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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs Finite Fields, Applications and Open Problems Daniel Panario School of Mathematics and Statistics Carleton University daniel@math.carleton.ca
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Finite Fields, Applications and Open Problems
Daniel Panario School of Mathematics and Statistics Carleton University daniel@math.carleton.ca LAWCI School, Campinas, July 2018
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Summary
Lecture 1: Applications in Combinatorics Brief review of finite fields. Introduction to combinatorics objects (designs, latin squares, several types of arrays). Classical results (latin squares and Sudoku; Costas arrays). Orthogonal arrays and their constructions based on finite fields. Some applications in cryptography/coding theory (brief):
secret sharing and combinatorial designs;
Orthogonal array variants (covering arrays, ordered orthogonal arrays) and their constructions based on finite fields.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Summary (cont.)
Lecture 2: Applications in cryptography Applications of finite fields (brief). Differential map, differential uniformity, and differential cryptanalysis (special functions and desired properties, nonlinearity and low differential uniformity). Examples of S-box functions and their characteristics. Perfect nonlinear (PN) and almost perfect nonlinear (APN) functions. Permutation polynomials and their cycle decomposition. Generating pseudorandom sequences: how random is a sequence, requirements for sequences in cryptography.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
History
Finite fields were originally treated, for some particular cases, by Fermat (1601-1665), Euler (1707-1783), Lagrange (1736-1813), Legendre (1752-1833) and Gauss (1777-1855). The crucial researcher for finite fields is ´ Evariste Galois (1811-1832). His paper Sur la th´ eorie des nombres marks the beginning of finite fields, or as they are also called, Galois fields. By the end of the 19th century all the structure of finite fields was
finite fields. Of course, the main reason for these applications to flourish was the appearance of computers.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Application Areas
Many projects undertaken in finite fields can be applied almost immediately to “real-world” problems. Finite fields are used extensively in areas such as: coding theory (for the recovery of errors), cryptography (for the secure transmission of data), communications and electrical engineering, computer science, · · · . The vast majority of these applications work on the finite field F2 that we introduce next.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Groups
(a) for all a, b ∈ G, a ∗ b ∈ G; (b) for all a, b, c ∈ G, a ∗ (b ∗ c) = (a ∗ b) ∗ c; (c) there exists an element e ∈ G such that a ∗ e = e ∗ a = a for all a ∈ G; (d) for all a ∈ G, there exists an element b ∈ G such that a ∗ b = b ∗ a = e. The group G is abelian if G is a group and (e) for all a, b ∈ G, a ∗ b = b ∗ a. Examples: (Z, +), and (Q \ {0}, ·).
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Finite Fields
and · such that: (1) (F, +) is an abelian group; (2) (F \ {0}, ·) is an abelian group; (3) Distributive laws hold, that is, for a, b, c ∈ F, we have a · (b + c) = a · b + a · c, (b + c) · a = b · a + c · a. If #F is finite, then we say that F is a finite field. Example: Z/pZ is a field if and only if p is a prime. p = 2: ({0, 1}, +, ·) is the field F2 of two elements!
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Background on Finite Fields I
(Existence and Uniqueness) Up to isomorphisms, there is exactly one finite field with q = pn elements, denoted Fpn = Fq for all primes p and positive integers n. The characteristic of the finite field Fq is p. In Fq, aq = a for all a ∈ Fq. (Freshman’s Dream) We have that for 0 < i < p p i
i! ≡ 0 (mod p). Hence, if α, β ∈ Fp, we have (α + β)p = αp + βp. This generalizes for powers pn.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Background on Finite Fields II
The multiplicative group of Fq is cyclic. The generators of this multiplicative group are primitive elements. A polynomial f ∈ Fq[x] is irreducible over Fq if f has positive degree and f = gh with g, h ∈ Fq[x] implies that g or h is a
with primitive roots is a primitive polynomial. (Subfield Criterion) Every subfield of Fqn is of the form Fqk for k dividing n. The extension field Fqn can be seen as a vector space of dimension n over Fq. Important in practice are polynomial and normal bases.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Examples: 1 + 1 = 0, and 1 + 1 + 1 = 0
The tables for addition and multiplication in F2 are: + 1 1 1 1 · 1 1 1
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Examples: 1 + 1 = 0, and 1 + 1 + 1 = 0
The tables for addition and multiplication in F2 are: + 1 1 1 1 · 1 1 1 The tables for addition and multiplication in F3 are: + 1 2 1 2 1 1 2 2 2 1 · 1 2 1 1 2 2 2 1 There are finite fields of order pn for prime p and integer n ≥ 1.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Combinatorial Designs
Combinatorial designs are combinatorial objects like arrays and set systems with some type of balance property. Examples: latin squares, Steiner triple systems, t-designs, block designs, orthogonal and covering arrays, finite geometry, etc.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Exemplo 1: Latin Squares
1 2 3 4 5 2 4 1 5 3 3 5 4 2 1 4 3 5 1 2 5 1 2 3 4 Definition A latin square of order n is a matrix n×n such that each symbol of {1, 2, . . . , n} appear exactaly once in each row and in each column.
