Codes, matroids and trellises occur at several levels. Do you (a) - PDF document

✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✂ ✂ ✂ ✂ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✄ ✄ ✄ ✄ ✁ ✁ ✁ ✁ ✁ ✁ � ✁ � � ✁ ✁ � ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ Factorial design You are investigating a process whose yield is affected by a number of factors, each of which can Codes, matroids and trellises occur at several levels. Do you (a) change one factor at a time? Peter J Cameron (with many contributions from C. Papadopoulos, R. A. Bailey and C. G. Rutherford) School of Mathematical Sciences Queen Mary and Westfield College London E1 4NS p.j.cameron@qmw.ac.uk (b) Use a design? Combinatorics 2000, Gaeta 1 3 Who discovered the Hamming codes? Was it Connections R. W. Hamming? Linear codes and factorial designs are almost the same concept, even though their theories have developed quite separately. M. J. E. Golay? Similarly, representations of matroids and point sets R. A. Fisher? in projective spaces are almost the same concept. The theme of these lecture is that in fact the two J. J. Sylvester? concepts just descried are almost the same . See “Hamming and Golay, Fisher and Bose” on this Web page for more about this. 2 4

✝ ✟ � ✝ � ✌ ✠ Codes A linear code C of length n and dimension k over a Exchange axiom field F is a k -dimensional subspace of F n . The weight ☎ v F n is the number of non-zero wt ✆ of a word v The exchange axiom states: If A and B are coordinates, and the minimum weight of C is the ✎ B ✎✑✏✒✎ A independent sets such that ✎ , then there exists smallest weight of a non-zero vector in C . b B ✓ A such that A ✔✖✕ b ✗ is independent. Codes C and C ✞ are monomial equivalent if C ✞ is This guarantees that all bases have the same obtained from C by permuting the coordinates and cardinality, and so makes the definition of rank multiplying them by non-zero scalars. sensible. Theorem 1 A code with minimum weight d can ☎ d 1 ✆☛✡ 2 correct up to ☞ errors. 5 7 The code-matroid connection Matroids Let A be a k ✘ n matrix over a field F having rank k . A matroid M on a set E is a family of subsets of E From A we construct called independent sets , closed under taking subsets and satisfying the exchange property. ☎ A a code C ✆ generated by the rows of A ; ☎ A The rank ρ ✆ of a subset A of E is the size of the largest independent subset of A . An independent ☎ A ☎ E ✆ represented in F k by the columns a matroid M subset of E of size ρ ✆ is called a basis of M . of A . ☎ k Example . The uniform matroid U ✍ n ✆ : the independent sets are all subsets having cardinality at The equivalence relation on such matrices given by most k . arbitary row operations and monomial column operations mirrors the natural notions of equivalence A representation of E over a field F is a map of E into for linear codes and representations of matroids. an F -vector space which preserves independence. Two representations are equivalent if they are related Note to geometers: Linear MDS codes in projective ☎ q by an invertible linear transformation between the space over GF ✆ correspond to representations of ☎ n ☎ q vector spaces. the uniform matroid U ✍ k ✆ over GF ✆ . 6 8

