Arrangements, matroids and codes

Transcription

1 Arrangements, matroids and codes first lecture Ruud Pellikaan joint work with Relinde Jurrius ACAGM summer school Leuven Belgium, 18 July 2011

2 References 2/43 1. Codes, arrangements and matroids by Relinde Jurrius and Ruud Pellikaan, in Series on Coding Theory and Cryptology vol. 8 Algebraic geometry modelling in information theory E. Martinez-Moro Ed., World Scientific ruudp/paper/57.pdf 2. Error-correcting codes and cryptology by R. Pellikaan, X.-W. Wu and S. Bulygin Book in preparation, February 2011 To be published by Cambridge University Press ruudp/courses/2wc11/2wc11-book.pdf

3 Introduction 3/43 M A C C = error-correcting codes (extended) weight enumerator polynomial W C (X, Y), W C (X, Y, T) M = matroids generalization of linear algebra and graph theory (di)chromatic polynomial and Tutte polynomial t M (X, Y) A = arrangements of hyperplanes topology, combinatorics characteristic polynomial χ(t), coboundary polynomial χ(s, T)

4 Content 5 lectures 4/43 1. Error-correcting codes Weight enumerator 2. q-ary symmetric channel and the probability of undetected error Arrangements and projective systems 3. Extended weight enumerator Graph theory and colorings 4. Matroids Tutte-Whitney polynomial 5. Geometric lattices Characteristic polynomial

5 Content first lecture 5/43 1. Error-correcting codes: Shannon, Hamming distance 2. Linear codes: Generator and parity check matrix, inner product and dual code Hamming and simplex codes 3. Singleton bound and MDS codes: Vandermonde matrices and generalized Reed-Solomon codes 4. Weight enumerator: MacWilliams identity and examples 5. Exercises

6 6/43 Error-correcting codes

7 Shannon s block diagram 7/43 message sender source encoding receiver message target decoding noise Shannon s block diagram of a communication system

8 Hamming code 8/43 4 message bits: (m 1, m 2, m 3, m 4 ) 3 redundant bits: (r 1, r 2, r 3 ) Rule: number of ones in every circle is even r 3 m 2 m 1 m 4 r 1 m 3 r 2 Venn diagram of the Hamming code

9 Block codes 9/43 The message words have a fixed length of k symbols the encoded words have a fixed length of n symbols both from the same alphabet Q Add redundant symbols to the message in a clever way An error-correcting code C of length n over Q is a non-empty subset of Q n The elements of C are called codewords If C contains M codewords, then M is called the size n log q (M) is called the redundancy R = log q (M)/n is the information rate

10 Block codes 9/43 The message words have a fixed length of k symbols the encoded words have a fixed length of n symbols both from the same alphabet Q Add redundant symbols to the message in a clever way An error-correcting code C of length n over Q is a non-empty subset of Q n The elements of C are called codewords If C contains M codewords, then M is called the size n log q (M) is called the redundancy R = log q (M)/n is the information rate

11 Block codes 9/43 The message words have a fixed length of k symbols the encoded words have a fixed length of n symbols both from the same alphabet Q Add redundant symbols to the message in a clever way An error-correcting code C of length n over Q is a non-empty subset of Q n The elements of C are called codewords If C contains M codewords, then M is called the size n log q (M) is called the redundancy R = log q (M)/n is the information rate

12 Hamming distance 10/43 The Hamming distance d(x, y) on Q n is defined by d(x, y) = {i : x i = y i } It is a metric: 1. d(x, y) 0 and equality hods if and only if x = y 2. d(x, y) = d(y, x) (symmetry) 3. d(x, z) d(x, y) + d(y, z) (triangle inequality) x y d(y,z) d(x,y) z d(x,z)

