Greedy Codes. Theodore Rice

Transcription

1 Greedy Codes Theodore Rice 1

2 Key ideas These greedy codes come from certain positions of combinatorial games. These codes will be identical to other well known codes. Little sophisticated mathematics is needed to generate them. 2

3 Combinatorial Games A combinatorial game is a two player game of perfect information. The game consists of a set of positions and a set of rules which delineate the allowable moves from each position. The first player unable to move loses. In impartial games the players have the same set of options from each position. 3

4 Game Theory Grundy s game is played with a heap(s) of counters. A move consists of dividing a heap into two non-empty heaps of different sizes. The first player unable to move loses. A general heap game is one where the moves are given by turning sets {h, p 1,..., p n } where a pile of size h may be replaced by piles of size p 1,..., p n. 4

5 Grundy s game is such a game where the turning sets are {3, 1, 2}; {4, 3, 1}; {5, 4, 1}; {5, 3, 2};... We can define a function, called the Grundy value or nim-value of a game (position), G(G) which assigns integer values to positions. G has the following properties: the value of a losing position is 0. The value of a position is the smallest value not already taken by an accessible position. 5

6 The Grundy value of a sum of positions is as follows: G(P 1 +P P n ) = G(P 1 ) G(P 2 )... G(P n ) where is binary addition without carrying. The first few Grundy values of Grundy s game are:

7 A general position may be represented by a vector: (..., a 3, a 2, a 1 ) where a i is the number of piles of size i. Now since two identical piles cancel each other out, (x x = 0) all we need to know is the parity of each number. Therefore we can represent our vectors in binary: (..., ζ 3, ζ 2, ζ 1 ). Winning positions in the game are positions for which ζi G(a i ) = 0. For example 3, 6 is a zero game. 7

8 Codes (at last) We call the collection of winning positions the winning code for the game. It consists of all positions for which the previous player wins the game. The winning code is linear (since 0 0 = 0.) 8

9 word heaps , , ,2,1 9

10 Theorem 1. The winning code for a game is a linear code over GF(2). Although these code words have infinitely many coordinates, we can get a code of any length by only using the code words that vanish outside our desired length. 10

11 For a general heap game in base B, a legal move from M = (ζ n,..., ζ 1 ) to M = (ζ n,..., ζ 1) so that N < N lexicographically and {i ζ i ζ i } is a turning set. Note turning sets only specify where to successive positions differ, not by how much. For each family of turning sets and base, B, we can create a code of vectors (ζ i ), ζ i < B of possible words considered in lexicographic order. 11

12 A word, N = (ζ i ), is rejected if there is an N = (ζ i ) in the code for which {i ζ i ζ i } is a turning set. Otherwise put the word in the code. Such a code is called a lexicode. Theorem 2. For any turning set and any base the winning moves in the game are to move to positions corresponding to the codewords of the lexicode. 12

13 Example 3. Let B = 8 and the turning sets have size 1,2. The code is: This code is linear. So 1456 = is in the code. 13

14 Example 4. Let B = 4, d = 4 where d is the Hamming distance. (Turning sets size 1,2,3) The code is: This is closed under but also by 0,1,2,3 which we will define later. 14

15 This code if truncated at length 6 is the Golay code over GF(4). This is the start of the table for The above code is closed under by 0,1,2,3, so 3 (010123) = (030312) is in the code. 15

16 Theorem 5. If B = 2 the lexicode defined by any family of turning sets is a binary linear code. Proof. This is a consequence of the two previous theorems. 16

17 This theorem can be generalized: Theorem 6. If B = 2 n the lexicode defined for any family of turning sets is linear under componentwise. Proof. We convert base B vectors into base 2 vectors by replacing each digit with its binary representation. That is ζ i is replaced by ζ in,..., ζ i1 The original game becomes a binary game in which 17

18 the turning sets are all T where {[ ] } i : i T n was a turning set in the original game. Turning sets are in groups of n. Now apply Theorem 5 to the new game. If not, then the lexicode is not closed under. 18

