-- Copyright (c) David Amos, 2008. All rights reserved. module Math.Combinatorics.Design where import Data.Maybe (fromJust, isJust) import qualified Data.List as L import qualified Data.Map as M import qualified Data.Set as S import Math.Common.ListSet (intersect, symDiff) import Math.Algebra.Field.Base import Math.Algebra.Field.Extension import Math.Algebra.Group.PermutationGroup hiding (elts, order, isMember) import Math.Algebra.Group.SchreierSims as SS import Math.Combinatorics.Graph as G hiding (to1n) import Math.Combinatorics.GraphAuts (refine', isSingleton, graphAuts, incidenceAuts) -- , removeGens) import Math.Combinatorics.FiniteGeometry -- Cameron & van Lint, Designs, Graphs, Codes and their Links {- set xs = map head $ group $ sort xs -} isSubset xs ys = all (`elem` ys) xs -- DESIGNS data Design a = D [a] [[a]] deriving (Eq,Ord,Show) design (xs,bs) | isValid d = d where d = D xs bs toDesign (xs,bs) = D xs' bs' where xs' = L.sort xs bs' = L.sort $ map L.sort bs -- in fact don't require that the blocks are in order isValid (D xs bs) = (xs == L.sort xs || error "design: points are not in order") && (all (\b -> b == L.sort b) bs || error "design: blocks do not have points in order") -- could also check that each block is a subset of xs, etc points (D xs bs) = xs blocks (D xs bs) = bs -- FINDING DESIGN PARAMETERS noRepeatedBlocks (D xs bs) = all ( (==1) . length ) $ L.group $ L.sort bs -- Note that the design parameters functions don't check no repeated blocks, so they're also valid for t-structures -- given t and a t-(v,k,lambda) design, return (v,k,lambda) tDesignParams t d = case findvk d of Nothing -> Nothing Just (v,k) -> case findlambda t d of Nothing -> Nothing Just lambda -> Just (v,k,lambda) findvk (D xs bs) = let k:ls = map length bs in if all (==k) ls then Just (v,k) else Nothing where v = length xs findlambda t (D xs bs) = let lambda:ls = [length [b | b <- bs, ts `isSubset` b] | ts <- combinationsOf t xs] in if all (==lambda) ls then Just lambda else Nothing -- given (xs,bs), return design parameters t-(v,k,lambda) with t maximal designParams d = case findvk d of Nothing -> Nothing Just (v,k) -> case reverse (takeWhile (isJust . snd) [(t, findlambda t d) | t <- [0..k] ]) of [] -> Nothing (t,Just lambda):_ -> Just (t,(v,k,lambda)) -- Note that a 0-(v,k,lambda) design just means that there are lambda blocks, all of size k, with no other regularity isStructure t d = isJust $ tDesignParams t d isDesign t d = noRepeatedBlocks d && isStructure t d is2Design d = isDesign 2 d -- square 2-design (more often called "symmetric" in the literature) isSquare d@(D xs bs) = is2Design d && length xs == length bs -- incidence matrix of a design -- (rows and columns indexed by blocks and points respectively) -- (this follows Cameron & van Lint, though elsewhere in the literature it is sometimes the other way round) incidenceMatrix (D xs bs) = [ [if x `elem` b then 1 else 0 | x <- xs] | b <- bs] -- SOME FAMILIES OF DESIGNS -- the following is trivially a k-(v,k,lambda) design subsetDesign v k = design (xs,bs) where xs = [1..v] bs = combinationsOf k xs -- Cameron & van Lint, p30 -- the pair design on n points is the complete graph on n points considered as a 2-(n,2,1) design pairDesign n = D vs es where graph = G.k n vs = vertices graph es = edges graph -- the affine plane AG(2,Fq) - a 2-(q^2,q,1) design ag2 fq = design (points, lines) where points = ptsAG 2 fq lines = map line $ tail $ ptsPG 2 fq line [a,b,c] = [ [x,y] | [x,y] <- points, a*x+b*y+c==0 ] -- the projective plane PG(2,Fq) - a square 2-(q^2+q+1,q+1,1) design pg2 fq = design (points, lines) where points = ptsPG 2 fq lines = map line points line u = [v | v <- points, u <.