13 Hamming distance 10/43 The Hamming distance d(x, y) on Q n is defined by d(x, y) = {i : x i = y i } It is a metric: 1. d(x, y) 0 and equality hods if and only if x = y 2. d(x, y) = d(y, x) (symmetry) 3. d(x, z) d(x, y) + d(y, z) (triangle inequality) x y d(y,z) d(x,y) z d(x,z)

14 Minimum distance 11/43 The minimum (Hamming) distance of a code C is defined as d = d(c) = min{d(x, y) : x, y C, x = y} The main problem of error-correcting codes from Hamming s point of view is: to construct for a given length and number of codewords a code with the largest possible minimum distance and to find efficient encoding and decoding algorithms

15 Minimum distance 11/43 The minimum (Hamming) distance of a code C is defined as d = d(c) = min{d(x, y) : x, y C, x = y} The main problem of error-correcting codes from Hamming s point of view is: to construct for a given length and number of codewords a code with the largest possible minimum distance and to find efficient encoding and decoding algorithms

16 Examples 12/43 The triple repetition binary code has length 3 and 2 codewords its information rate is 1/3 its minimum distance is 3. The binary Hamming code has length 7 and 2 4 codewords therefore its rate is 4/7 its minimum distance is 3

17 13/43 Linear codes

18 Linear codes 14/43 If the alphabet Q is the finite field F q with q elements then Q n is a vector space Therefore it is natural to look at codes in Q n that are linear subspaces A linear code C is a linear subspace of F n q Its dimension is denoted by k = k(c) and its minimum distance by d = d(c) Then [n, k, d] q or [n, k, d] denote the parameters of the code Size: M = q k Information rate: R = k/n Redundancy: n k

19 Support and minimal weight 15/43 The support of x in F n q is defined by supp(x) = {j : x j = 0} The weight of x is defined by wt(x) = supp(x) that is the number of nonzero entries of x Let C be an F q -linear code, then d(c) = min{wt(c) : 0 = c C}

20 Support and minimal weight 15/43 The support of x in F n q is defined by supp(x) = {j : x j = 0} The weight of x is defined by wt(x) = supp(x) that is the number of nonzero entries of x Let C be an F q -linear code, then d(c) = min{wt(c) : 0 = c C}

21 Generator and parity check matrix 16/43 C an F q -linear code of dimension k A k n matrix G with entries in F q is called generator matrix of C if C = { xg x F k q } A (n k) n matrix H with entries in F q is called a parity check matrix of C if C = { c F n q ch T = 0 }

22 Generator and parity check matrix 16/43 C an F q -linear code of dimension k A k n matrix G with entries in F q is called generator matrix of C if C = { xg x F k q } A (n k) n matrix H with entries in F q is called a parity check matrix of C if C = { c F n q ch T = 0 }

23 Example 17/43 The binary Hamming code with parameters [7, 4, 3] has generator matrix G: G = and parity check matrix H: H =

24 Example 17/43 The binary Hamming code with parameters [7, 4, 3] has generator matrix G: G = and parity check matrix H: H =

25 Proposition 18/43 Suppose C is a [n, k] code Let I k be the k k identity matrix Let P be a k (n k) matrix Then (I k P) is a generator matrix of C if and only if ( P T I n k ) is a parity check matrix of C

26 Proposition 18/43 Suppose C is a [n, k] code Let I k be the k k identity matrix Let P be a k (n k) matrix Then (I k P) is a generator matrix of C if and only if ( P T I n k ) is a parity check matrix of C

27 Inner product and dual code 19/43 The inner product on F n q is defined by x y = x 1 y x n y n This inner product is bilinear, symmetric and nondegenerate but the notion of positive definite makes no sense over a finite field For an [n, k] code C we define the dual or orthogonal code C as C = {x F n q : c x = 0 for all c C}.

28 Inner product and dual code 19/43 The inner product on F n q is defined by x y = x 1 y x n y n This inner product is bilinear, symmetric and nondegenerate but the notion of positive definite makes no sense over a finite field For an [n, k] code C we define the dual or orthogonal code C as C = {x F n q : c x = 0 for all c C}.