19 For example: If B = 4 and we have the vector (..., a 4, a 3, a 2, a 1, a 0 ) = (..., 1, 2, 3, 1, 0) we convert it to the vector ( ). If {4, 2, 1} was a turning set, {9, 8, 5, 4, 3, 2} and {8, 5, 4, 3, 2} will be turning sets. Let s apply the above turning set to the above vector: 19

20 (..., 1, 2, 3, 1, 0) (0, 2, 4, 2, 0) (0, 2, 0, 2, 0) Now in the binary case this is: (..., 0, 1, 1, 0, 1, 1, 0, 1, 0, 0) (..., 0, 0, 1, 0, 0, 0, 1, 0, 0, 0) while applying the binary turning set does the same thing. 20

21 The Nim Field In an additive group, if a a and b b then a + b is not a + b, a + b. We can define on the set of integers as a b = mex a<a,b <b{a + b, a + b} Now if (a a )(b b ) 0, we have ab a b + ab a b or in a field of char 2: ab a b + ab + a b. a b = mex a <a,b <b{(a b) (a b ) (a b )} In this way we get a field of characteristic 2. 21

22 Proposition 7. Properties of the field: 2 n 2 m = 2 n + 2 m and N N = 0. Let N = 2 2n. N n = Nn for n < N. N N = 3 2 N. Theorem 8. If B = 2 2n the lexicode derived over any family of turning sets is closed under component wise, by numbers less than B. That is the field is linear over GF(2 2n ). 22

23 We now look at some particular families of lexicodes in detail. We specify a base, B, the desired mininal Hamming distance, d, and take the turning sets to be all sets of size 1, 2,..., d 1. The lexicode is formed by taking the zero word and repeatedly joining the earliest word a least d from all previous words in the code. If B = 2 a the code is closed under and if B = 2 2a the code is closed under, by scalars. That is it is linear over GF (B). 23

24 To obtain a code of length n, only use codewords that vanish outside the desired size. How good are these codes? Ho do they stack up against other codes? 24

25 Considering binary lexicodes and comparing with other codes, the lexicodes have dimensions with in one or two of the best known codes,and often have the same dimension. We now look at some examples. 25

26 Example 9. Zero-sum codes. For d = 2 and any B the lexicode of length n is the zero-sum code, consisting of all vectors, (a n 1,..., a 2, a 1 ) for which the nim sum ai = 0. In the binary case, the is the even-weight code. 26

27 Example 10. Hamming codes. When B = 2, d = 3 the turning sets have size 1,2 and the game is nim. Nim is played with several heaps of counters. A turn consists of removing some or all of the counters in one heap. The first player unable to move loses. The corresponging lexicodes have length 2 m 1 and correspond to binary Hamming codes. 27

28 Example 11. Similarly, when B = 2, d = 4 and m = 2 m, we obtain extended Hamming codes. This code corresponds to Mock Turtles. Mock Turtles is played with turtles numbered 0,1,2,3... A turn consists of turning over 1,2,3 turtles subject to the condition that the highest numbered turtle turned must be turned from its feet to its back. 28

29 Example 12. Extended Quadratic Redidue code of length 18. The lexicode with B = 2, d = 6, n = 18 is the [18,9,6] binary extended quadratic residue code. This correspond to Mock Turtles where you can turn up to 5 turtles. Example 13. The lexicode with B = 2, d = 8, n = 24 is the [24,12,8] binary Golay code. This corresponds to Mock turtles turning up to 7 turtles. 29

30 Specifying the code For B = 2 n the lexicode may be efficiently specified by giving the values for the G value of a position with a single non-zero digit at coordinate i. This function is written f(ζ, i). We write it in base B. For the position (...0, x, 0, 0, 0) in base 8 (say) we can move to any of (0, x, 0, 0, y); (0, x, 0, y, 0); (0, x, y, 0, 0), where x = 0, 1 and y = 0, 1,..., 7, since turning 30

31 sets are size 1 or 2. Thus f(2, 3) is the mex of f(0, 3) abc = 000 abc f(1, 3) abc = 012 abc Where abc = f(y, 0), f(y, 1), f(y, 2). 31

