛❧✉❡s ❛r❡ r♦✉♥❞❡❞ t♦ ③❡r♦✳
❉✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❇♦♦❧❡❛♥ ❋✉♥❝t✐♦♥s ♦❢
n
✷✸
❱❛r✐❛❜❧❡s ✳ ✳ ✳
Bn ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s✱
n = 3, 4, . . . , 10
❚❛❜❧❡ ✷✳ P❛rt✐❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❢r♦♠
❢♦r
❚❤❡ ✈❛❧✉❡s ♦❢
k =n−3
n
k =n−2
p(n, k)✱ ❢♦r✿
k =n−1
k=n
✸
✵✳✵✵✸✾✵✻✷✺
✵✳✵✺✹✻✽✼✺
✵✳✹✸✼✺
✵✳✺
✹
✵✳✵✵✵✹✺✼✼✻✸✼
✵✳✵✸✵✼✻✶✼✶✽✼
✵✳✹✻✽✼✺
✵✳✺
✺
✵✳✵✵✵✵✶✺✷✹✸✾
✵✳✵✶✺✻✵✾✼✹✶✷
✵✳✹✽✹✸✼✺
✵✳✺
✻
✵✳✵✵✵✵✵✵✷✸✽✹
✵✳✵✵✼✽✶✷✷✻✶✻
✵✳✹✾✷✶✽✼✺
✵✳✺
✼
✵✳✵✵✵✵✵✵✵✵✶✾
✵✳✵✵✸✾✵✻✷✹✽✶
✵✳✹✾✻✵✾✸✼✺
✵✳✺
✽
✵
✵✳✵✵✶✾✺✸✶✷✺✵
✵✳✹✾✽✵✹✻✽✼✺
✵✳✺
✾
✵
✵✳✵✵✵✾✼✻✺✻✷✺
✵✳✹✾✾✵✷✸✹✸✼✺
✵✳✺
✶✵
✵
✵✳✵✵✵✹✽✽✷✽✶✷
✵✳✹✾✾✺✶✶✼✶✽✼
✵✳✺
✶✳ ❚❤❡ ❦♥♦✇❧❡❞❣❡ ❛❜♦✉t t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❢r♦♠
Bn
❛❝❝♦r❞✐♥❣ t♦
t❤❡✐r ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s ✐s ✈❡r② ✉s❡❢✉❧ ✐♥ t❤❡ ❞❡s✐❣♥ ❛♥❞ ❛♥❛❧②s✐s ♦❢ ❛❧❣♦r✐t❤♠s
❢♦r ❡✣❝✐❡♥t ❝♦♠♣✉t✐♥❣ t❤❡ ❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡ ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s✳ ❙✉❝❤ ❛❧❣♦✲
r✐t❤♠s ❛r❡ ❝♦♥s✐❞❡r❡❞ ✐♥ ❬✻✱ ✷❪✳ ❚❤❡ ❝♦♥t✐♥✉❛t✐♦♥s ♦✉t❧✐♥❡❞ ✐♥ ❬✷❪ ✭♣r❡s❡♥t❡❞
❛t t❤❡ ❝♦♥❢❡r❡♥❝❡ ❈❆■ ✷✵✶✾✱ ❜✉t st✐❧❧ ✉♥♣✉❜❧✐s❤❡❞✮ ❞❡♠♦♥str❛t❡ t❤❡ ❜❡♥❡✜t
♦❢ t❤✐s ❦♥♦✇❧❡❞❣❡✳
♣r♦❜❛❜✐❧✐st✐❝ ❛❧❣♦r✐t❤♠ ❢♦r ❢❛st ❝♦♠♣✉t✐♥❣ t❤❡ ❛❧❣❡✲
f ∈ Bn ✱ ❣✐✈❡♥ ❜② ✐ts T T (f )✳ ■t ❝❛♥ ❜❡ ❞❡s❝r✐❜❡❞ ✐♥ s❤♦rt
t❤❡ ✇❡✐❣❤t ♦❢ T T (f )✳ ■❢ ✐t ✐s ❛♥ ♦❞❞ ♥✉♠❜❡r✱ r❡t✉r♥ n✱ ❡❧s❡✱
✷✳ ❆ ❢❛st ❛♥❞ ✈❡r② s✐♠♣❧❡
❜r❛✐❝ ❞❡❣r❡❡ ♦❢