29 Generator and parity check matrix of dual code 20/43 Proposition If C has length n, then dim(c) + dim(c ) = n Furthermore G is a generator matrix of C if and only if G is a parity check matrix of C H is a parity check matrix of C if and only if H is a generator matrix of C

30 Generator and parity check matrix of dual code 20/43 Proposition If C has length n, then dim(c) + dim(c ) = n Furthermore G is a generator matrix of C if and only if G is a parity check matrix of C H is a parity check matrix of C if and only if H is a generator matrix of C

31 Parity check matrix and minimum distance 21/43 Proposition Let H be a parity check matrix of a code C Then the minimum distance d of C is the smallest integer d such that d columns of H are linearly dependent Proof H parity check matrix with entries h i,j c a codeword of weight d with nonzero entries c ji Hc T = 0 c j1 c jd h 1,j1 h 1,jd.. h (n k),j1 h (n k),jd

32 Parity check matrix and minimum distance 21/43 Proposition Let H be a parity check matrix of a code C Then the minimum distance d of C is the smallest integer d such that d columns of H are linearly dependent Proof H parity check matrix with entries h i,j c a codeword of weight d with nonzero entries c ji Hc T = 0 c j1 c jd h 1,j1 h 1,jd.. h (n k),j1 h (n k),jd

33 Nondegenerate codes 22/43 Let G be a generator matrix of C C is called nondegenerate if for every position j there is codeword c such that c j = 0 The following statements are equivalent: 1. C is nondegenerate 2. G has no zero column 3. d(c ) 2

34 Nondegenerate codes 22/43 Let G be a generator matrix of C C is called nondegenerate if for every position j there is codeword c such that c j = 0 The following statements are equivalent: 1. C is nondegenerate 2. G has no zero column 3. d(c ) 2

35 q-ary Hamming and simplex codes 23/43 Let n = (q r 1)/(q 1) Let H r (q) be a r n matrix over F q such that no two columns are dependent The code H r (q) with H r (q) as parity check matrix is called a q-ary Hamming code The code with H r (q) as generator matrix is called a q-ary simplex code and is denoted by S r (q) S r (q) and H r (q) are dual codes

36 q-ary Hamming and simplex codes 23/43 Let n = (q r 1)/(q 1) Let H r (q) be a r n matrix over F q such that no two columns are dependent The code H r (q) with H r (q) as parity check matrix is called a q-ary Hamming code The code with H r (q) as generator matrix is called a q-ary simplex code and is denoted by S r (q) S r (q) and H r (q) are dual codes

37 Parameters of Hamming and Simplex codes 24/43 Let r 2 The q-ary Hamming code H r (q) has parameters [(q r 1)/(q 1), (q r 1)/(q 1) r, 3] The q-ary simplex code S r (q) is a constant weight code with parameters [(q r 1)/(q 1), r, q r 1 ]

38 Parameters of Hamming and Simplex codes 24/43 Let r 2 The q-ary Hamming code H r (q) has parameters [(q r 1)/(q 1), (q r 1)/(q 1) r, 3] The q-ary simplex code S r (q) is a constant weight code with parameters [(q r 1)/(q 1), r, q r 1 ]

39 Example S 3 (3) 25/43 Consider the ternary Hamming code S 3 (3) of redundancy 3 of length 13 with parity check matrix H 3 (3) = The code H 3 (3) has parameters [13, 10, 3] The code S 3 (3) has parameters [13, 3, 9] All rows of H 3 (3) have weight 9 In fact all nonzero codewords have weight 9.

40 Example S 3 (3) 25/43 Consider the ternary Hamming code S 3 (3) of redundancy 3 of length 13 with parity check matrix H 3 (3) = The code H 3 (3) has parameters [13, 10, 3] The code S 3 (3) has parameters [13, 3, 9] All rows of H 3 (3) have weight 9 In fact all nonzero codewords have weight 9.

