Some bridges between codes and A problem designs What is the - PDF document

✠ ✆ ✟ ✟ ✟ ✞ ✞ ✄☎ ✞ ✞ ✆ ✆ ✆ ✟ ✆ ✆ ✆ ✝ ✝ ✝ ✝ ✝ ✝ ✞ ✟ ✟ ✞ ✒ ✗ ✖ ✖ � ✏ ✖ ✕ ✔ ✒ � ✒ ✠ ✎ ✌ ☞ ☛ ✡ ✠ ✠ ✠ ✠ ✠ ✞ Some bridges between codes and A problem designs What is the smallest number m of subsets (blocks) of ✑ 1 ✎ n ✓ such that Peter J. Cameron School of Mathematical Sciences (a) any two blocks meet in at most two points; Queen Mary and Westfield College London E1 4NS, U.K. (b) any two points lie in at least two blocks? p.j.cameron@qmw.ac.uk 1 3 Remembering Hamming and Assmus Some results ✁✂ 0 0 0 1 1 1 1 Theorem (i) m n , with equality if and only if the 0 1 1 0 0 1 1 H blocks form a biplane. 1 0 1 0 1 0 1 ✍ 2 ✍ 1 (ii) m o ✏ n . Hamming code Proof (i) Count incidences between point-pairs and The dual code has one non-zero weight. block pairs. q 2 1 , q a prime power, and let D be a (ii) Let n q ✍ n planar difference set in Z ✏ . Take all translates of D ✘ D . and The Assmus–Mattson theorem gives 2 -designs, ✍ 7 ✍ 7 ✎ 3 ✎ 1 ✎ 4 ✎ 2 ✏ and ✏ . 2 4

✕ ✎ ✖ ✕ ✘ ✔ ✜ � ✣ ✖ ✘ ✔ ✘ ✣ ✥ ✖ � ✦ ✒ ✏ ✖ ✛ Some values Two approaches 3 4 5 6 7 8 9 n Scheme 1. Choose a family of sets as in the earlier 4 4 7 7 7 10 11 m problem and test these sets. Condition (b) guarantees at least two positive results, and 10 11 12 13 14 15 16 n condition (a) guarantees that these determine the 11 11 16 16 16 16 16 m ✍ 2 ✍ 1 o ✏ n tests are required. active pair. ✚ 1 ✙ r ✙ r 4 and 2 If r n ✛ , then 2 2 Scheme 2. It is possible to identify the active pair in 2 k 1 at most 8log 2 n tests (and at most 4log 2 n if n 1 ) ✢ r ✢ n ✢ r min 1 ✤ 2 1 m ✣ / 2 2 2 using coding theory. 5 7 The coding scheme A problem 2 k 1 . There is a 2 -error-correcting Suppose that n BCH code of length n and codimension 2 k . Imagine We are given a set of n objects, and we know that that the zero codeword was transmitted, and two one pair is ‘active’. We can test any subset of the errors were made in the positions of the active pair. objects; the test result will be positive if and only if We can correct the errors, i.e. identify the active pair. the active pair is contained in the set being tested. Let H be a 2 k ✧ n parity check matrix for the code. Use syndrome decoding , i.e. calculate vH ★ , where v How many tests are required? is the characteristic vector of the active pair. Then v can be recovered from its syndrome. 6 8

Example The coding scheme Suppose that we are trying to identify an active pair from a set of size 1000 . The 2 -error-correcting BCH The i th bit in the syndrome is the inner product of v code of length 1023 has codimension 20 , so that 40 with the i th row of H , i.e. it is 1 if the active pair is tests are required. separated by the support of the i th row. We can determine this with two tests A 0 i and A 1 i , where A ε i is If we suspect that some tests will give the wrong the set of positions where ε occurs in the i th row. So result, but (say) not more than 3% of all tests in a 4 k tests are required. sequence will be wrong, we could use a 2 -error-correcting shortened BCH code of length 30 The number may be smaller: if A 0 i tests positive, then and dimension 20 , which will yield the required A 1 i doesn’t have to be tested. information in 60 tests, correctly if at most two test results are wrong. 9 11 Who found the Hamming codes? Variations R. A. Fisher, The theory of confounding in factorial experiments in relation to the theory of groups, Ann. What if some test results are incorrect? Eugenics 11 (1942), 341–353. Choose a code C of length m and dimension 2 k that will correct the maximum number of errors likely to R. A. Fisher, A system of confounding for factors with occur in the tests. Let G be its 2 k ✧ m generator more than two alternatives, giving completely ★ H instead of H in Scheme 2. 2 m matrix. Then use G orthogonal cubes and higher powers, Ann. Eugenics 12 (1945), 2283–290. tests are required. M. J. E. Golay, Notes on digital coding, Proc. IEEE What if we have to identify subsets of other sizes? 37 (1949), 657. Just choose a code correcting the appropriate R. W. Hamming, Error detecting and error correcting number of errors. codes, Bell Systems Tech. J. 29 (1950), 147–160. 10 12