❛s✿ ✏❈♦♠♣✉t❡
n − 1✳✑ ❋r♦♠ ❚❤❡♦r❡♠ ✶✱ ❈♦r♦❧❧❛r② ✶ ❛♥❞ ❚❛❜❧❡ ✷✱ ✐t ❢♦❧❧♦✇s t❤❛t ❢♦r
f ∈ Bn t❤✐s ❛❧❣♦r✐t❤♠ ✇✐❧❧ r❡t✉r♥ ❛ ❝♦rr❡❝t ♦✉t♣✉t ✐♥ ≈ 100% ♦❢
❛❧❧ s✉❝❤ ❢✉♥❝t✐♦♥s✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ✏ t❤❡ ❛❧❣♦r✐t❤♠ r❡t✉r♥s ❛
❝♦rr❡❝t ♦✉t♣✉t ✑ t❡♥❞s t♦ ✶ ✇❤❡♥ n → ∞✳
r❡t✉r♥
❛r❜✐tr❛r②
■♥ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥s ♦❢ ❬✷❪✱ ✇❡ s❤♦✇❡❞ t❤❛t ✐❢ t❤❡
T T (f )
❤❛s ❛ ❜②t❡✲✇✐s❡
T T (f ) ✇✐❧❧ ❤❛✈❡
T T (f ) ❤❛s ❛ ❜✐t✇✐s❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐♥
64 = 26 ✲❜✐t ❝♦♠♣✉t❡r ✇♦r❞s✱ ✐t ♦❝❝✉♣✐❡s s = 2n−6 s✉❝❤ ✇♦r❞s✳ ❚❤✉s✱ ❜②
16
✉s✐♥❣ ❛ ❧♦♦❦✲✉♣ t❛❜❧❡ ✕ ❛♥ ❛rr❛② a ♦❢ s✐③❡ 2
❡❧❡♠❡♥ts ✇❤✐❝❤ ❛r❡ ♣r❡❝♦♠♣✉t❡❞
16
✇❡✐❣❤ts ♦❢ ✐♥t❡❣❡rs ✭✐✳ ❡✳✱ a[i] = weight(i)✱ ❢♦r i = 0, 1, . . . , 2
− 1✮ ✕ t❤❡
n−4
✇❡✐❣❤t ♦❢ T T (f ) ✇✐❧❧ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ Θ(4.s) = Θ(2
) st❡♣s✳ ❇✉t ✇❡
r❡❛❧✐③❡❞ t❤❛t ✇❡ ♥❡❡❞ t❤❡ ♣❛r✐t② ❝❤❡❝❦ ♦❢ T T (f ) ✐♥st❡❛❞ ♦❢ ✐ts ✇❡✐❣❤t✳ ❚❤❡
♣❛r✐t② ❝❤❡❝❦ ♦❢ T T (f ) ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ s✐❣♥✐✜❝❛♥t❧② ♠♦r❡ ❡✣❝✐❡♥t❧②✖✐♥
Θ(s − 1 + 6) = Θ(2n−6 ) st❡♣s✳ ❚❤❡ ❧❛st ❝♦♠♠❡♥ts s❤♦✇ ❤♦✇ ❢❛st ❝❛♥ ❜❡ t❤❡
r❡♣r❡s❡♥t❛t✐♦♥✱ ❛♥ ❛❧❣♦r✐t❤♠ ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ ✇❡✐❣❤t ♦❢
❛ t✐♠❡ ❝♦♠♣❧❡①✐t②
Θ(2n )✳
❲❤❡♥ t❤❡
♣r♦❜❛❜✐❧✐st✐❝ ❛❧❣♦r✐t❤♠ ❛♥❞ ✇❤❛t ✐♠♣r♦✈❡♠❡♥t t♦ ❡①♣❡❝t ✐♥ t❤❡ ❡✣❝✐❡♥❝②
♦❢ ❡①❛❝t ❛❧❣♦r✐t❤♠s ✇❤❡♥ t❤❡② ✉s❡ ❛ ❜✐t✇✐s❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢
T T (f )
❛♥❞
❱❛❧❡♥t✐♥ ❇❛❦♦❡✈
✷✹
❜✐t✇✐s❡ ♦♣❡r❛t✐♦♥s✳
✸✳ ■t ✐s ❦♥♦✇♥ t❤❛t ✏❲✐t❤ t♦❞❛②✬s ❝♦♠♣✉t❛t✐♦♥ ♣♦✇❡r✱ ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❝♦♥✲
❞✉❝t ❛♥ ❡①❤❛✉st✐✈❡ s❡❛r❝❤ ♦♥ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♦❢ ✻ ✈❛r✐❛❜❧❡s ♦r ♠♦r❡✑ ❬✼❪✳
❚❤❛t ✐s ✇❤② t❤❡ ❛❧❣♦r✐t❤♠s ❢♦r ✐♥✈❡st✐❣❛t✐♦♥ ♦❢ s✉❝❤ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ✉s❡
s❛♠♣❧❡s ♦❢ t❤❡♠ ✇❤✐❝❤ s❤♦✉❧❞ ❜❡ r❡♣r❡s❡♥t❛t✐✈❡✳ ❚❤❡ r❡s✉❧ts ♦❜t❛✐♥❡❞ ❤❡r❡
✇❡r❡ ✉s❡❞ t♦ ❝❤❡❝❦ ❢♦✉r ✜❧❡s ❢♦r r❡♣r❡s❡♥t❛t✐✈❡♥❡ss✳ ❚❤❡s❡ ✜❧❡s ❝♦♥t❛✐♥
7
10
✱
8
10
❛♥❞
9
10
106 ✱
r❛♥❞♦♠❧② ❣❡♥❡r❛t❡❞ ✉♥s✐❣♥❡❞ ✐♥t❡❣❡rs ✐♥ ✻✹✲❜✐ts ❝♦♠♣✉t❡r
✇♦r❞s✳ ❚❤❡s❡ ✐♥t❡❣❡rs ✇❡r❡ ♦❜t❛✐♥❡❞ ❜② ✉s✐♥❣ t❤❡ ✉s✉❛❧
r❛♥❞ ✭✮ ❢✉♥❝t✐♦♥ ✐♥
❈✴❈✰✰ ♣r♦❣r❛♠♠✐♥❣ ❧❛♥❣✉❛❣❡✱ ✇✐t❤♦✉t ❛♥② ❛❞❞✐t✐♦♥❛❧ ❝❤❡❝❦s ❛♥❞ ❝♦♥❞✐✲
t✐♦♥s✳ ❚❤❡ ✜❧❡s ✇❡r❡ ✉s❡❞ ❛s t❡st ❡①❛♠♣❧❡s ✐♥ ❬✶✱ ✷❪✳ ❲❤❡♥ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s
n ≥ 6 ✈❛r✐❛❜❧❡s ❛r❡ ✉s❡❞✱ 2n−6 ✐♥t❡❣❡rs ❛r❡ r❡❛❞ ❢r♦♠ t❤❡ s❡❧❡❝t❡❞ ✜❧❡ ❛♥❞
s♦ t❤❡② ❢♦r♠ t❤❡ T T (f ) ✭✐♥ ❛ ❜✐t✇✐s❡ r❡♣r❡s❡♥t❛t✐♦♥✮ ♦❢ t❤❡ s❡r✐❛❧ ❇♦♦❧❡❛♥
♦❢
❢✉♥❝t✐♦♥✳
❲❡ ✉s❡❞ ❡❛❝❤ ♦❢ t❤❡s❡ ✜❧❡s ❛s ❛♥ ✐♥♣✉t ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♦❢
n = 6, 8, 10, 12, 14, 16
✈❛r✐❛❜❧❡s✳ ❲❡ ❝♦♠♣✉t❡❞ t❤❡✐r ❆◆❋s✱ t❤❡r❡❛❢t❡r t❤❡
❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s ♦❢ ❡❛❝❤ ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s✱ ❛♥❞ ✇❡ ❡♥✉♠❡r❛t❡❞ ❛❧❧ ❢✉♥❝t✐♦♥s
