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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

  1. 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 Murty T he Sudoku puzzle has become a very Recall that a Latin square of rank n is an n × n popular puzzle that many newspapers array consisting of the numbers such that each carry as a daily feature. The puzzle con- row and column has all the numbers from 1 to sists of a 9 × 9 grid in which some of the n . In particular, every Sudoku square is a Latin entries of the grid have a number from square of rank 9, but not conversely because of 1 to 9. One is then required to complete the grid the condition on the nine 3 × 3 sub-grids. Figure 2 (taken from [6]) shows one such puzzle with in such a way that every row, every column, and seventeen entries given. 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. 1 4 2 5 4 7 8 3 1 9 3 4 2 5 1 8 6 Figure 2. A Sudoku puzzle with 17 entries. Figure 1. A Sudoku grid. For anyone trying to solve a Sudoku puzzle, several questions arise naturally. For a given puz- zle, does a solution exist? If the solution exists, is Agnes M. Herzberg is professor emeritus of mathemat- it unique? If the solution is not unique, how many ics at Queen’s University, Canada. Her email address is solutions are there? Moreover, is there a system- herzberg@post.queensu.ca . atic way of determining all the solutions? How M. Ram Murty is professor of mathematics at Queen’s many puzzles are there with a unique solution? University, Canada. His email address is murty@mast. What is the minimum number of entries that can queensu.ca . be specified in a single puzzle in order to ensure Research of both authors is partially supported by Natu- a unique solution? For instance, Figure 2 shows ral Sciences and Engineering Research Council (NSERC) that the minimum is at most 17. (We leave it to grants. 708 Notices of the AMS Volume 54 , Number 6 Finite Fields, Applications and Open Problems Daniel Panario

  2. 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): 0 1 2 0 1 2 A = 1 2 0 , B = 2 0 1 2 0 1 1 2 0 código capaz de corrigir 2 erros g construction: 1 erro location s A s B codewords (0 , 0) 0 0 0000 (0 , 1) 1 1 0111 (0 , 2) 2 2 0222 (1 , 0) 1 2 1012 − → (1 , 1) 2 0 1120 (1 , 2) 0 1 1201 (2 , 0) 2 1 2021 (2 , 1) 0 2 2102 (2 , 2) 1 0 2210 Finite Fields, Applications and Open Problems Daniel Panario

  3. 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 v t × k array with each entry from a set V of size v and satisfying the following property: for any v t × t subarray each t -tuple of V t appears exactly once as a row. 0000   0122     1220     2202     OA (2 , 4 , 3) = 2021     0211     2110     1101   1012 Finite Fields, Applications and Open Problems Daniel Panario

  4. 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

  5. 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 3 4 = 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

  6. 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 of X appears in exactely one of the triples. 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

  7. 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 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 Kirkman problem “girl school parade” (1850) 03, 08, 13 08, 09, 12 09, 10, 13 12, 13, 01 07, 09, 15 09, 11, 02 15, 02, 08 balanced scheduling working groups. 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

  8. 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

  9. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs Threshold Secret Sharing Scheme Application shadows keys group of 3 managers (3 people) (secret) 2 managers can open box 1, 2, 3 a 1 manager cannot have info 4, 5, 6 a 1 2 3 7, 8, 9 a 1, 4, 7 b 4 5 6 2, 5, 8 b 3, 6, 9 b 1, 5, 9 c 7 8 9 2, 6, 7 c Example: key b 3, 4, 8 c M 1 receives “5” 1, 6, 8 d M 2 receives “2” 2, 4, 9 d M 3 receives “8” 3, 5, 7 d Finite Fields, Applications and Open Problems Daniel Panario

  10. 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 2 q + 1 . Proof: Let q = 6 t + 1 and let α ∈ F q a primitive element. Let θ = ( α t + 1)2 − 1 . We define X = ( F q × { 1 , 2 } ) ∪ {∞} . Let us construct the first parallel class with the following ste of blocks: Π 0 = {{∞ , (0 , 1) , (0 , 2) }} ∪ {{ ( α i , 1) , ( α i + t , 1) , ( θα i , 2) } : 0 ≤ i ≤ t − 1 , 2 t ≤ i ≤ 3 t − 1 , 4 t ≤ i ≤ 5 t − 1 } ∪ {{ θα i + t , 2) , ( θα i +3 t , 2) , ( θα i +5 t , 2) } : 0 ≤ i ≤ t − 1 } . The other parallel classes are constructed developing through F q (add 1 to the elements of F q in each block, sucessively). Finite Fields, Applications and Open Problems Daniel Panario