experimental farm medication test Sudoku game
Sudoku Squares and Chromatic Polynomials
Agnes M. Herzberg and M. Ram MurtyT
he Sudoku puzzle has become a very popular puzzle that many newspapers carry as a daily feature. The puzzle con- sists of a 9×9 grid in which some of the entries of the grid have a number from 1 to 9. One is then required to complete the grid in such a way that every row, every column, and every one of the nine 3 × 3 sub-grids contain the digits from 1 to 9 exactly once. The sub-grids are shown in Figure 1. Figure 1. A Sudoku grid. Agnes M. Herzberg is professor emeritus of mathemat- ics at Queen’s University, Canada. Her email address is herzberg@post.queensu.ca.Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Applications: Latin Squares
experimental designs (statistics) mathematical puzzles: Sudoku error correcting codes (orthogonal latin squares):
A = 1 2 1 2 2 1 , B = 1 2 2 1 1 2 g construction: location sA sB (0, 0) (0, 1) 1 1 (0, 2) 2 2 (1, 0) 1 2 (1, 1) 2 (1, 2) 1 (2, 0) 2 1 (2, 1) 2 (2, 2) 1 − → codewords 0000 0111 0222 1012 1120 1201 2021 2102 2210
código capaz de corrigir 2 erros 1 erro Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example 2: Orthogonal and Covering Arrays
Definition An orthogonal array OA(t, k, v) is a vt × k array with each entry from a set V of size v and satisfying the following property: for any vt × t subarray each t-tuple of V t appears exactly once as a row. OA(2, 4, 3) = 0000 0122 1220 2202 2021 0211 2110 1101 1012
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Covering Arrays
Definition A covering array CA(N; t, k, v) is an N × k array with each entry from a set V of size v and satisfying the following property: for any N × t subarray each t-tuple of V t appears at least once as a row. CAN(t, k, v) = min
N∈N{N : ∃ CA(N; t, k, v)}.
We comment on relations of orthogonal arrays to other objects like MOLS (mutually orthogonal latin squares) and codes, as well as on constructions of covering arrays based on finite fields.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Application of Covering Arrays to Software Testing
Test a system with k = 4 components, each one with g = 3 options:
Component Web Browser Operating Connection Printer System Type Config Config: Netscape(0) Windows(0) LAN(0) Local (0) IE(1) Macintosh(1) PPP(1) Networked(1) Other(2) Linux(2) ISDN(2) Screen(2)
Exaustively testing all options requires 34 = 81 tests. In general, errors are caused by the “interaction” of t components (t << k). A covering array with t = 2, k = 4, g = 3 covers all possible pairs with just 9 tests.
Test Case Browser OS Connection Printer 1 NetScape Windows LAN Local 2 NetScape Linux ISDN Networked 3 NetScape Macintosh PPP Screen 4 IE Windows ISDN Screen 5 IE Macintosh LAN Networked 6 IE Linux PPP Local 7 Other Windows PPP Networked 8 Other Linux LAN Screen 9 Other Macintosh ISDN Local
(Example due to Colbourn, 2004.)
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Exemplo 3: Steiner Triple Systems
Definition A Steiner triple systems STS(n) of order n is a set of triples, subsets of X = {1, 2, . . . , n}, such that for each pair of elements
STS(7) : {1, 2, 4}, {1, 3, 7}, {1, 5, 6}, {2, 3, 5}, {2, 6, 7}, {3, 4, 6}, {4, 5, 7}
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Steiner Triple Systems
The next examples of STS are “resolvable” in “parallel classes” STS(9) has 9 points and 12 triples, 4 parallel classes
1 2 3 4 5 6 7 8 9
STS(15) has 15 points, 35 triples and 7 parallel classes
Kirkman problem “girl school parade” (1850) balanced scheduling working groups.