32 i, ζ Table 1: G values f(ζ, i) for B = 8, d = 3 32

33 Note that an entry in column 3 is the sum of the entries in columns 1,2 and the entry in column 5 is the sum of the entries in columns 1,4. The codewords are the positions where the G value is 0. Example 14. For what values of x, y is xy a code word? Since f(2, 3) = 023 and f(x, 1) = 0x0, f(y, 0) = 00y (from the table), x = 2, y = 3. 33

34 Example 15. For what value of x, y is 024xy in the code? Since f(2, 4) = 023 and f(4, 2) = 044 we have = 067. Thus x = 6, y = 7 as above. These codes are comparable in efficiency with Reed-Solomon codes but exist for all lengths, where Reed-Solomon codes do not. 34

35 The multiplicative property seen earlier makes it easy to construct lexicodes when B = 2 2n since f(ζ, P ) = ζ f(p ) where f(p ) = f(1, P ). Theorem 16. In the case B = 2 2n, d = 3 the lexicodes of length 1 + B + B B m 1 are Hamming codes. 35

36 What if, say, B = 3, d = 3? Unfortunately, this code seems to have little structure. Conway et. al. have been unable to discover any structure in the code or solve to corresponding game. Lexicodes in general have little or no known structure, but for the bases discussed they do and are identical to known codes. 36

37 Fixed Weight Codes We can insist that the code has a fixed weight. Consider all words of the desired weight in lexicographic order and accept a word if it is at Hamming distance from all previous words. I will only consider binary codes. The Hamming distance is necessarily even. 37

38 These codes also correspond to a game: Consider a set {a n,..., a 1 } of distinct integers. A turn consists of decreasing 1, 2,..., t 1 of these, keeping the integers distinct. The first player unable to move loses. Remark: These games will not be described by turning sets. But we can define a winning code. 38

39 Theorem 17. For any d = 2t 4 and any weight w, the winning code for this game is the constant weight lexicode with the same parameters. The proof is analogous to that of previous theorems. Example 18. For d = 2 we get the lexicode of all words of weight w for any w, since they must differ at at least two places. 39

40 Welter s Game Welter s Game is usually played on a semi-infinite strip. Coins are placed at various places on the strip. A move consists of moving the coin down the strip, possibly jumping other coins, with the caveat the two coins may not be on the same square at any time. The first player unable to move loses. Thus the game ends when the coins are jammed in the lowest number squares. 40

41 41

42 Example 19. Consider the constant weight lexicode with d = 4. This is actually the winning code for Welter s game! In fact, this is the only other case other than d = 2 where a complete theory exists. (as of 1986). Welter s Game is actually the case t = 2 of the above situation. Analysis of Welter s Game is more complex than that of other games like Nim, but it is possible. 42

43 We can compute values in Welter s Game using Welter s (remarkable) function. Suppose there are coins at (a 1,..., a n ). Compute W (a 1,..., a n ) as follows: W (a 1 ) = a 1 W (a 1, a 2 ) = (a 1 a 2 ) 1 W (a 1,..., a k+1 ) = W (a 2,..., a k ) ((W (a 1,..., a k ) W (a 2,..., a k+1 )) 1) 43

44 Theorem 20. If W (a 1,..., a w ) = n and n n,!a 1,..., a w so that a 1,..., a w, a 1,..., a w are distinct and: W (a 1, a 2,..., a w ) = n W (a 1, a 2,..., a w ) = n W (a 1, a 2,..., a w) = n. 44

45 More generally, if an even number of a i are replaced by the prime letters, the new vector has W value n. Furthermore an even number a 1 < a 1. a w < a w n < n This is referred to as the Even Alteration Property 45

46 The Even Alteration Property implies the following theorem: Theorem 21. (a 1, a 2,..., a w ) is a winning position for Welter s game, or equivalently, the vector with 1 s in positions a 1,..., a w is in the lexicode with constant weight w and minimal distance 4 iff W (a 1,..., a w ) = 0. 46