  11. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs Example for q = 7 : α = 3 is primitive in F 7 and θ = (3 + 1) × 2 − 1 = 4 × 4 = 2 . Blocks of Π 0 : {∞ , (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 Π 1 : {∞ , (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 Π 6 : {∞ , (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

  12. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs Latin Squares and Sudoku Finite Fields, Applications and Open Problems Daniel Panario

  13. 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

  14. 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

  15. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs A Sudoku square: 0 4 8 7 2 3 5 6 1 5 6 1 0 4 8 7 2 3 7 2 3 5 6 1 0 4 8 8 0 4 3 7 2 1 5 6 1 5 6 8 0 4 3 7 2 3 7 2 1 5 6 8 0 4 4 8 0 2 3 7 6 1 5 6 1 5 4 8 0 2 3 7 2 3 7 6 1 5 4 8 0 Finite Fields, Applications and Open Problems Daniel Panario

  16. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs Here is a Sudoku puzzle from the above Sudoku square 2 3 1 5 7 2 3 5 0 4 8 0 3 2 6 1 6 8 0 7 7 1 5 8 3 7 6 1 5 6 0 2 1 5 8 Finite Fields, Applications and Open Problems Daniel Panario

  17. 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 once in every row and column. Here are two examples: 0 1 2 0 1 2 L 1 = 1 2 0 L 2 = 2 0 1 2 0 1 1 2 0 Two Latin squares are called orthogonal if when superimposed each of the n 2 pairs appear exactly once. (0,0) (1,1) (2,2) ( L 1 , L 2 ) : (1,2) (2,0) (0,1) (2,1) (0,2) (1,0) Finite Fields, Applications and Open Problems Daniel Panario

  18. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs A set { L 1 , . . . , L t } of Latin squares is mutually orthogonal (MOLS) if L i is orthogonal to L j 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

  19. 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 ): L 1 , . . . , L s , and assume without loss of generality that the first row of each L i is [1 , 2 , . . . , n ] . The values L 1 (2 , 1) , . . . , L s (2 , 1) are all distinct since if L i (2 , 1) = L j (2 , 1) = x the pair ( x, x ) would appear in positions (2 , 1) and (1 , x ) . Since L (1 , 1) = 1 , then L i (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

  20. 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 L α ( i, j ) = i + αj , where i, j ∈ F q . The set of Latin squares { L α : 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

  21. 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

  22. 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 , . . . , n 2 exactly once, and has every row, every column and every diagonal summing to a constant value (magic sum) n ( n 2 + 1) / 2 . Finite Fields, Applications and Open Problems Daniel Panario

  23. 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

  24. 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 16 3 2 13 11 7 6 9 5 10 11 8 8 10 10 5 9 6 7 12 13 2 3 15 4 15 14 1 Finite Fields, Applications and Open Problems Daniel Panario

  25. 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

  26. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs Some Sudoku Math Tidbits 1 L 9 = # LSs order 9 is 9!8!377 , 597 , 570 , 964 , 258 , 816 2 # Sudoku sqs. is 6,670,903,752,021,072,936,960 = L 9 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

  27. 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 of today); here is one of them 1 4 2 5 4 7 8 3 1 9 3 4 2 5 1 8 6 Finite Fields, Applications and Open Problems Daniel Panario

  28. 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

  29. 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 application. These arrays have low auto-ambiguity function, used 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

  30. 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

  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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 experiments. There are Costas arrays for infinitely many n , but not for all n ; the smallest not known size is n = 32 . Welch Construction: n = p − 1 , α a primitive element in F p . Finite Fields, Applications and Open Problems Daniel Panario

  37. 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 experiments. There are Costas arrays for infinitely many n , but not for all n ; the smallest not known size is n = 32 . Welch Construction: n = p − 1 , α a primitive element in F p . The multiplicative group of F q is cyclic. The generators of this multiplicative group are called primitive elements. Example: 2 is not primitive in F 7 since 2 1 = 2 , 2 2 = 4 , 2 3 = 1 , 2 4 = 2 , 2 5 = 4 , 2 6 = 1 , but 3 is primitive in F 7 since 3 1 = 3 , 3 2 = 2 , 3 3 = 6 , 3 4 = 4 , 3 5 = 5 , 3 6 = 1 . Finite Fields, Applications and Open Problems Daniel Panario