Sun. Mon. Tues. Wed. Thurs. Fri. Sat. 01, 06, 11 01, 02, 05 02, 03, 06 05, 06, 09 03, 05, 11 05, 07, 13 11, 13, 04 02, 07, 12 03, 04, 07 04, 05, 08 07, 08, 11 04, 06, 12 06, 08, 14 12, 14, 05 03, 08, 13 08, 09, 12 09, 10, 13 12, 13, 01 07, 09, 15 09, 11, 02 15, 02, 08 04, 09, 14 10, 11, 14 11, 12, 15 14, 15, 03 08, 10, 01 10, 12, 03 01, 03, 09 05, 10, 15 13, 15, 06 14, 01, 07 02, 04, 10 13, 14, 02 15, 01, 04 06, 07, 10
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Kirkman Problem (“Girl School Parade”)
Kirkman (1850) problem (how to arrange a girl school parade) deals with balanced scheduling work groups:
Sun. Mon. Tues. Wed. Thurs. Fri. Sat. 01, 06, 11 01, 02, 05 02, 03, 06 05, 06, 09 03, 05, 11 05, 07, 13 11, 13, 04 02, 07, 12 03, 04, 07 04, 05, 08 07, 08, 11 04, 06, 12 06, 08, 14 12, 14, 05 03, 08, 13 08, 09, 12 09, 10, 13 12, 13, 01 07, 09, 15 09, 11, 02 15, 02, 08 04, 09, 14 10, 11, 14 11, 12, 15 14, 15, 03 08, 10, 01 10, 12, 03 01, 03, 09 05, 10, 15 13, 15, 06 14, 01, 07 02, 04, 10 13, 14, 02 15, 01, 04 06, 07, 10
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Threshold Secret Sharing Scheme Application
group of 3 managers 2 managers can open box 1 manager cannot have info
1 2 3 4 5 6 7 8 9
Example: key b M1 receives “5” M2 receives “2” M3 receives “8” shadows keys (3 people) (secret) 1, 2, 3 a 4, 5, 6 a 7, 8, 9 a 1, 4, 7 b 2, 5, 8 b 3, 6, 9 b 1, 5, 9 c 2, 6, 7 c 3, 4, 8 c 1, 6, 8 d 2, 4, 9 d 3, 5, 7 d
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Theorem If q ≡ 1 (mod 6) is a prime power, then there exists a Kirkman triple system of order 2q + 1. Proof: Let q = 6t + 1 and let α ∈ Fq a primitive element. Let θ = (αt + 1)2−1. We define X = (Fq × {1, 2}) ∪ {∞}. Let us construct the first parallel class with the following ste of blocks: Π0 = {{∞, (0, 1), (0, 2)}} ∪ {{(αi, 1), (αi+t, 1), (θαi, 2)} : 0 ≤ i ≤ t − 1, 2t ≤ i ≤ 3t − 1, 4t ≤ i ≤ 5t − 1} ∪ {{θαi+t, 2), (θαi+3t, 2), (θαi+5t, 2)} : 0 ≤ i ≤ t − 1}. The other parallel classes are constructed developing through Fq (add 1 to the elements of Fq in each block, sucessively).
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example for q = 7: α = 3 is primitive in F7 and θ = (3 + 1) × 2−1 = 4 × 4 = 2. Blocks of Π0: {∞, (0, 1), (0, 2)}, {(1, 1), (3, 1), (2, 2)}, {(2, 1), (6, 1), (4, 2)}, {(4, 1), (5, 1), (1, 2)} {(6, 2), (5, 2), (3, 2)} Blocks of Π1: {∞, (1, 1), (1, 2)}, {(2, 1), (4, 1), (3, 2)}, {(3, 1), (0, 1), (5, 2)}, {(5, 1), (6, 1), (2, 2)} {(0, 2), (6, 2), (4, 2)} . . . Blocks of Π6: {∞, (6, 1), (6, 2)}, {(0, 1), (2, 1), (1, 2)}, {(1, 1), (5, 1), (3, 2)}, {(3, 1), (4, 1), (0, 2)} {(5, 2), (4, 2), (2, 2)}
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Sudoku History
1 Modern puzzle designed by Howard Garns (at age 74), and
first published by Dell Magazines in 1979 with the name number place. (This was only rediscovered around 2005.)
2 In Japan, Nikoli, Inc. first published puzzles in the Monthly
Nikolist in 1984.
3 Maki Kaji (Nikoli President) originally named named the
puzzle Suuji Wa Dokushin Ni Kagiru (”the numbers must be single”), then abbreviated it to “Sudoku” (Su = number, Doku = single).
4 International hit by 2005. Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Sudoku Definition
A Sudoku square is a 9 × 9 array using the numbers 1, . . . , 9 arranged so that
1 Each row has each number once. 2 Each column has each number once. 3 Each of the 9, 3 × 3 subsquares has each number once.