47 Proof. When n 0 if we take n = 0 we see there is always possible to move from a position for which W (a 1,..., a w ) 0 to one for which W (a 1,..., a w ) = 0 since 0 = n < n 0 by the Even Alteration Property, we have at least one a i < a i and that is the move. 47

48 To calculate Welter s function, we use a frieze pattern: W (a 1 ) W(a 2 )... W(a k ) W (a k+1 ) W (a 1, a 2 )... W (a k, a k+1 ). W (a 1,..., a k ) W (a 2,..., a k+1 W (a 1,..., a k+1 ) 48

49 Where whenever we have a diamond: A B C D the equality A D = (B C) 1 is satisfied. 49

50 Theorem 17 guarantees that if a vector ζ is not in the code, there is a codeword within Hamming distance 3 occurring before ζ in the lexicographic order. To find the winning move a i a i, and hence decode ζ, we will again use a frieze pattern. I will demonstrate with an example. 50

51 Example 22. Set up the frieze pattern: a b c d e a b c d e n n n n n n 51

52 Suppose we have received the word and we want to decode it. This word corresponds to the position (2, 3, 5, 7, 11) in Welter s Game. To decode, we find the winning move. (If there is no winning move, we have a codeword.) The first order of business to to compute W (2, 3, 5, 7, 11). Note that we will set n = 0. 52

53 Fill in the left half working down using the diamond property: a b c d e n n n n n 53

54 For instance in: ? the? may be computed by realizing that (6 14) 1 = 8 1 = 7. So 1? = 7 or? = 7 1 = 6 as seen in the table. When we get to the bottom, we see that W (2, 3, 5, 7, 11) = 4. We want to decode to a word of value 0. So let n = 0, n = 4. 54

55 a b c d e

56 Now work up the table until all of a, b, c, d, e are found. For instance the first dot can be computed as follows: 14 6 Now 14 0 = (6 ) 1 so 15 = 6. So = 6 15 = 9. Fill in the array in a similar way. 0 56

57 So: a = 6, b = 23, c = 1, d = 19, e = 15. The only one of the primes less than the non primes is c < c. Thus moving from c to c is the unique winning move. 57

58 Thus we move from (2, 3, 5, 7, 11) to (2, 3, 1, 7, 11) = (1, 2, 3, 7, 11) Thus the word is decoded to

59 Generalizations So far we have been considering vectors in the standard lexicographic order. What if we consider them in a different order? We do in fact get codes similar to the ones we got before. The first question is how to determine an order. 59

60 B-orders Let B = {y 1, y 2,..., y n } be an order basis of F2 n. This basis induces an order on the vectors of F2 n defined recursively: Let V 0 = (0, 0,..., 0) and let V 1 = v 1, v 2,..., v i. Now suppose the vectors in V i 1 are ordered, x 1, x 2,..., x m where m = 2 n 1. Then we have the partition V i = V i 1 (y i V i 1 ). 60

61 We order the vectors x 1, x 2,..., x m, y i x 1, y 2, x 2,..., y i x m. Call this order the B-order of F2 n. In this way we can obtain an order of F2 n than the lexicographic order other 61

62 Example 23. Suppose we take B = (0,..., 0, 1); (0,..., 1, 0);,..., (1, 0,..., 0). In this case the B-order is the standard lexicographic order Example 24. Let B = (0,..., 0, 1); (0,..., 1, 1); (0,..., 1, 1, 0);...; (1, 1, 0,..., 0). In this case the B-order corresponds to the reflected Gray code of order n. Call B the Gray ordered basis of F n 2. 62

63 Both the lexicographic and Gray orders are what may be called triangular orders of F n 2. Consider an order basis B = y 1, y 2,..., y n of F n 2 for which each V i = y 1,..., y i is the coordinate subspace consisting of all n-tuples with 0 s in the n i leftmost positions. 63

64 y 1 y 2. y n = Such a basis is called a triangular ordered basis of F n 2. 64

65 We can now take an ordered basis B of F2 n and take 0 d n to be a desired distance. We apply a greedy algorithm as before to the B order to obtain a code with minimal distance d. The code obtained is a greedy code. The lexicodes from before are a special types of greedy codes. 65