  38. 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 , a i,j has a dot iff α j = i . · · · · · · 3 2 6 4 5 1 We have α j + k − α j = α i + k − α i implies that either i = j or k = 0 . Finite Fields, Applications and Open Problems Daniel Panario

  39. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs Orthogonal, Covering and Ordered Orthogonal Arrays Finite Fields, Applications and Open Problems Daniel Panario

  40. 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 v t × k array with entries from a set V of size v satisfying that for any v t × 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 n 2 × 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. Finite Fields, Applications and Open Problems Daniel Panario

  41. 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 L 1 , . . . , L s , construct tuples ( i, j, L 1 ( i, j ) , . . . , L s ( 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 L b − 2 is a permutation) a = 2 , b ≥ 3 (a column of L b − 2 is a permutation) a, b ≥ 3 ( L a − 2 and L b − 2 are orthogonal). Finite Fields, Applications and Open Problems Daniel Panario

  42. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs Example: OA ( n + 1 , n ) from n − 1 MOLS( n ) 0 1 2 0 1 2 A = 1 2 0 , B = 2 0 1 2 0 1 1 2 0 código capaz de corrigir 2 erros 1 erro g construction: location s A s B codewords (0 , 0) 0 0 0000 (0 , 1) 1 1 0111 (0 , 2) 2 2 0222 (1 , 0) 1 2 1012 − → (1 , 1) 2 0 1120 (1 , 2) 0 1 1201 (2 , 0) 2 1 2021 (2 , 1) 0 2 2102 (2 , 2) 1 0 2210 This gives an MDS (maximum distance separable) code: length n + 1 , minimum distance d = n , alphabet size n , and number for codewords: M = n 2 . Finite Fields, Applications and Open Problems Daniel Panario

  43. 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 a 1 , . . . , a k distinct elements in F q (they exist since k ≤ q ). Let us consider v 1 , v 2 ∈ ( F q ) k : v 1 = (1 , . . . , 1) , v 2 = ( a 1 , . . . , a k ) , and define the rows of A , with indexes in F q × F q , by row ( i, j ) of A : iv 1 + jv 2 . To prove that A is an orthogonal array, pick any two columns c and d , 1 ≤ c < d ≤ k , and let x, y ∈ F q . 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 : Finite Fields, Applications and Open Problems Daniel Panario

  44. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs i + ja c = x i + ja d = y. Subtracting we get j ( a c − a d ) = x − y . Since ( a c − a d ) � = 0 , there exists a multiplicative inverse ( a c − a d ) − 1 ∈ F q . We conclude that j = ( a c − a d ) − 1 ( x − y ) , and substituting we have i = x − ja c = x − a c ( a c − a d ) − 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 ) . Finite Fields, Applications and Open Problems Daniel Panario

  45. 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 n t 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. Finite Fields, Applications and Open Problems Daniel Panario

  46. 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 secret. A ( t − 1 , t, n ) -ramp scheme is a ( t, n ) threshold scheme. 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 orthogonal arrays. Finite Fields, Applications and Open Problems Daniel Panario

  47. 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 = q m − 1 is the smallest positive integer such that f divides x k − 1 . A shift-register sequence with characteristic polynomial f ( x ) = x m + � m − 1 i =0 c i x i is the sequence a = ( a 0 , a 1 , . . . ) defined by the recurrence relation m − 1 � a n + m = − c i a i + n , for n ≥ 0 . i =0 If f is primitive over F q , the sequence has period q m − 1 . A subset C of F n q is an orthogonal array of strength t if for any t -subset T = { i 1 , i 2 , . . . , i t } of { 1 , 2 , . . . , n } and any t -tuple q , there exists exactly | C | /q t elements ( b 1 , b 2 , . . . , b t ) ∈ F t c = ( c 1 , c 2 , . . . , c n ) of C such that c i j = b j for all 1 ≤ j ≤ t . Finite Fields, Applications and Open Problems Daniel Panario