✜ ✧ ✭ ✗ ✜ ✠ ✠ ✯ ✝ ✝ ✣ ✡ ✝ ✛ ✝ ✣ ✝ ✝ ✧ Internal and external activity There is an equivalent definition as follows. Suppose First and last base that M is a matroid on the set E , which is totally ordered. Let B be a base of M . An element b B is Let the ground set E of the matroid M be totally internally active with respect to B if, for all c B , we ☎ E ordered, and let ρ k . Let have B ✔✱✕ c ✗✚✓✲✕ b c b . The internal activity of be the set of bases ✆✚✙ ✗✳✝✖✛✵✴ of M . When we write a base as ✕ b 1 ✍☛✜✢✜☛✜☛✍ b k ✗ , we a base is the number of internally active elements assume that b 1 b k . associated with it. ✣✥✤☛✤☛✤✦✣ Dually, an element e B is externally active with The (lexicographically) first base F ✙✒✕ f 1 ✍☛✜☛✜☛✜✢✍ f k respect to B if, for all f B , we have satisfies f i b i for any base B ✙✒✕ b 1 ✍☛✜☛✜✢✜☛✍ b k ✗ . ☎ e f ✝ C ✍ B f e .The external activity of a base is ✆✚✴ the number of externally active elements associated Dually the last base L ✙★✕ l 1 ✍☛✜☛✜✢✜☛✍ l k ✗ satisfies b i l i for with it. any base B ✙✒✕ b 1 ✍☛✜☛✜☛✜✢✍ b k ✗ . Then we have These properties express the relationship of matroids ☎ M ; x ∑ ✸ j x i y i T ✍ y t i to the greedy algorithm . ✆✩✙ B ✪✷✶ where t i ✸ j is the number of bases with i internally active elements and j externally active elements. 9 11 First and last; internal and external A loop in a matroid is an element e E which is contained in no basis. Weight enumerator and Tutte polynomial A coloop is an element e E which is contained in The weight enumerator of a code C of length n is every basis. given by ☎ x ✭ y wt ✫ wt ∑ x n ✬ c ✬ c W C ✍ y Note that ✆✩✙ c ✪ C The Tutte polynomial of a matroid M on E with rank (a) The internal activity of the first base is the function ρ is given by number of coloops of M , while its external ☎ E ρ ☎ M ; x ☎ x ☎ y ✎ E activity is equal to ✆ . ∑ ✆ ρ E ✫ ρ A ✆✰✯ A ✫ ρ A 1 1 T ✍ y ✎✹✠ ✆✩✙ A ✮ E ☎ E (b) The internal activity of the last base is ρ ✆ , while its external activity is equal to the number of loops of M . 10 12

✻ ☎ ✺ ✻ ✠ ✠ ✼ ✜ ✠ ✝ ✙ ✠ ✙ ✙ ❄ ❂ ✜ ✙ ✠ ✭ An example Greene’s Theorem Suppose that we are using the binary dual Hamming code of length 7 to send information. The codewords Curtis Greene showed in 1975 that the weight ☎ A are: enumerator of C C ✆ is a specialisation of the ☎ A 0000000 Tutte polynomial of M M ✆ : 0011011 0101101 Theorem 2 ☎ q 0110110 ☎ x ☎ x M ; x 1 ✆ y ✍ x ✭ T y n ✫ dim ✬ C ✆ dim ✬ C W C ✍ y y 1001110 x y y ✆✩✙ 1010101 1100011 ☎ A In particular, the Tutte polynomial of M ✆ determines 1111000 ☎ A the minimum weight of C ✆ . The minimum weight is 4 , so we can correct one error and detect two errors. 13 15 Analog errors Duality In practice, the received word is an analog signal, The dual of a matroid M on E is the matroid M ✽ on E sampled at seven time points, i.e. seven real whose bases are the complements of the bases of M . numbers. Suppose that we receive 7 The dual of a code C is the code ✠ 0 ✜ 1 ✍ 0 ✜ 0 ✍ 0 ✜ 2 ✍ 0 ✜ 9 ✍ 1 ✜ 8 ✍ 0 ✜ 9 ✍ 1 ✜ 4 w ✆✚✝✖❄ F n : v 0 for all c C ✾✿✙✒✕ v ✤ c ✝ C If we round each value to the nearest of zero and ✗❀✍ one, we obtain 0001111 , which is at distance 2 from where ✤ is the usual dot product. the second, third and fifth codewords in the list, so we have a decoding failure. Under the code–matroid connection, dual codes correspond to dual matroids. Also, it is trivial that If we make the (physically realistic) assumptions that ☎ M ☎ M ; y ✽ ; x T ✍ y T ✍ x the errors at the sampling points are independent ✆✩✙ ✆❁✍ identically distributed Gaussian variables, then it can from which we obtain the MacWilliams relation be shown that the most likely codeword to have been ☎ x 1 ☎ x ☎ q 1 W C ✍ y ✎ W C ✆ y ✍ x y transmitted is the one at smallest Euclidean distance ✎ C ✆❃✙ ✆❁✜ 7 , which turns out to be 0101101 . from w in 14 16