✪ ✩ ✪ ✪ � ✩ ✧ ✪ ✪ ✪ ✩ ✧ ✒ ✧ ✒ ✒ ✖ � ✔ ✪ ✪ ✩ ✩ � ✒ Coding theory We wish to send words of length n over an alphabet A with ✩ A q over a noisy channel where errors can Factorial design occur. Let C be the annihilator of B in A ✧ A ✫ n . (Here A We assume that, with high probability, not too many ✫ 1 ✫ i is the group of characters of A i ; so C is the set of all errors occur during transmission of a word. characters of A i ✧ A n which are trivial on B .) The strategy is to send words from a code C , a subset of A n . We require: Elements of C represent combinations of treatments which are confounded in the experiment. (For 2 e 1 , we can (a) large minimum distance d : if d example, if an element of C has support in correct up to e errors; A ✬ A ✬ A ✫ k , then the interaction of factors j and k ✫ i ✫ j cannot be distinguished from the main effect of factor i .) (b) many codewords (subject to (a)): the transmission rate is log q ✩ C ✗ n ; (c) computationally efficient encoding and decoding (subject to (a) and (b)). 13 15 Factorial design Factorial design We want We are investigating n factors which can affect the (a) Large weight in C so that potentially significant yield of some process. The i th factor can take any combinations of factors are not confounded; one of a set A i of levels, with ✩ A i q i . (b) Few trials (subject to (a)): trials are expensive! We assume that only the interactions of small This means small B , and so large C : note that numbers of factors affect the yield significantly. q 1 ✪ q n ✩ C ✩ B We impose the structure of an abelian group on A i , and test treatment combinations lying in a subgroup (c) simple description which can be explained to B of A 1 ✧ A n . experimenters and for which results can be analysed (subject to (a) and (b)). 14 16

✒ ✪ ✧ ✷ ✘ ✷ ✏ ✗ ✷ ✘ ✮ ✕ ✒ ✕ ✪ ✪ ✕ ✏ ✏ ✘ ✴ � ✮ ✯ ✚ ✱ ✲ ✚ ✱ ✳ ✒ ✎ ✘ ✩ ✒ ✩ ✪ ✪ ✪ ✧ ✘ ✗ � ✕ ✩ ✹ ✒ ✏ ✘ � ✘ ✒ ✏ ✗ ✕ ✘ Fisher’s Theorem on Minimal Confounding Comparison Fisher (1942) proved that: Design theorists and coding theorists are both A 2 n factorial scheme can be arranged in 2 n ✭ p looking for subsets C of A 1 ✧ A n with large blocks of 2 p plots each, without confounding either main effects or 2 -factor interactions, provided that minimum distance and large cardinality. n 2 p . Coding theorists have n large, all A i of the same size Subsequently (1945), he generalized this theorem and proved that: (almost always 2 ), and don’t insist on group structure A π n factorial scheme can be arranged in π n ✭ p (though it does help to use a linear code). blocks of π p plots each, without confounding either main effects or 2 -factor interactions, provided that Statisticians have n fairly small, varying alphabet ✰ π p 1 1 n ✰ π size, and do require group structure. D. J. Finney, An Introduction to the Theory of Experimental Hamming codes satisfy both specifications! Design , University of Chicago Press, Chicago, 1960. (Here π is a prime power.) 17 19 Mixed alphabets C is a code of length n and minimum distance d over Hamming codes ✍ d 1 ✗ 2 alphabets of size q 1 ✎ q n . Let e ✵ , and assume that q 1 q n . ✍ q ✏ k . Partition the non-zero vectors in V GF Let V Sphere-packing bound: into equivalence classes, where two vectors are n equivalent if one is a non-zero scalar multiple of the ✍ q k ✍ q ∏ q i 1 1 other. There are ✏ equivalence classes. i ✶ 1 ✩ C e k ✍ q i j ∑ ∑ ∏ 1 Choose one vector from each equivalence class, and ✍ q k ✍ q ✶ 0 ✶ 1 k i 1 ✮ i k j 1 1 let H be the k ✏ matrix having these vectors as columns. (For simplicity, take all vectors Singleton bound: whose first non-zero entry is 1 .) Then any two ✸ 1 n ✚ d columns of H are linearly independent. ∏ ✩ C q i ✶ 1 i The code C with parity check matrix H thus has minimum weight 3 and so is 1 -error-correcting. This Plotkin bound: Let ✍ k n ✍ 1 is the Hamming code H ✎ q ✏ . ∑ α 1 ✗ q i ✶ 1 i ✍ d α then α If d ✩ C d ✏ . 18 20