  48. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs Let q = 2 , n = 7 , C ⊆ F 7 2 with | C | = 8 , and t = 2 . Thus, we want an orthogonal array with | C | / 2 t = 2 and any 2-tuple of 2 = { (0 , 0) , (0 , 1) , (1 , 0) , (1 , 1) } appearing exactly | C | / 2 t = 2 F 2 times: 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 1 . 1 1 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 Theorem. Let C be a linear code over F q . Then, C is an orthogonal array of maximal strength t if and only if C ⊥ , its dual code, has minimum weight t + 1 . Finite Fields, Applications and Open Problems Daniel Panario

  49. 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 F q and let 2 ≤ n ≤ q m − 1 . Let C f 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 C f n is given by n − 1 n ) ⊥ = { ( b 1 , . . . , b n ) : b i +1 x i is divisible by f } . � ( C f i =0 Theorem (Munemasa). Let f ( x ) = x m + x l + 1 be a trinomial over F 2 such that gcd( m, l ) = 1 . If g is a trinomial of degree at most 2m that is divisible by f , then g ( x ) = x deg g − m f ( x ) , g ( x ) = f ( x ) 2 , or g ( x ) = x 5 + x 4 + 1 = ( x 2 + x + 1)( x 3 + x + 1) or, its reciprocal, g ( x ) = x 5 + x + 1 = ( x 2 + x + 1)( x 3 + x 2 + 1) . C f n corresponds to an orthogonal array of strength 2 that has a property very close to being an orthogonal array of strength 3. Finite Fields, Applications and Open Problems Daniel Panario

  50. 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 ) = x 3 + x + 1 over F 2 : x 0 x 1 x 2 x 3 x 4 x 5 x 6 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 0 1 1 . 0 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 0 0 1 1 1 0 0 1 We have strength 3 for many columns, but we do not have strength 3 for shifts of f ( x ) = x 3 + x + 1 and f ( x ) 2 = x 6 + x 2 + 1 . Check: x ( x 3 + x + 1) = x 4 + x 2 + x and f ( x ) 2 = x 6 + x 2 + 1 . Finite Fields, Applications and Open Problems Daniel Panario

  51. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs Orthogonal Arrays Definition An orthogonal array OA λ ( 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 v t times as a row. λ : the index of the array. N : Number of rows t : Strength k : Number of columns v : Number of symbols Finite Fields, Applications and Open Problems Daniel Panario

  52. 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 Applications. Springer, 1999. Munemasa, A.: Orthogonal arrays, primitive trinomials, and shift-register sequences, Finite Fields Appl. 4, 252–260 (1998). Finite Fields, Applications and Open Problems Daniel Panario

  53. 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 B i of cardinality s , where B i = { b is , . . . , b ( i +1) s − 1 } for i = 0 , . . . , m − 1 . Each block has the total ordering: b is ≺ b is +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. Finite Fields, Applications and Open Problems Daniel Panario

  54. 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 B i of cardinality s , where B i = { b is , . . . , b ( i +1) s − 1 } for i = 0 , . . . , m − 1 . Each block has the total ordering: b is ≺ b is +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. Finite Fields, Applications and Open Problems Daniel Panario

  55. 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 } B 0 = { 0 , 1 , 2 } , B 1 = { 3 , 4 , 5 } , B 2 = { 6 , 7 , 8 } Antiideals of size 3 { 0 , 1 , 2 } { 1 , 2 , 5 } 2 5 8 { 1 , 2 , 8 } { 2 , 4 , 5 } { 2 , 5 , 8 } 1 4 7 { 2 , 7 , 8 } { 3 , 4 , 5 } 0 3 6 { 4 , 5 , 8 } { 5 , 7 , 8 } { 6 , 7 , 8 } Finite Fields, Applications and Open Problems Daniel Panario

  56. 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 } B 0 = { 0 , 1 , 2 } , B 1 = { 3 , 4 , 5 } , B 2 = { 6 , 7 , 8 } Antiideals of size 3 { 0 , 1 , 2 } { 1 , 2 , 5 } 2 5 8 { 1 , 2 , 8 } { 2 , 4 , 5 } { 2 , 5 , 8 } 1 4 7 { 2 , 7 , 8 } { 3 , 4 , 5 } 0 3 6 { 4 , 5 , 8 } { 5 , 7 , 8 } { 6 , 7 , 8 } Finite Fields, Applications and Open Problems Daniel Panario