♦❢ ❡q✉❛❧ ❞❡❣r❡❡s✳ ❋✐♥❛❧❧②✱ ✇❡ ❝♦♠♣✉t❡ t❤❡ ❞❡✈✐❛t✐♦♥s ✕ t❤❡ ❛❜s♦❧✉t❡ ✈❛❧✉❡s
♦❢ t❤❡ ❞✐✛❡r❡♥❝❡s ❜❡t✇❡❡♥ t❤❡ t❤❡♦r❡t✐❝❛❧ ❛♥❞ ❝♦♠♣✉t❡❞ ❞✐str✐❜✉t✐♦♥s✳ ❚❤❡
♦❜t❛✐♥❡❞ r❡s✉❧ts ❛r❡ ❣✐✈❡♥ ✐♥ ❚❛❜❧❡ ✸✳
❚❛❜❧❡ ✸✳ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts ❛❜♦✉t t❤❡ t❡st ✜❧❡s r❡♣r❡s❡♥t❛t✐✈❡♥❡ss✖
t❤❡ ♠❛①✐♠❛❧ ❞❡✈✐❛t✐♦♥s ✐♥ ♣❡r❝❡♥ts
◆✉♠❜❡r ♦❢
▼❛①✐♠❛❧ ❞❡✈✐❛t✐♦♥s ✐♥ ✪ ❢♦r ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ♦❢✿
✐♥t❡❣❡rs✿ ✻ ✈❛rs
✽ ✈❛rs
✶✵ ✈❛rs
✶✷ ✈❛rs ✶✹ ✈❛rs ✶✻ ✈❛rs
106
✵✳✻✹✼✼✺ ✵✳✵✶✽
✵✳✶✺✵✹
✵✳✵✾✷✷ ✵✳✽✹✼✾ ✶✳✷✸✵✸
7
10
✵✳✻✷✽✺✸ ✵✳✵✹✸✾✸ ✵✳✶✵✹✷✾ ✵✳✶✼✷✶✻ ✵✳✷✾✾✺✷ ✵✳✶✵✼✺✸
108
✵✳✻✷✸✵✻✺ ✵✳✵✵✼✻✵✹ ✵✳✵✷✼✾✼✷ ✵✳✵✸✶✾✸✻ ✵✳✵✵✸✷✷✷ ✵✳✷✻✶✽✽✹
109
✵✳✻✷✸✷✸✷ ✵✳✵✵✼✾✾✻✼ ✵✳✵✵✾✾✼✹✹ ✵✳✵✶✹✺✶✶ ✵✳✵✶✼✹✸ ✵✳✵✾✷✾✷✸
❲❡ ❤❛✈❡ t♦ ♥♦t❡ t❤❛t t❤❡ t✇♦ t❡st ✜❧❡s ♦❢ s♠❛❧❧❡r s✐③❡s ❤❛✈❡ ❜❡❡♥ ✉s❡❞ ✐♥
t❤❡ ❞❡✈❡❧♦♣♠❡♥t ❛♥❞ ❞❡❜✉❣❣✐♥❣ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠s ✐♥ ❬✶✱ ✷❪✱ ✇❤❡r❡❛s t❤❡ t✇♦
t❡st ✜❧❡s ♦❢ ❧❛r❣❡r s✐③❡s ❤❛✈❡ ❜❡❡♥ ✉s❡❞ ❢♦r t❤❡ tr✉❡ t❡sts ❛♥❞ r❡s✉❧ts✳ ❙♦ ✇❡
❝♦♥s✐❞❡r t❤❛t t❤❡ ❛❧❣♦r✐t❤♠s ✇♦r❦ ✇✐t❤ s❛♠♣❧❡s ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ✇❤✐❝❤
❛r❡ r❡♣r❡s❡♥t❛t✐✈❡ ❡♥♦✉❣❤ ❛♥❞ t❤❡✐r r❡s✉❧ts ❛r❡ r❡❛❧✳
✹✳ ❚❤❡ r❡❝❡♥t r❡s✉❧ts ❝❛♥ ❜❡ ✉s❡❞ ✐♥ ❝r❡❛t✐♥❣ ❜❡♥❝❤♠❛r❦s ✜❧❡s ❝♦♥t❛✐♥✐♥❣ s❛♠✲
♣❧❡s ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ✇❤♦s❡ ❞✐str✐❜✉t✐♦♥ ✭❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❛❧❣❡❜r❛✐❝
❞❡❣r❡❡s✮ ✇✐❧❧ ❜❡ ❛s ❝❧♦s❡ ❛s ♥❡❝❡ss❛r② t♦ t❤❡ t❤❡♦r❡t✐❝❛❧ ❞✐str✐❜✉t✐♦♥✳ ❚❤✉s