66 Example 25. Let the length be 5 and the desired distance be 3. We chose our basis to be B = (0, 0, 0, 0, 1); (0, 0, 0, 1, 1); (0, 0, 1, 1, 0); (0, 1, 1, 0, 0); (1, 1, 0, 0, 0). The code obtained is 00000, 00111, 11001,

67 We can define an analog of the G value on this order. Define g : F2 n Z 0 as follows. Assume Fn 2 has been ordered z a, z 2,..., z 2 n. Define g(z 1 ) = 0. Then define g(z i ) to be the smallest t such that ρ(z i, x) d for all x {z 1,..., z i 1 } such that g(x) = t. If there is no such t, let g(z) = mex(g(z 1 ),..., g(z i 1 )). 67

68 That is g(z i ) is the smallest value possible so that z i is d away from all other vectors with the same value. The greedy code is the set {z F 2 n : g(z) = 0}. The smallest integer m, such that g(z) 2 m 1 for all z F2 n is called the index of the given ordering F2 m relative to the distance d. Each g(z) can be seen as a vector in F2 m. 68

69 Hence we can see g as a map g : F2 n F2 m. Here, F2 m is ordered in the lexicographic way. It can be checked that the set of all vectors with a particular g-value is a coset of the Gray-greedy code. Thus g : F2 5 F2 3 is a homomorphism, whose kernel is the code. 69

70 The 3 by 5 matrix: [ ] H = g(e 5 ) g(e 4 ) g(3 2 ) g(e 2 ) g(e 1 ) is a parity check matrix for the code. The g value of a vector is its syndrome relative to this check matrix H =

71 Example 26. Suppose we receive z = = A syndrome of 011 corresponds to an error is the second place so decodes to

72 Linearity of Greedy Codes Theorem 27. Let B be an ordered basis of F n 2 and let d be an integer with 0 d n. Let m be the index of the B-order of F n 2. Then g : F n 2 F m 2 is a surjective homomorphism whose kernel equals the B-greedy code C of length n and distance d. In particular, C is a linear code of dimension n m. 72

73 A parity check matrix for C is given by: [ ] H = g(e n )... g(e 2 ) g(e 1 ) and for each x F n 2, g(z) is the syndrome of z relative to H 73

74 We observe that every linear code C with minimal distance d and covering radius d 1 is a B-greedy code of designated distance d and some ordered basis B. We may choose ; B to be any ordered basis whose first k vectors are a basis of C where k is the dimension of C. Since B-greedy codes of distance d has covering radius d 1 they obtain the Varshamov-Gilbert bound for binary linear codes. 74

75 If y F n 2, then let ŷ F n+1 2 be the vector obtained from y be adding a parity check digit. Theorem 28. Let B = y 1,..., y n be an ordered basis of F2 n and let d be odd. Let z be an odd weight vector of F2 n+1 and let B be the ordered basis ŷ 1,..., ŷ n, z. Then the B -greedy code of distance d + 1 is obtained from the B greedy code of distance d by adding a parity check. 75

76 We can play a game analogous to these codes as well. The game is played on the binary n-tuples. A legal move consists of moving to a position earlier in the B order subject to the condition that the new vector is with in d of the old. The last player to move wins. 76

77 Note that if g(z) = 0 then the only moves are to vectors x such that g(x) 0. And if g(z) 0, there is a vector x such that g(x) = 0. So the winning positions of the game are exactly the 0 positions which form the code. 77

78 Conclusions There is a strong connection between games and codes. Using greedy algorithms allows is to get familiar codes. The greedy algorithms are easy to understand and implement. 78

79 Buraldi, Richard and Pless, Vera Greedy Codes Journal of Combinatorial Theory Ser A 1993 pp Conway, John H. and Sloane N.J.A. Lexicographic Codes: Error-Correcting Codes from Game Theory IEEE Transactions on Information Theory May 1986 pp Conway, John H Integral Lexicographic Codes Discrete Mathematics 1990 pp