41 26/43 Singleton bound and MDS codes

42 Singleton bound 27/43 Theorem If C is an [n, k, d] code, then d n k + 1 Proof Let H be a parity check matrix of C This is an (n k) n matrix There exist n k + 1 dependent columns in fact every (n k + 1)-tuple of columns is dependent Now d(c) is the smallest integer d such that H has d linearly dependent columns So d n k + 1

43 Singleton bound 27/43 Theorem If C is an [n, k, d] code, then d n k + 1 Proof Let H be a parity check matrix of C This is an (n k) n matrix There exist n k + 1 dependent columns in fact every (n k + 1)-tuple of columns is dependent Now d(c) is the smallest integer d such that H has d linearly dependent columns So d n k + 1

44 MDS codes 28/43 Definition Let C be a [n, k, n k + 1] code Then C is called a maximum distance separable (MDS) code From the Singleton bound, an MDS code achieves the maximum possible value for the minimum distance given its length and dimension Examples The minimum distance of the zero code of length n is n + 1, by definition Hence the zero code has parameters [n, 0, n + 1] and is MDS Its dual is the whole space F n q with parameters [n, n, 1] and is also MDS The n-fold repetition code has parameters [n, 1, n] and its dual is an [n, n 1, 2] code and both are MDS

45 MDS codes 28/43 Definition Let C be a [n, k, n k + 1] code Then C is called a maximum distance separable (MDS) code From the Singleton bound, an MDS code achieves the maximum possible value for the minimum distance given its length and dimension Examples The minimum distance of the zero code of length n is n + 1, by definition Hence the zero code has parameters [n, 0, n + 1] and is MDS Its dual is the whole space F n q with parameters [n, n, 1] and is also MDS The n-fold repetition code has parameters [n, 1, n] and its dual is an [n, n 1, 2] code and both are MDS

46 MDS codes 28/43 Definition Let C be a [n, k, n k + 1] code Then C is called a maximum distance separable (MDS) code From the Singleton bound, an MDS code achieves the maximum possible value for the minimum distance given its length and dimension Examples The minimum distance of the zero code of length n is n + 1, by definition Hence the zero code has parameters [n, 0, n + 1] and is MDS Its dual is the whole space F n q with parameters [n, n, 1] and is also MDS The n-fold repetition code has parameters [n, 1, n] and its dual is an [n, n 1, 2] code and both are MDS

47 Dual of an MDS code 29/43 Proposition Let C be an [n, k, d] code over F q Then the following statements are equivalent: 1. C is an MDS code, 2. every n k columns of a parity check matrix H of C are linearly independent, 3. every k columns of a generator matrix G of C are linearly independent. Corollary The dual of an MDS code is again MDS

48 Dual of an MDS code 29/43 Proposition Let C be an [n, k, d] code over F q Then the following statements are equivalent: 1. C is an MDS code, 2. every n k columns of a parity check matrix H of C are linearly independent, 3. every k columns of a generator matrix G of C are linearly independent. Corollary The dual of an MDS code is again MDS

49 Vandermonde matrices and MDS codes 30/43 Proposition Let n q Let a = (a 1,..., a n ) be an n-tuple of mutually distinct elements of F q Let k be an integer such that 0 k n Define the matrix G k (a) by 1 1 a 1 a n G k (a) =..... a1 k 1 an k 1 The code with generator matrix G k (a) is MDS Proof All k k submatrices are Vandermonde matrices and their determinants are not zero

50 Vandermonde matrices and MDS codes 30/43 Proposition Let n q Let a = (a 1,..., a n ) be an n-tuple of mutually distinct elements of F q Let k be an integer such that 0 k n Define the matrix G k (a) by 1 1 a 1 a n G k (a) =..... a1 k 1 an k 1 The code with generator matrix G k (a) is MDS Proof All k k submatrices are Vandermonde matrices and their determinants are not zero