✇❡ ❝❛♥ ♦❜t❛✐♥ s✐❣♥✐✜❝❛♥t❧② ♠♦r❡ r❡♣r❡s❡♥t❛t✐✈❡ s❛♠♣❧❡s t❤❛♥ t❤♦s❡ ✐♥ t❤❡
t❡st ✜❧❡s ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✳
❉✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❇♦♦❧❡❛♥ ❋✉♥❝t✐♦♥s ♦❢
n
❱❛r✐❛❜❧❡s ✳ ✳ ✳
✷✺
❚❤❡ ♠❛✐♥ ❣♦❛❧ ♦❢ t❤✐s ♣❛♣❡r ✐s t♦ ❝♦♥tr✐❜✉t❡ ❦♥♦✇❧❡❞❣❡
❛❜♦✉t t❤❡ ❡♥✉♠❡r❛t✐♦♥ ❛♥❞ ❞✐str✐❜✉t✐♦♥ ♦❢ ❇♦♦❧❡❛♥ ❢✉♥❝t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r
❛❧❣❡❜r❛✐❝ ❞❡❣r❡❡s✱ ❛s ✇❡❧❧ ❛s t♦ ♠❛❦❡ t❤❡ ❦♥♦✇❧❡❞❣❡ ♦❢ t❤✐s ♠❛tt❡r ♠♦r❡ ♣♦♣✉❧❛r✳
❲❡ ❞❡♠♦♥str❛t❡❞ ❢♦✉r ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ✐t ❛s ✐❧❧✉str❛t✐♦♥s ♦❢ ✐ts ✉s❡✳ ❲❡ ❤♦♣❡ t❤✐s
❦♥♦✇❧❡❞❣❡ ✇✐❧❧ ❛❧s♦ ✜♥❞ ♦t❤❡r ❛♣♣❧✐❝❛t✐♦♥s✳
✺✳ ❈♦♥❝❧✉s✐♦♥s✳
❘❊❋❊❘❊◆❈❊❙
❬✶❪ ❇❛❦♦❡✈ ❱✳ ❋❛st ❇✐t✇✐s❡ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❆❧❣❡❜r❛✐❝ ◆♦r♠❛❧ ❋♦r♠
❚r❛♥s❢♦r♠✳ ❙❡r❞✐❝❛ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t✐♥❣✱ ✶✶ ✭✷✵✶✼✮✱ ◆♦ ✶✱ ✹✺✕✺✼✳
❬✷❪ ❇❛❦♦❡✈ ❱✳ ❋❛st ❈♦♠♣✉t✐♥❣ t❤❡ ❆❧❣❡❜r❛✐❝ ❉❡❣r❡❡ ♦❢ ❇♦♦❧❡❛♥ ❋✉♥❝t✐♦♥s✳
✁ ❝✱ ▼✳ ❉r♦st❡✱ ❏✲❊✳
✁ P✐♥ ✭❡❞s✮✳ ❆❧❣❡❜r❛✐❝ ■♥❢♦r♠❛t✐❝s✳ ✽t❤ ■♥t❡r♥❛✲
■♥✿ ▼✳ ❈✐r✐✁
t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡✱ ❈❆■ ✷✵✶✾✱ ◆✐✞s✱ ❙❡r❜✐❛✳ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱
✶✶✺✹✺ ✭✷✵✶✾✮✱ ✺✵✕✻✸✳
❬✸❪ ❈❛♥t❡❛✉t ❆✳ ▲❡❝t✉r❡ ♥♦t❡s ♦♥ ❈r②♣t♦❣r❛♣❤✐❝ ❇♦♦❧❡❛♥ ❋✉♥❝t✐♦♥s✳ ■♥r✐❛✱
P❛r✐s✱ ❋r❛♥❝❡✱ ✷✵✶✻✳
❬✹❪ ❈❛r❧❡t ❈✳ ❇♦♦❧❡❛♥ ❋✉♥❝t✐♦♥s ❢♦r ❈r②♣t♦❣r❛♣❤② ❛♥❞ ❊rr♦r ❈♦rr❡❝t✐♥❣