> v == 0] u <.> v = sum (zipWith (*) u v) -- Remember that the points and lines of PG(2,Fp) are really the lines and planes of AG(3,Fp). -- A line in AG(3,Fp) defines a plane orthogonal to it. -- The points and i-flats of PG(n,fq), 1<=i<=n-1, form a 2-design -- For i==1, this is a 2-((q^(n+1)-1)/(q-1),q+1,1) design -- For i==n-1, this is a 2-((q^(n+1)-1)/(q-1),(q^n-1)/(q-1),(q^(n-1)-1)/(q-1)) design -- Cameron & van Lint, p8 flatsDesignPG n fq k = design (points, blocks) where points = ptsPG n fq blocks = map closurePG $ flatsPG n fq k -- the closurePG replaces the generators of the flat by the list of points of the flat -- The projective point-hyperplane design is also denoted PG(n,q) pg n fq = flatsDesignPG n fq (n-1) -- (Cameron & van Lint don't actually state that this is a design except when k == n-1) flatsDesignAG n fq k = design (points, blocks) where points = ptsAG n fq blocks = map closureAG $ flatsAG n fq k -- the closureAG replaces the generators of the flat by the list of points of the flat -- The affine point-hyperplane design is also denoted AG(n,q) -- It a 2-(q^n,q^(n-1),(q^(n-1)-1)/(q-1)) design -- Cameron & van Lint, p17 ag n fq = flatsDesignAG n fq (n-1) -- convert a design to be defined over the set [1..n] to1n (D xs bs) = (D xs' bs') where mapping = M.fromList $ zip xs [1..] -- the mapping from vs to [1..n] xs' = M.elems mapping bs' = [map (mapping M.!) b | b <- bs] -- the blocks will already be sorted correctly by construction -- Cameron & van Lint p10 paleyDesign fq | length fq `mod` 4 == 3 = design (xs,bs) where xs = fq qs = set [x^2 | x <- xs] L.\\ [0] -- the non-zero squares in Fq bs = [L.sort (map (x+) qs) | x <- xs] fanoPlane = paleyDesign f7 -- isomorphic to PG(2,F2) -- NEW DESIGNS FROM OLD -- Dual of a design. Cameron & van Lint p11 dual (D xs bs) = design (bs, map beta xs) where beta x = filter (x `elem`) bs -- Derived design relative to a point. Cameron & van Lint p11 -- Derived design of a t-(v,k,lambda) is a t-1-(v-1,k-1,lambda) design. derivedDesign (D xs bs) p = design (xs L.\\ [p], [b L.\\ [p] | b <- bs, p `elem` b]) -- Residual design relative to a point. Cameron & van Lint p13 -- Point-residual of a t-(v,k,lambda) is a t-1-(v-1,k,mu). pointResidual (D xs bs) p = design (xs L.\\ [p], [b | b <- bs, p `notElem` b]) -- Complementary design relative to a point. Cameron & van Lint p13 -- Complement of a t-(v,k,lambda) is a t-(v,v-k,mu). complementaryDesign (D xs bs) = design (xs, [xs L.\\ b | b <- bs]) -- Residual design relative to a block. Cameron & van Lint p13 -- This is only a design if (xs,bs) is a square design -- It may have repeated blocks - but if so, residuals of the complement will not -- Block-residual of a 2-(v,k,lambda) is a 2-(v-k,k-lambda,lambda). blockResidual d@(D xs bs) b | isSquare d = design (xs L.