51 Reed-Solomon codes 31/43 f(x) = f 0 + f 1 X + + f i X i + + f k 1 X k 1 F q [X] a 1 a n f f 1 a 1 a n X f i a1 i an i X i f k 1 a1 k 1 an k 1 f(a 1 ) f(a n ) f(x) X k 1 The linear combination of the rows of G k (a) is equal to the evaluation of f(x) at a 1,..., a n

52 Reed-Solomon codes 31/43 f(x) = f 0 + f 1 X + + f i X i + + f k 1 X k 1 F q [X] a 1 a n f f 1 a 1 a n X f i a1 i an i X i f k 1 a1 k 1 an k 1 f(a 1 ) f(a n ) f(x) X k 1 The linear combination of the rows of G k (a) is equal to the evaluation of f(x) at a 1,..., a n

53 Generalized Reed-Solomon codes 32/43 a = (a 1,..., a n ) an n-tuple of mutually distinct elements of F q b = (b 1,..., b n ) an n-tuple of nonzero elements of F q GRS k (a, b) = { (f(a 1 )b 1,..., f(a n )b n ) f(x) F q [X], deg(f(x)) < k } Generator matrix: G k (a, b) = b 1 b j b n a 1 b 1 a j b j a n b n... a1 k 1 b 1 a k 1 j b j an k 1 b n MDS code with parameters: [n, k, n k + 1] if k n

54 Generalized Reed-Solomon codes 32/43 a = (a 1,..., a n ) an n-tuple of mutually distinct elements of F q b = (b 1,..., b n ) an n-tuple of nonzero elements of F q GRS k (a, b) = { (f(a 1 )b 1,..., f(a n )b n ) f(x) F q [X], deg(f(x)) < k } Generator matrix: G k (a, b) = b 1 b j b n a 1 b 1 a j b j a n b n... a1 k 1 b 1 a k 1 j b j an k 1 b n MDS code with parameters: [n, k, n k + 1] if k n

55 33/43 Weight enumerator

56 Weight enumerator 34/43 Let C be a code of length n The weight spectrum or weight distribution is the set {(w, A w ) : w = 0, 1,..., n} where A w denotes the number of codewords in C of weight w The weight enumerator is the polynomial: W C (Z) = n A w Z w. w=0 The homogeneous weight enumerator is: n W C (X, Y) = A w X n w Y w. w=0

57 Weight enumerator 34/43 Let C be a code of length n The weight spectrum or weight distribution is the set {(w, A w ) : w = 0, 1,..., n} where A w denotes the number of codewords in C of weight w The weight enumerator is the polynomial: W C (Z) = n A w Z w. w=0 The homogeneous weight enumerator is: n W C (X, Y) = A w X n w Y w. w=0

58 Example: trivial codes 35/43 The zero code has one codeword and its weight is zero Hence W {0} (X, Y) = X n The number of words of weight w in the trivial code F n q is So W F n q (X, Y) = n w=0 A w = ( ) n (q 1) w w ( ) n (q 1) w X n w Y w = (X + (q 1)Y) n w

59 Example: trivial codes 35/43 The zero code has one codeword and its weight is zero Hence W {0} (X, Y) = X n The number of words of weight w in the trivial code F n q is So W F n q (X, Y) = n w=0 A w = ( ) n (q 1) w w ( ) n (q 1) w X n w Y w = (X + (q 1)Y) n w

60 Example: repetition and even weight code 36/43 The n-fold repetition code C has homogeneous weight enumerator W C (X, Y) = X n + (q 1)Y n In the binary case its dual is the even weight code Hence it has homogeneous weight enumerator: W C (X, Y) = n/2 t=0 ( ) n X n 2t Y 2t = 1 ( (X + Y) n + (X Y) n) 2t 2