We also use numbers 0, . . . , 8 for convenience. There are innumerous generalizations of Sudoku including diagonal Sudoku, even-odd Sudoku, colored Sudoku, geometry Sudoku (irregular regions), and many more.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
A Sudoku square:
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Here is a Sudoku puzzle from the above Sudoku square
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Latin Squares
Let n be a positive integer. A Latin square of order n is an n × n array on n distinct symbols such that every symbol appears exactly
L1 = 1 2 1 2 2 1 L2 = 1 2 2 1 1 2 Two Latin squares are called orthogonal if when superimposed each of the n2 pairs appear exactly once. (L1, L2): (0,0) (1,1) (2,2) (1,2) (2,0) (0,1) (2,1) (0,2) (1,0)
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
A set {L1, . . . , Lt} of Latin squares is mutually orthogonal (MOLS) if Li is orthogonal to Lj for all i = j. Mutually orthogonal Latin squares were originally considered by Euler (1779) for military parade arrangements: Six different regiments have six officers, each one holding a different rank (of six different ranks altogether). Can these 36 officers be arranged in a square formation so that each row and column contains one officer of each rank and one from each regiment? The solution requires a pair of MOLS of order 6. The answer is negative: we cannot have this arrangement for n = 6 (or n = 2). For n = 3 (3 regiments and 3 officers), see the previous slide!
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Let N(n) be the maximum number of MOLS of orden n. Theorem Given n ≥ 2, there does not exist n MOLS(n), that is, N(n) ≤ n − 1. Proof: Let s MOLS(n): L1, . . . , Ls, and assume without loss of generality that the first row of each Li is [1, 2, . . . , n]. The values L1(2, 1), . . . , Ls(2, 1) are all distinct since if Li(2, 1) = Lj(2, 1) = x the pair (x, x) would appear in positions (2, 1) and (1, x). Since L(1, 1) = 1, then Li(2, 1) = 1, for all 1 ≤ i ≤ s. Since we have s distinct elements of {2, . . . , n}, we have s ≤ n − 1. Check the case n = 3: why there can not be more than n − 1 = 2 Latin squares?
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Bose (1938) proved that if q is a prime power, N(q) = q − 1. Idea: Let α ∈ F∗
q and define the Latin square Lα(i, j) = i + αj,
where i, j ∈ Fq. The set of Latin squares {Lα : α ∈ F∗
q} is a set of
q − 1 MOLS of order q. We only know that this is true in the prime power case where we can use finite fields. Big open problem: (Prime Power Conjecture) There are n − 1 MOLS order n iff n is prime power.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Relations
Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square. Sudoku imposes the additional restriction that nine particular 3 adjacent subsquares must also contain the digits 1 to 9.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Relations
Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square. Sudoku imposes the additional restriction that nine particular 3 adjacent subsquares must also contain the digits 1 to 9. One can construct some classes of Sudokus using ideas from Latin squares like rotations. One can also use 3 × 3 subsquares close to “magic” squares. . . like in our previous example! However these are easy Sudokus. A magic square of order n has each of the numbers 1, . . . , n2 exactly once, and has every row, every column and every diagonal summing to a constant value (magic sum) n(n2 + 1)/2.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Albrecht D¨ urer ‘Melencolia’ (1514)
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
The Passion: Fa¸ cade of the Sagrada Familia : 33
1 14 14 4 11 7 6 9 8 10 10 5 13 2 3 15 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Magic Squares as Stamps
Macau 2014 stamps (The Guardian Science, November 3, 2014):
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Some Sudoku Math Tidbits
1 L9 = # LSs order 9 is 9!8!377, 597, 570, 964, 258, 816 2 # Sudoku sqs. is 6,670,903,752,021,072,936,960 =
L9 828186
3 # “essentially different” Sudoku sqs. (rotations, reflections,
permutations and relabellings) is 5,472,730,538.
4 Can have 77 of 81 cells filled, but no unique solution. Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs 5 17 of 81 is minimum number of known cells filled for which
the puzzle has unique solution; 49151 such puzzles known (as
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs 6 Problem. It was proved (through an exhaustive search) in
2011 that there are no unique solution with 16 of the 81 numbers given. It took a computational year; however, there is no mathematical proof of this fact yet.
7 Problem. Given a Sudoku solution square, how does one
delete numbers so that the resulting Sudoku puzzle always has a unique solution?