  57. 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 } B 0 = { 0 , 1 , 2 } , B 1 = { 3 , 4 , 5 } , B 2 = { 6 , 7 , 8 } Antiideals of size 3 { 0 , 1 , 2 } { 1 , 2 , 5 } 2 5 8 { 1 , 2 , 8 } { 2 , 4 , 5 } { 2 , 5 , 8 } 1 4 7 { 2 , 7 , 8 } { 3 , 4 , 5 } 0 3 6 { 4 , 5 , 8 } { 5 , 7 , 8 } { 6 , 7 , 8 } Finite Fields, Applications and Open Problems Daniel Panario

  58. 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 } B 0 = { 0 , 1 , 2 } , B 1 = { 3 , 4 , 5 } , B 2 = { 6 , 7 , 8 } Antiideals of size 3 { 0 , 1 , 2 } { 1 , 2 , 5 } 2 5 8 { 1 , 2 , 8 } { 2 , 4 , 5 } { 2 , 5 , 8 } 1 4 7 { 2 , 7 , 8 } { 3 , 4 , 5 } 0 3 6 { 4 , 5 , 8 } { 5 , 7 , 8 } { 6 , 7 , 8 } Finite Fields, Applications and Open Problems Daniel Panario

  59. 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 ordered orthogonal array OOA ( t, m, s, v ) is a v t × ms array A 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 ) . Finite Fields, Applications and Open Problems Daniel Panario

  60. 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 } B 0 = { 0 , 1 , 2 } , B 1 = { 3 , 4 , 5 } , B 2 = { 6 , 7 , 8 } OOA ( t = 3 , m = 3 , s = 3 , v = 2) Antiideals of size 3 { 0 , 1 , 2 } 0 1 2 3 4 5 6 7 8   { 1 , 2 , 5 } 1 0 0 1 1 1 0 1 1   { 1 , 2 , 8 }   0 0 1 0 1 1 1 0 0   { 2 , 4 , 5 }   0 1 1 1 0 1 1 0 1   { 2 , 5 , 8 }   1 1 1 0 1 0 1 1 0   { 2 , 7 , 8 }   1 1 0 0 0 1 0 1 0   { 3 , 4 , 5 }   1 0 1 1 0 0 1 1 1   { 4 , 5 , 8 }   0 1 0 1 1 0 0 0 1   { 5 , 7 , 8 } 0 0 0 0 0 0 0 0 0 { 6 , 7 , 8 } Finite Fields, Applications and Open Problems Daniel Panario

  61. 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 } B 0 = { 0 , 1 , 2 } , B 1 = { 3 , 4 , 5 } , B 2 = { 6 , 7 , 8 } OOA ( t = 3 , m = 3 , s = 3 , v = 2) Antiideals of size 3 { 0 , 1 , 2 } 0 1 2 3 4 5 6 7 8   { 1 , 2 , 5 } 1 0 0 1 1 1 0 1 1   { 1 , 2 , 8 }   0 0 1 0 1 1 1 0 0   { 2 , 4 , 5 }   0 1 1 1 0 1 1 0 1   { 2 , 5 , 8 }   1 1 1 0 1 0 1 1 0   { 2 , 7 , 8 }   1 1 0 0 0 1 0 1 0   { 3 , 4 , 5 }   1 0 1 1 0 0 1 1 1   { 4 , 5 , 8 }   0 1 0 1 1 0 0 0 1   { 5 , 7 , 8 } 0 0 0 0 0 0 0 0 0 { 6 , 7 , 8 } Finite Fields, Applications and Open Problems Daniel Panario

  62. 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 } B 0 = { 0 , 1 , 2 } , B 1 = { 3 , 4 , 5 } , B 2 = { 6 , 7 , 8 } OOA ( t = 3 , m = 3 , s = 3 , v = 2) Antiideals of size 3 { 0 , 1 , 2 } 0 1 2 3 4 5 6 7 8   { 1 , 2 , 5 } 1 0 0 1 1 1 0 1 1   { 1 , 2 , 8 }   0 0 1 0 1 1 1 0 0   { 2 , 4 , 5 }   0 1 1 1 0 1 1 0 1   { 2 , 5 , 8 }   1 1 1 0 1 0 1 1 0   { 2 , 7 , 8 }   1 1 0 0 0 1 0 1 0   { 3 , 4 , 5 }   1 0 1 1 0 0 1 1 1   { 4 , 5 , 8 }   0 1 0 1 1 0 0 0 1   { 5 , 7 , 8 } 0 0 0 0 0 0 0 0 0 { 6 , 7 , 8 } Finite Fields, Applications and Open Problems Daniel Panario