\\ b, [b' L.\\ b | b' <- bs, b' /= b]) -- DESIGN AUTOMORPHISMS isDesignAut (D xs bs) g | supp g `isSubset` xs = all (`S.member` bs') [b -^ g | b <- bs] where bs' = S.fromList bs incidenceGraph (D xs bs) = G vs es where -- graph (vs,es) where vs = L.sort $ map Left xs ++ map Right bs es = L.sort [ [Left x, Right b] | x <- xs, b <- bs, x `elem` b ] designAuts d = incidenceAuts $ incidenceGraph d -- We find design auts by finding graph auts of the incidence graph of the design -- In a square design, we need to watch out for graph auts which are mapping points <-> blocks designAuts1 d = filter (/=1) $ map points $ graphAuts $ incidenceGraph d where points h = fromPairs [(x,y) | (Left x, Left y) <- toPairs h] -- This implicitly filters out (Right x, Right y) action on blocks, -- and also (Left x, Right y) auts taking points to blocks. -- The filter (/=1) is to remove points <-> blocks auts -- The incidence graph is a bipartite graph, so the distance function naturally partitions points from blocks -- MATHIEU GROUPS AND WITT DESIGNS alphaL2_23 = p [[-1],[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]] -- t -> t+1 betaL2_23 = p [[-1],[0],[1,2,4,8,16,9,18,13,3,6,12],[5,10,20,17,11,22,21,19,15,7,14]] -- t -> 2*t gammaL2_23 = p [[-1,0],[1,22],[2,11],[3,15],[4,17],[5,9],[6,19],[7,13],[8,20],[10,16],[12,21],[14,18]] -- t -> -1/t l2_23 = [alphaL2_23, betaL2_23, gammaL2_23] -- Mathieu group M24 -- Conway and Sloane p274ff -- This is the automorphism group of the extended binary Golay code G24 -- or alternatively of the unique Steiner system S(5,8,24) (which consists of the weight 8 codewords of the above) deltaM24 = p [[-1],[0],[1,18,4,2,6],[3],[5,21,20,10,7],[8,16,13,9,12],[11,19,22,14,17],[15]] -- this is t -> t^3 / 9 (for t a quadratic residue), t -> 9 t^3 (t a non-residue) m24 = [alphaL2_23, betaL2_23, gammaL2_23, deltaM24] m24sgs = sgs m24 -- order 244823040 m23sgs = filter (\g -> (-1).^g == -1) m24sgs -- order 10200960 m22sgs = filter (\g -> 0.^g == 0) m23sgs -- order 443520 -- sgs uses the base implied by the Ord instance, which will be [-1,0,..] -- Steiner system S(5,8,24) octad = [0,1,2,3,4,7,10,12] -- Conway&Sloane p276 - this is a weight 8 codeword from Golay code G24 s_5_8_24 = design ([-1..22], octad -^^ l2_23) -- S(5,8,24) constructed as the image of a single octad under the action of PSL(2,23) -- 759 blocks ( (24 `choose` 5) `div` (8 `choose` 5) ) -- Automorphism group is M24 s_4_7_23 = derivedDesign s_5_8_24 (-1) -- 253 blocks ( (23 `choose` 4) `div` (7 `choose` 4) ) -- Automorphism group is M23 s_3_6_22 = derivedDesign s_4_7_23 0 -- 77 blocks -- Automorphism group is M22 -- Derived design of s_3_6_22 is PG(2,F4) -- An alternative construction s_5_8_24' = D xs bs where xs = [1..24] bs = sift [] (combinationsOf 8 xs) sift ls (r:rs) = if all ((<=4) . length) [r `intersect` l | l <- ls] then r : sift (r:ls) rs else sift ls rs sift ls [] = [] -- Could test that m22sgs are all designAuts of s_3_6_22 -- S(5,6,12) and M12 alphaL2_11 = p [[-1],[0,1,2,3,4,5,6,7,8,9,10]] -- t -> t+1 betaL2_11 = p [[-1],[0],[1,3,9,5,4],[2,6,7,10,8]] -- t -> 3*t gammaL2_11 = p [[-1,0],[1,10],[2,5],[3,7],[4,8],[6,9]] -- t -> -1/t l2_11 = [alphaL2_11, betaL2_11, gammaL2_11] deltaM12 = p [[-1],[0],[1],[2,10],[3,4],[5,9],[6,7],[8]] -- Conway&Sloane p271, 327 hexad = [0,1,3,4,5,9] -- the squares (quadratic residues) in F11 -- http://en.wikipedia.org/wiki/Steiner_system s_5_6_12 = design ([-1..10], hexad -^^ l2_11) -- S(5,6,12) constructed as the image of a single hexad under the action of PSL(2,11) -- 132 blocks ( (12 `choose` 5) `div` (6 `choose` 5) ) -- Automorphism group is M12 s_4_5_11 = derivedDesign s_5_6_12 (-1) -- 66 blocks -- Automorphism group is M11 m12 = [alphaL2_11, betaL2_11, gammaL2_11, deltaM12] m12sgs = sgs m12 -- order 95040 m11sgs = filter (\g -> (-1).^g == -1) m12sgs -- order 7920 {- -- WITT DESIGNS -- S(5,8,24) AND S(5,6,12) -- Let D be a square 2-design. -- An n-arc is a set of n points of D, no three of which are contained in a block arcs n (D xs bs) = map reverse $ dfs n [] xs where dfs 0 ys _ = [ys] dfs i ys xs = concat [dfs (i-1) (x:ys) (dropWhile (<=x) xs) | x <- xs, isCompatible (x:ys)] isCompatible ys = all ((<=2) . length) [ys `L.intersect` b | b <- bs] tangents (D xs bs) arc = [b | b <- bs, length (arc `L.intersect` b) == 1] -- !! NOT QUITE AS EXPECTED -- Cameron van Lint implies that ovals should have n = 1+(k-1)/lambda, whereas I'm finding that they're one bigger than that -- eg length $ ovals $ ag2 f3 should be 54 -- But ag2 f3 isn't a *square* design ovals d = let Just (_,k,lambda) = tDesignParams 2 d (q,r) = (k-1) `quotRem` lambda n = 2+q -- == 1+(k-1)/lambda in if r == 0 then [arc | arc <- arcs n d, arc == L.sort (concat $ map (L.intersect arc) $ tangents d arc)] -- each point has a unique tangent else [] hyperovals d = let Just (_,k,lambda) = tDesignParams 2 d (q,r) = k `quotRem` lambda n = 1+q -- == 1+k/lambda in if r == 0 then filter (null . tangents d) $ arcs n d else [] -- Cameron & van Lint, p22 -- s_5_8_24 = [length (intersect (head h) (head s)) | h <- [h1,h2,h3], s <- [s1,s2,s3]] where s_5_8_24 = design (points,lines) where points = map Left xs ++ map Right [1,2,3] lines = [map Left b ++ map Right [1,2,3] | b <- bs] ++ -- line plus three points at infinity [map Left h ++ map Right [2,3] | h <- h1] ++ -- hyperoval plus two points at infinity [map Left h ++ map Right [1,3] | h <- h2] ++ [map Left h ++ map Right [1,2] | h <- h3] ++ [map Left s ++ map Right [1] | s <- s1] ++ -- Baer subplanes plus one point at infinity [map Left s ++ map Right [2] | s <- s2] ++ [map Left s ++ map Right [3] | s <- s3] ++ [map Left (l1 `symDiff` l2) | l1 <- bs, l2 <- bs, l1 < l2] d@(D xs bs) = pg2 f4 hs = hyperovals d [h1,h2,h3] = evenClasses hs [s2,s1,s3] = oddClasses baerSubplanes -- we have to number the ss so that if h <- hi, s <- sj, then |h intersect s| is even <=> i == j evenClasses (h:hs) = let (ys,ns) = partition (even . length . L.intersect h) hs in (h:ys) : evenClasses ns evenClasses [] = [] oddClasses (h:hs) = let (ys,ns) = partition (odd . length . L.intersect h) hs in (h:ys) : oddClasses ns oddClasses [] = [] baerSubplanes = [s | s <- baerSubplanes', and [length (L.intersect s b) `elem` [1,3] | b <- bs] ] baerSubplanes' = map reverse $ dfs 7 [] xs where dfs 0 ys _ = [ys] dfs i ys xs = concat [dfs (i-1) (x:ys) (dropWhile (<=x) xs) | x <- xs, isCompatible (x:ys)] isCompatible ys = all ((<=3) . length) [ys `L.intersect` b | b <- bs] -}