8 Problem Same thing for a given a Sudoku solution puzzle:
what are the different numbers of cells that can be left filled, and still have unique solution. For example in our earlier example, we had 35 clues given in the original puzzle.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Costas Arrays
Costas arrays were introduced by John Costas in 1965 for a sonar
to counter-attack echo. This make them useful in applications in sonar and radar communications, as well as CDMA (code-division multiple access) fiber-optic local area networks. A Costas array of order n is an n × n array of dots and blanks which satisfies n dots, n(n − 1) blanks, with exactly one dot in each row and column; and all segments between pairs of dots are different.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example n = 3: · · · · · · · · · · · · Question: how many Costas arrays are there of order n = 4? Shifted left-right in time and up-down in frequency, copies of the pattern can only agree with the original in one dot, no dots, or all n dots at once. This allows the recovery of the information.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Radar or Sonar Echo
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Radar or Sonar Echo
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Radar or Sonar Echo
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Radar or Sonar Echo
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Radar or Sonar Echo
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Constructions
All known constructions of Costas arrays (Welch, Lempel and Golomb) are based on finite fields. Then, there are computational
for all n; the smallest not known size is n = 32. Welch Construction: n = p − 1, α a primitive element in Fp.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Constructions
All known constructions of Costas arrays (Welch, Lempel and Golomb) are based on finite fields. Then, there are computational
for all n; the smallest not known size is n = 32. Welch Construction: n = p − 1, α a primitive element in Fp. The multiplicative group of Fq is cyclic. The generators of this multiplicative group are called primitive elements. Example: 2 is not primitive in F7 since 21 = 2, 22 = 4, 23 = 1, 24 = 2, 25 = 4, 26 = 1, but 3 is primitive in F7 since 31 = 3, 32 = 2, 33 = 6, 34 = 4, 35 = 5, 36 = 1.
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Constructions (cont)
Example Let p = 7, n = 6, α = 3; for 1 ≤ j ≤ 6, ai,j has a dot iff αj = i. · · · · · · 3 2 6 4 5 1 We have αj+k − αj = αi+k − αi implies that either i = j or k = 0.
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Orthogonal Arrays
Previously, we consider an orthogonal array OA(t, k, v) as a vt × k array with entries from a set V of size v satisfying that for any vt × t subarray each t-tuple of V t appears exactly once as a row. Let us consider t = 2, and index (number of repetitions) equal to 1. Definition Let us consider integers k ≥ 2 and n ≥ 1. An orthogonal array OA(k, n) is an array A with dimension n2 × k and entries from a set X of cardinality n, such that in any two columns every ordered pair of symbols from X appears exactly in 1 row of A. Orthogonal arrays are related to various combinatorial objects including MOLS and codes.
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Orthogonal arrays and MOLS
Theorem An OA(s + 2, n) exists if and only if s MOLS(n) exist. Idea of proof: Given L1, . . . , Ls, construct tuples (i, j, L1(i, j), . . . , Ls(i, j)) as rows of the OA for all 1 ≤ i, j ≤ n. Each pair of symbols occurs in each pair of columns (a, b): a = 1, b = 2 (by construction) a = 1, b ≥ 3 (a row of Lb−2 is a permutation) a = 2, b ≥ 3 (a column of Lb−2 is a permutation) a, b ≥ 3 (La−2 and Lb−2 are orthogonal).
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example: OA(n + 1, n) from n − 1 MOLS(n)
A = 1 2 1 2 2 1 , B = 1 2 2 1 1 2 g construction: location sA sB (0, 0) (0, 1) 1 1 (0, 2) 2 2 (1, 0) 1 2 (1, 1) 2 (1, 2) 1 (2, 0) 2 1 (2, 1) 2 (2, 2) 1 − → codewords 0000 0111 0222 1012 1120 1201 2021 2102 2210
código capaz de corrigir 2 erros 1 erro
This gives an MDS (maximum distance separable) code: length n + 1, minimum distance d = n, alphabet size n, and number for codewords: M = n2.
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
OA Constructions via Finite Fields
Theorem Let q be a prime power and 2 ≤ k ≤ q. Then, there exists an OA(k, q). Proof: Let a1, . . . , ak distinct elements in Fq (they exist since k ≤ q). Let us consider v1, v2 ∈ (Fq)k: v1 = (1, . . . , 1), v2 = (a1, . . . , ak), and define the rows of A, with indexes in Fq × Fq, by row (i, j) of A: iv1 + jv2. To prove that A is an orthogonal array, pick any two columns c and d, 1 ≤ c < d ≤ k, and let x, y ∈ Fq. We need to show that there exist a unique row (i, j) of A such that A((i, j), c) = x and A((i, j), d) = y. This gives a system in the unknowns i and j:
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
i + jac = x i + jad = y. Subtracting we get j(ac − ad) = x − y. Since (ac − ad) = 0, there exists a multiplicative inverse (ac − ad)−1 ∈ Fq. We conclude that j = (ac − ad)−1(x − y), and substituting we have i = x − jac = x − ac(ac − ad)−1(x − y). We can extend the above OA(q, q) to construct an OA(q + 1, q). Theorem Let q be a prime power. Then, there exists an OA(q + 1, q). Prova: Construct an OA(q, q) as above. Include a column (q + 1) with A((i, j), q + 1) = j for all i, j. We get an OA(q + 1, q).