  63. 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 } B 0 = { 0 , 1 , 2 } , B 1 = { 3 , 4 , 5 } , B 2 = { 6 , 7 , 8 } OOA ( t = 3 , m = 3 , s = 3 , v = 2) Antiideals of size 3 { 0 , 1 , 2 } 0 1 2 3 4 5 6 7 8   { 1 , 2 , 5 } 1 0 0 1 1 1 0 1 1   { 1 , 2 , 8 }   0 0 1 0 1 1 1 0 0   { 2 , 4 , 5 }   0 1 1 1 0 1 1 0 1   { 2 , 5 , 8 }   1 1 1 0 1 0 1 1 0   { 2 , 7 , 8 }   1 1 0 0 0 1 0 1 0   { 3 , 4 , 5 }   1 0 1 1 0 0 1 1 1   { 4 , 5 , 8 }   0 1 0 1 1 0 0 0 1   { 5 , 7 , 8 } 0 0 0 0 0 0 0 0 0 { 6 , 7 , 8 } Finite Fields, Applications and Open Problems Daniel Panario

  64. 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 } B 0 = { 0 , 1 , 2 } , B 1 = { 3 , 4 , 5 } , B 2 = { 6 , 7 , 8 } OOA ( t = 3 , m = 3 , s = 3 , v = 2) Antiideals of size 3 { 0 , 1 , 2 } 0 1 2 3 4 5 6 7 8   { 1 , 2 , 5 } 1 0 0 1 1 1 0 1 1   { 1 , 2 , 8 }  0 0 1 0 1 1 1 0 0    { 2 , 4 , 5 }   0 1 1 1 0 1 1 0 1   { 2 , 5 , 8 }   1 1 1 0 1 0 1 1 0   { 2 , 7 , 8 }   1 1 0 0 0 1 0 1 0   { 3 , 4 , 5 }   1 0 1 1 0 0 1 1 1   { 4 , 5 , 8 }   0 1 0 1 1 0 0 0 1   { 5 , 7 , 8 } 0 0 0 0 0 0 0 0 0 { 6 , 7 , 8 } Finite Fields, Applications and Open Problems Daniel Panario

  65. 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 OA b t (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 OOA b t ( m − t, s, m − t, b ) . Ordered orthogonal arrays are a combinatorial characterization of ( t, m, s ) -nets. Finite Fields, Applications and Open Problems Daniel Panario

  66. 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 ) . Finite Fields, Applications and Open Problems Daniel Panario

  67. 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 ordered orthogonal array definition. 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, A. Castoldi, L. Moura, D. Panario and B. Stevens, IEEE Transactions on Information Theory, 63, 1336-1347, 2017. A general construction of ordered orthogonal arrays using LFSRs, D. Panario, M. Saaltink, B. Stevens and D. Wevrick, submitted. Finite Fields, Applications and Open Problems Daniel Panario

  68. 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 ) = c 0 + c 1 x + . . . + c t − 1 x t − 1 + x t be a degree- t primitive polynomial over F q = { 0 , β 1 , . . . , β q − 1 } and α ∈ F q t a root of f . Label the columns of the subinterval array M ( f ) by Z k = { 0 , 1 , . . . , k − 1 } , where k = q t − 1 q − 1 . For each i = 1 , . . . , q − 1 , let k β i ∈ Z q t − 1 such that α k βi ( α − β i ) = 1 . Choose the columns of the subinterval array M ( f ) labeled by the following indexes modulo k : Finite Fields, Applications and Open Problems Daniel Panario

  69. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs t + k β 1 t + k β q − 1 t − 1 t t + 2 k β 1 t + 2 k β q − 1 t − 2 t + 1 t + ( t − 1) k β q − 1 2 t − 2 t + ( t − 1) k β 1 1 t + tk β 1 t + tk β q − 1 0 2 t − 1 Finite Fields, Applications and Open Problems Daniel Panario