61 Example: repetition and even weight code 36/43 The n-fold repetition code C has homogeneous weight enumerator W C (X, Y) = X n + (q 1)Y n In the binary case its dual is the even weight code Hence it has homogeneous weight enumerator: W C (X, Y) = n/2 t=0 ( ) n X n 2t Y 2t = 1 ( (X + Y) n + (X Y) n) 2t 2

62 Example: [7, 4, 3] binary Hamming code 37/43 The nonzero entries of the weight distribution are A 0 = 1, A 3 = 7, A 4 = 7, A 7 = 1 by inspection of all the 16 codewords Hence its homogeneous weight enumerator is X 7 + 7X 4 Y 3 + 7X 3 Y 4 + Y 7

63 Example: [7, 4, 3] binary Hamming code 37/43 The nonzero entries of the weight distribution are A 0 = 1, A 3 = 7, A 4 = 7, A 7 = 1 by inspection of all the 16 codewords Hence its homogeneous weight enumerator is X 7 + 7X 4 Y 3 + 7X 3 Y 4 + Y 7

64 Example: simplex code S r (q) 38/43 This is a constant weight code with parameters [(q r 1)/(q 1), r, q r 1 ] Hence its homogeneous weight enumerator is W Sr (q)(x, Y) = X n + (q r 1)X n qr 1 Y qr 1

65 Example: simplex code S r (q) 38/43 This is a constant weight code with parameters [(q r 1)/(q 1), r, q r 1 ] Hence its homogeneous weight enumerator is W Sr (q)(x, Y) = X n + (q r 1)X n qr 1 Y qr 1

66 MacWilliams identity 39/43 Theorem Let C be a [n, k] code over F q Then W C (X, Y) = q k W C (X + (q 1)Y, X Y) Proof Several proofs are known One will be given at the end by means of the Tutte polynomial

67 MacWilliams identity 39/43 Theorem Let C be a [n, k] code over F q Then W C (X, Y) = q k W C (X + (q 1)Y, X Y) Proof Several proofs are known One will be given at the end by means of the Tutte polynomial

68 Example: dual repetition code 40/43 The n-fold repetition code C has homogeneous weight enumerator W C (X, Y) = X n + (q 1)Y n Using MacWilliams identity gives the homogeneous weight enumerator of its dual: W C (X, Y) = q 1 W C (X + (q 1)Y, X Y) = q 1 ((X + (q 1)Y) n + (q 1)(X Y) n ) n ( ) n (q 1) w + (q 1)( 1) w = X n w Y w w q w=0

69 Example: dual repetition code 40/43 The n-fold repetition code C has homogeneous weight enumerator W C (X, Y) = X n + (q 1)Y n Using MacWilliams identity gives the homogeneous weight enumerator of its dual: W C (X, Y) = q 1 W C (X + (q 1)Y, X Y) = q 1 ((X + (q 1)Y) n + (q 1)(X Y) n ) n ( ) n (q 1) w + (q 1)( 1) w = X n w Y w w q w=0

70 41/43 Exercises

71 Exercises first lecture 42/43 1. Check MacWilliams identity for the binary [7, 4, 3] Hamming code and its dual the [7, 3, 4] simplex code. 2. Compute the weight enumerators of the ternary Hamming H 3 (3) code and its dual the Simplex S 3 (3) code. Show that MacWilliams identity holds. 3. Compute the weight spectrum of the dual of the q-ary n-fold repetition code directly, without using MacWilliams identity. Compare this result as given in the lecture. 4. Consider the code with generator matrix (I k I k ). Show that its weight enumerator is equal (X 2 + (q 1)Y 2 ) k. Show that the code is self dual if and only if q is even. Verify that this code is formally self-dual, that is the code and its dual have the same weight enumerator.

72 Exercises first lecture 43/43 5. Let C be the code over F q with generator matrix G given by G = Determine the finite fields F q such that the code C contains a word of weight 7?