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Applications of orthogonal arrays
OAs can be seen as maximum distance separable (MDS) codes (see, for example, Paterson and Stinson, 2014). These are codes that meet the Singleton bound (d + k = n + 1). OAs can also be used for secret sharing, as we commented. An OA(t, k, n) is used to distribute n shares with threshold t, having nt possible keys. The number of possible shares in a threshold scheme must be greater than or equal to the number of possible secrets. If the number of possible secrets in a threshold scheme equals the number of possible shares, the scheme is ideal. Ideal threshold schemes are equivalent to combinatorial orthogonal arrays and maximum distance separable (MDS) codes.
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Orthogonal Arrays and Ramp Schemes
A (s, t, n)-ramp is a generalization of threshold schemes in which there are two thresholds: s is the lower threshold value and t is the upper threshold. In a ramp scheme, any t of the n players can compute the secret, and no subset of s players can determine the
The relation between ramp schemes and combinatorial arrays is less clear; see the recent article by Stinson (Discrete Mathematics, 2018) where he connects these schemes with augmented
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Orthogonal Arrays, LFSRs and Primitive Polynomials
A polynomial f of degree m is called primitive if k = qm − 1 is the smallest positive integer such that f divides xk − 1. A shift-register sequence with characteristic polynomial f(x) = xm + m−1
i=0 cixi is the sequence a = (a0, a1, . . .)
defined by the recurrence relation an+m = −
m−1
ciai+n, for n ≥ 0. If f is primitive over Fq, the sequence has period qm − 1. A subset C of Fn
q is an orthogonal array of strength t if for any
t-subset T = {i1, i2, . . . , it} of {1, 2, . . . , n} and any t-tuple (b1, b2, . . . , bt) ∈ Ft
q, there exists exactly |C|/qt elements
c = (c1, c2, . . . , cn) of C such that cij = bj for all 1 ≤ j ≤ t.
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Let q = 2, n = 7, C ⊆ F7
2 with |C| = 8, and t = 2. Thus, we
want an orthogonal array with |C|/2t = 2 and any 2-tuple of F2
2 = {(0, 0), (0, 1), (1, 0), (1, 1)} appearing exactly |C|/2t = 2
times: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 .
dual code, has minimum weight t + 1.
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Theorem (Munemasa). Let f be a primitive polynomial of degree m over Fq and let 2 ≤ n ≤ qm − 1. Let Cf
n be the set of all
subintervals of the shift-register sequence with length n generated by f, together with the zero vector of length n. The dual code of Cf
n is given by
(Cf
n)⊥ = {(b1, . . . , bn) : n−1
bi+1xi is divisible by f}. Theorem (Munemasa). Let f(x) = xm + xl + 1 be a trinomial
most 2m that is divisible by f, then g(x) = xdeg g−mf(x), g(x) = f(x)2, or g(x) = x5 + x4 + 1 = (x2 + x + 1)(x3 + x + 1)
Cf
n corresponds to an orthogonal array of strength 2 that has a
property very close to being an orthogonal array of strength 3.
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example: Consider the orthogonal array constructed from the LFSR defined by the primitive polynomial f(x) = x3 + x + 1 over F2: x0 x1 x2 x3 x4 x5 x6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . We have strength 3 for many columns, but we do not have strength 3 for shifts of f(x) = x3 + x + 1 and f(x)2 = x6 + x2 + 1. Check: x(x3 + x + 1) = x4 + x2 + x and f(x)2 = x6 + x2 + 1.
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Orthogonal Arrays
Definition An orthogonal array OAλ(N; t, k, v) is a N × k array with each entry from a set V of size v and satisfying the following property: For any N × t subarray each t-tuple of V t appears exactly λ = N
vt
times as a row. λ: the index of the array. N: Number of rows t: Strength k: Number of columns v: Number of symbols
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
OAs were introduced by Rao (1946, 1947, 1949) for use in design of experiments in Statistics (medicine, agriculture and manufacturing). OAs are used in computer science and cryptography. Hedayat, Sloane and Stufken; Orthogonal Arrays: Theory and
Munemasa, A.: Orthogonal arrays, primitive trinomials, and shift-register sequences, Finite Fields Appl. 4, 252–260 (1998).
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Ordered Orthogonal Arrays
Let m and s be positive integers and Ω[m, s] be a set of size ms, partitioned into m blocks Bi of cardinality s, where Bi = {bis, . . . , b(i+1)s−1} for i = 0, . . . , m − 1. Each block has the total ordering: bis ≺ bis+1 ≺ . . . ≺ b(i+1)s−1. The set Ω[m, s] has the structure of partially ordered set (poset): the union of m totally ordered sets with s elements each.