  70. 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 F t q ? Table: OOA (3 , 4 , 3 , 3) Primitive Polynomial 3-sets covered Percentage 1 + 2 x 2 + x 3 163 0.740909 1 + x + 2 x 2 + x 3 156 0.709091 1 + 2 x + x 3 156 0.709091 1 + 2 x + x 2 + x 3 162 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

  71. 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. Finite Fields, Applications and Open Problems Daniel Panario

  72. Finite Fields Combinatorial Objects Latin Squares and Sudoku Costas Arrays OAs, CAs and OOAs Next: Covering Arrays Finite Fields, Applications and Open Problems Daniel Panario

  73. 2 1 0 1 1 0 2 2 0 2 2 1 2 2 0 2 2 0 1 1 2 1 2 0 1 2 1 0 1 1 0 1 0 1 1 2 0 0 0 0

  74. 2 1 0 1 1 0 2 2 0 2 2 1 2 2 0 2 2 0 1 1 2 1 2 0 1 2 1 0 1 1 0 1 0 1 1 2 0 0 0 0

  75. 2 1 0 1 1 0 2 2 0 2 2 1 2 2 0 2 2 0 1 1 2 1 2 0 1 2 1 0 1 1 0 1 0 1 1 2 0 0 0 0

  76. 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

  77. 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

  78. 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

  79. 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

  80. 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

  81. 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

  82. Example

  83. 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 Arrays from cyclic shifts 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 of m-sequences 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 See survey by Moura, Mullen and Panario 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 (DCC, 2016) 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

  84. 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 Arrays from cyclic shifts 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 of m-sequences 1 1 0 0 1 1 2 2 1 1 1 1 2 2 0 0 1 1 1 1 1 1 0 0 0 0 2 2 0 0 2 2 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 0 0 1 1 2 2 1 1 1 1 2 2 0 0 1 1 1 1 1 1 0 0 0 0 2 2 0 0 2 2 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 See survey by Moura, 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 Mullen and Panario 2 2 1 1 1 1 2 2 0 0 1 1 1 1 1 1 0 0 0 0 2 2 0 0 2 2 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 0 0 1 1 (DCC, 2016) 1 1 1 1 2 2 0 0 1 1 1 1 1 1 0 0 0 0 2 2 0 0 2 2 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 0 0 1 1 2 2 1 1 2 2 0 0 1 1 1 1 1 1 0 0 0 0 2 2 0 0 2 2 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 0 0 1 1 2 2 1 1 2 2 0 0 1 1 1 1 1 1 0 0 0 0 2 2 0 0 2 2 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 0 0 1 1 2 2 1 1 1 1 0 0 1 1 1 1 1 1 0 0 0 0 2 2 0 0 2 2 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 0 0 1 1 2 2 1 1 1 1 2 2 1 1 1 1 1 1 0 0 0 0 2 2 0 0 2 2 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 0 0 1 1 2 2 1 1 1 1 2 2 0 0 1 1 1 1 0 0 0 0 2 2 0 0 2 2 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 0 0 1 1 2 2 1 1 1 1 2 2 0 0 1 1 2 + = 1 1 0 0 0 0 2 2 0 0 2 2 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 0 0 1 1 2 2 1 1 1 1 2 2 0 0 1 1 1 1 0 0 0 0 2 2 0 0 2 2 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 0 0 1 1 2 2 1 1 1 1 2 2 0 0 1 1 1 1 1 1 0 0 2 2 0 0 2 2 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 0 0 1 1 2 2 1 1 1 1 2 2 0 0 1 1 1 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 2 2 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 0 0 1 1 2 2 1 1 1 1 2 2 0 0 1 1 1 1 1 1 0 0 0 0 2 2 0 0 1 1 2 2 2 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 0 0 1 1 2 2 1 1 1 1 2 2 0 0 1 1 1 1 1 1 0 0 0 0 2 2 0 0 2 2 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 1 1 0 0 2 2 2 2 2 2 0 0 0 0 1 1 0 0 1 1 2 2 1 1 1 1 2 2 0 0 1 1 1 1 1 1 0 0 0 0 2 2 0 0 2 2 1 1 2 2 2 2 Linearly dependent 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 columns do not have 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 the CA or OA property 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1 2 2 1 0 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

  85. Arrays from cyclic shifts of m-sequences: Orthogonal arrays

  86. Arrays from cyclic shifts of m-sequences: Orthogonal arrays

  87. Arrays from cyclic shifts of m-sequences: Orthogonal arrays

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