❈♦❞❡s✳ ■♥✿ ❨✳ ❈r❛♠❛ ❛♥❞ P✳ ▲✳ ❍❛♠♠❡r ✭❡❞s✮✳ ❇♦♦❧❡❛♥ ▼♦❞❡❧s ❛♥❞ ▼❡t❤✲
♦❞s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ❛♥❞ ❊♥❣✐♥❡❡r✐♥❣✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈✳
Pr❡ss✱ ✷✵✶✵✱ ✷✺✼✕✸✾✼✳
❬✺❪ ❈❛r❧❡t ❈✳ ❱❡❝t♦r✐❛❧ ❇♦♦❧❡❛♥ ❋✉♥❝t✐♦♥s ❢♦r ❈r②♣t♦❣r❛♣❤②✳ ■♥✿ ❨✳ ❈r❛♠❛
❛♥❞ P✳ ▲✳ ❍❛♠♠❡r ✭❡❞s✮✳ ❇♦♦❧❡❛♥ ▼♦❞❡❧s ❛♥❞ ▼❡t❤♦❞s ✐♥ ▼❛t❤❡♠❛t✐❝s✱
❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ❛♥❞ ❊♥❣✐♥❡❡r✐♥❣✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈✳ Pr❡ss✱ ✷✵✶✵✱ ✸✾✽✕✹✻✾✳
❬✻❪ ❈❧✐♠❡♥t ❏✳✲❏✳✱ ●❛r❝✁✙❛ ❋✳✱ ❘❡q✉❡♥❛ ❱✳ ❚❤❡ ❞❡❣r❡❡ ♦❢ ❛ ❇♦♦❧❡❛♥ ❢✉♥❝✲
t✐♦♥ ❛♥❞ s♦♠❡ ❛❧❣❡❜r❛✐❝ ♣r♦♣❡rt✐❡s ♦❢ ✐ts s✉♣♣♦rt✳ ■♥✿ ❉❛t❛ ▼❛♥❛❣❡♠❡♥t ❛♥❞
❙❡❝✉r✐t②✳ ❲■❚ Pr❡ss✱ ✷✵✶✸✱ ✷✺✕✸✻✳
❬✼❪ ❈❛❧✐❦
☛
❈✳
☛ ❈♦♠♣✉t✐♥❣ ❈r②♣t♦❣r❛♣❤✐❝ Pr♦♣❡rt✐❡s ♦❢ ❇♦♦❧❡❛♥ ❋✉♥❝t✐♦♥s ❢r♦♠
t❤❡ ❆❧❣❡❜r❛✐❝ ◆♦r♠❛❧ ❋♦r♠ ❘❡♣r❡s❡♥t❛t✐♦♥✳ P❤✳ ❉✳ ❚❤❡s✐s✱ ▼✐❞❞❧❡ ❊❛st ❚❡❝❤✲
♥✐❝❛❧ ❯♥✐✈❡rs✐t②✱ ❆♥❦❛r❛✱ ❚✉r❦❡②✱ ✷✵✶✸✳
❬✽❪ ▼❛❝❲✐❧❧✐❛♠s ❋✳ ❏✳✱ ❙❧♦❛♥❡ ◆✳ ❏✳ ❆✳ ❚❤❡ ❚❤❡♦r② ♦❢ ❊rr♦r✲❈♦rr❡❝t✐♥❣
❈♦❞❡s✳ ❆♠st❡r❞❛♠✱ ◆♦rt❤✲❍♦❧❧❛♥❞✱ ✶✾✼✽✳
✷✻
❱❛❧❡♥t✐♥ ❇❛❦♦❡✈
❬✾❪ ❖❊■❙ ❋♦✉♥❞❛t✐♦♥ ■♥❝✳ ❚❤❡ ❖♥✲❧✐♥❡ ❊♥❝②❝❧♦♣❡❞✐❛ ♦❢ ■♥t❡❣❡r ❙❡q✉❡♥❝❡s✳
❤tt♣s✿✴✴♦❡✐s✳♦r❣✴✱
✸✶ ❏❛♥✉❛r② ✷✵✷✵✳
❱❛❧❡♥t✐♥ ❇❛❦♦❡✈
❋❛❝✉❧t② ♦❢ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ■♥❢♦r♠❛t✐❝s
❙t✳ ❈②r✐❧ ❛♥❞ ❙t✳ ▼❡t❤♦❞✐✉s ❯♥✐✈❡rs✐t②
❱❡❧✐❦♦ ❚❛r♥♦✈♦✱ ❇✉❧❣❛r✐❛
❡✲♠❛✐❧✿
✈✳❜❛❦♦❡✈❅ts✳✉♥✐✲✈t✳❜❣
❘❡❝❡✐✈❡❞ ▼❛r❝❤ ✻✱ ✷✵✶✾
❋✐♥❛❧ ❆❝❝❡♣t❡❞ ❏✉❧② ✾✱ ✷✵✶✾
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