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Ordered Orthogonal Arrays
Let m and s be positive integers and Ω[m, s] be a set of size ms, partitioned into m blocks Bi of cardinality s, where Bi = {bis, . . . , b(i+1)s−1} for i = 0, . . . , m − 1. Each block has the total ordering: bis ≺ bis+1 ≺ . . . ≺ b(i+1)s−1. The set Ω[m, s] has the structure of partially ordered set (poset): the union of m totally ordered sets with s elements each. An antiideal is a subset of Ω[m, s] closed under followers.
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example: m = 3 and s = 3
Ω[3, 3] = {0, 1, 2, 3, 4, 5, 6, 7, 8} B0 = {0, 1, 2}, B1 = {3, 4, 5}, B2 = {6, 7, 8} 1 2 3 4 5 6 7 8 Antiideals of size 3 {0, 1, 2} {1, 2, 5} {1, 2, 8} {2, 4, 5} {2, 5, 8} {2, 7, 8} {3, 4, 5} {4, 5, 8} {5, 7, 8} {6, 7, 8}
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example: m = 3 and s = 3
Ω[3, 3] = {0, 1, 2, 3, 4, 5, 6, 7, 8} B0 = {0, 1, 2}, B1 = {3, 4, 5}, B2 = {6, 7, 8} 1 2 3 4 5 6 7 8 Antiideals of size 3 {0, 1, 2} {1, 2, 5} {1, 2, 8} {2, 4, 5} {2, 5, 8} {2, 7, 8} {3, 4, 5} {4, 5, 8} {5, 7, 8} {6, 7, 8}
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example: m = 3 and s = 3
Ω[3, 3] = {0, 1, 2, 3, 4, 5, 6, 7, 8} B0 = {0, 1, 2}, B1 = {3, 4, 5}, B2 = {6, 7, 8} 1 2 3 4 5 6 7 8 Antiideals of size 3 {0, 1, 2} {1, 2, 5} {1, 2, 8} {2, 4, 5} {2, 5, 8} {2, 7, 8} {3, 4, 5} {4, 5, 8} {5, 7, 8} {6, 7, 8}
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example: m = 3 and s = 3
Ω[3, 3] = {0, 1, 2, 3, 4, 5, 6, 7, 8} B0 = {0, 1, 2}, B1 = {3, 4, 5}, B2 = {6, 7, 8} 1 2 3 4 5 6 7 8 Antiideals of size 3 {0, 1, 2} {1, 2, 5} {1, 2, 8} {2, 4, 5} {2, 5, 8} {2, 7, 8} {3, 4, 5} {4, 5, 8} {5, 7, 8} {6, 7, 8}
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Definition Let t, m, s, v be positive integers such that s ≤ t ≤ ms. An
with entries from a set V of size v, columns labeled by Ω[m, s], and satisfying the property: For each antiideal I ⊂ Ω[m, s] of size t, each t-tuple of V t appears exactly once in the t columns of A labeled by I. When s = 1, OOA(t, m, 1, v) = OA(t, m, v).
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example
Ω[3, 3] = {0, 1, 2, 3, 4, 5, 6, 7, 8} B0 = {0, 1, 2}, B1 = {3, 4, 5}, B2 = {6, 7, 8} OOA(t = 3, m = 3, s = 3, v = 2) 1 2 3 4 5 6 7 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Antiideals of size 3 {0, 1, 2} {1, 2, 5} {1, 2, 8} {2, 4, 5} {2, 5, 8} {2, 7, 8} {3, 4, 5} {4, 5, 8} {5, 7, 8} {6, 7, 8}
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example
Ω[3, 3] = {0, 1, 2, 3, 4, 5, 6, 7, 8} B0 = {0, 1, 2}, B1 = {3, 4, 5}, B2 = {6, 7, 8} OOA(t = 3, m = 3, s = 3, v = 2) 1 2 3 4 5 6 7 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Antiideals of size 3 {0, 1, 2} {1, 2, 5} {1, 2, 8} {2, 4, 5} {2, 5, 8} {2, 7, 8} {3, 4, 5} {4, 5, 8} {5, 7, 8} {6, 7, 8}
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example
Ω[3, 3] = {0, 1, 2, 3, 4, 5, 6, 7, 8} B0 = {0, 1, 2}, B1 = {3, 4, 5}, B2 = {6, 7, 8} OOA(t = 3, m = 3, s = 3, v = 2) 1 2 3 4 5 6 7 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Antiideals of size 3 {0, 1, 2} {1, 2, 5} {1, 2, 8} {2, 4, 5} {2, 5, 8} {2, 7, 8} {3, 4, 5} {4, 5, 8} {5, 7, 8} {6, 7, 8}
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example
Ω[3, 3] = {0, 1, 2, 3, 4, 5, 6, 7, 8} B0 = {0, 1, 2}, B1 = {3, 4, 5}, B2 = {6, 7, 8} OOA(t = 3, m = 3, s = 3, v = 2) 1 2 3 4 5 6 7 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Antiideals of size 3 {0, 1, 2} {1, 2, 5} {1, 2, 8} {2, 4, 5} {2, 5, 8} {2, 7, 8} {3, 4, 5} {4, 5, 8} {5, 7, 8} {6, 7, 8}
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Example
Ω[3, 3] = {0, 1, 2, 3, 4, 5, 6, 7, 8} B0 = {0, 1, 2}, B1 = {3, 4, 5}, B2 = {6, 7, 8} OOA(t = 3, m = 3, s = 3, v = 2) 1 2 3 4 5 6 7 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Antiideals of size 3 {0, 1, 2} {1, 2, 5} {1, 2, 8} {2, 4, 5} {2, 5, 8} {2, 7, 8} {3, 4, 5} {4, 5, 8} {5, 7, 8} {6, 7, 8}
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Motivation
Niederreiter (1987) introduced (t, m, s)-nets in base b; there are several applications of this object to numerical integration (quasi-Monte Carlo methods). Niederreiter (1987) showed that a (t, t + 2, s)-net in base b is equivalent to an OAbt(2, s, b). Lawrence (1996) and Mullen and Schmid (1996) show that there exists a (t, m, s)-net in base b if and only if there exists an OOAbt(m − t, s, m − t, b). Ordered orthogonal arrays are a combinatorial characterization of (t, m, s)-nets.
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Rosenbloom and Tsfasman (1997) and Skriganov (2002) constructed a class of maximum distance separable (MDS) codes with respect to the NRT metric. For q a prime power and s ≤ t they show that there exists an MDS code with respect to the NRT metric with length (q + 1)s, dimension t, and minimum distance (q + 1)s − t + 1. This class of MDS codes is known as Reed-Solomon m-codes; they are equivalent to an OOA(t, q + 1, t, q).
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
The existing OOA constructions prior to 2002 essentially repeated columns of existing orthogonal arrays in clever ways so that the resulting arrays satisfied the required column coverage for the
The work of Fuji-Hara and Miao (2002) for t = 3, 4 and the OOA construction of Castoldi et al (2017) for arbitrary t are the first constructions of OOAs which did not simply repeat columns.
Ordered orthogonal array construction using LFSR sequences,
IEEE Transactions on Information Theory, 63, 1336-1347, 2017. A general construction of ordered orthogonal arrays using LFSRs,
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Theorem For q a prime power and t ≥ 3, there exists an OOA(t, q + 1, t, q). Let f(x) = c0 + c1x + . . . + ct−1xt−1 + xt be a degree-t primitive polynomial over Fq = {0, β1, . . . , βq−1} and α ∈ Fqt a root of f. Label the columns of the subinterval array M(f) by Zk = {0, 1, . . . , k − 1}, where k = qt−1
q−1 .
For each i = 1, . . . , q − 1, let kβi ∈ Zqt−1 such that αkβi(α − βi) = 1. Choose the columns of the subinterval array M(f) labeled by the following indexes modulo k:
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
1 t − 2 t − 1 2t − 1 2t − 2 t + 1 t t + tkβ1 t + (t − 1)kβ1 t + 2kβ1 t + kβ1 t + tkβq−1 t + (t − 1)kβq−1 t + 2kβq−1 t + kβq−1
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
How many t-subsets of columns of an OOA(t, q + 1, t, q) have the property of cover all the t-tuples of Ft
q?
Table: OOA(3, 4, 3, 3)
Primitive Polynomial 3-sets covered Percentage 1 + 2x2 + x3 163 0.740909 1 + x + 2x2 + x3 156 0.709091 1 + 2x + x3 156 0.709091 1 + 2x + x2 + x3 162 0.736364 RT construction 120 0.545455 Number of 3-antiideals 20 Number of 3-subsets of a 12-set 220
Finite Fields, Applications and Open Problems Daniel Panario
Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Summary
In this lecture we revised several combinatorial objects where finite fields play a role in their construction. We covered some historic results related to several types of designs, latin squares and Costas arrays. We also showed an example where finite fields did not produce interesting Sudokus. Then we focused on combinatorial arrays such as orthogonal arrays, covering arrays and ordered orthogonal arrays. The finite fields constructions here are more sophisticated leading to competitive arrays.
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Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs
Finite Fields, Applications and Open Problems Daniel Panario
1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2
1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2
1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2
Covering and orthogonal arrays m-sequences Orthogonal arrays from m-sequences Covering arrays from m-sequences
1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2
1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2
1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 2 1 1 2 1 1 1 2 2 1 2 2 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1