linear-code (empty) → 0.1.0
raw patch · 10 files changed
+1880/−0 lines, 10 filesdep +HaskellForMathsdep +QuickCheckdep +basesetup-changed
Dependencies added: HaskellForMaths, QuickCheck, base, combinat, containers, data-default, ghc-typelits-knownnat, ghc-typelits-natnormalise, linear-code, matrix, random, smallcheck, tasty, tasty-hunit, tasty-quickcheck, tasty-smallcheck
Files
- ChangeLog.md +7/−0
- LICENSE +674/−0
- README.md +57/−0
- Setup.hs +2/−0
- linear-code.cabal +77/−0
- src/Math/Algebra/Code/Linear.hs +529/−0
- src/Math/Algebra/Field/Instances.hs +73/−0
- src/Math/Algebra/Field/Static.hs +103/−0
- src/Math/Algebra/Matrix.hs +237/−0
- test/Main.hs +121/−0
+ ChangeLog.md view
@@ -0,0 +1,7 @@+0.1.0+-----++* Initial release+ - Includes trivial, hamming and random codes+ - Implements syndrome decoding+
+ LICENSE view
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Of course, your program's commands+might be different; for a GUI interface, you would use an "about box".++ You should also get your employer (if you work as a programmer) or school,+if any, to sign a "copyright disclaimer" for the program, if necessary.+For more information on this, and how to apply and follow the GNU GPL, see+<https://www.gnu.org/licenses/>.++ The GNU General Public License does not permit incorporating your program+into proprietary programs. If your program is a subroutine library, you+may consider it more useful to permit linking proprietary applications with+the library. If this is what you want to do, use the GNU Lesser General+Public License instead of this License. But first, please read+<https://www.gnu.org/licenses/why-not-lgpl.html>.
+ README.md view
@@ -0,0 +1,57 @@+# linear-code+Library to handle linear codes from coding theory.++The library is designed to carry the most important bits of information in the+type system while still keeping the types sane.++This library is based roughly on [_Introduction to Coding Theory_ by _Yehuda Lindell_](http://u.cs.biu.ac.il/~lindell/89-662/coding_theory-lecture-notes.pdf)++# Usage example+## Working with random codes+```Haskell+> :m + Math.Code.Linear System.Random+> :set -XDataKinds+> c <- randomIO :: IO (LinearCode 7 4 F5)+> c+[7,4]_5-Code+> generatorMatrix c+( 1 0 1 0 0 2 0 )+( 0 2 0 0 1 2 0 )+( 0 1 0 1 0 1 0 )+( 1 0 0 0 0 1 1 )+> e1 :: Vector 4 F5+( 1 0 0 0 )+> v = encode c e1+> v+( 1 0 1 0 0 2 0 )+> 2 ^* e4 :: Vector 7 F3+( 0 0 0 2 0 0 0 )+> vWithError = v + 2 ^* e4+> vWithError+( 1 0 1 2 0 2 0 )+> isCodeword c v+True+> isCodeword c vWithError+False+> decode c vWithError+Just ( 1 0 2 2 2 2 0 )+```+Notice, the returned vector is NOT the one without error. The reason for this+is that a random code most likely does not have a distance >2 which would be+needed to correct one error. Let's try with a hamming code++## Correcting errors with hamming codes+```Haskell+> c = hamming :: BinaryCode 7 4+> generatorMatrix c+( 1 1 0 1 0 0 0 )+( 1 0 1 0 1 0 0 )+( 0 1 1 0 0 1 0 )+( 1 1 1 0 0 0 1 )+> v = encode c e2+> vWithError = v + e3+> Just v' = decode c vWithError+> v' == v+True+```+
+ Setup.hs view
@@ -0,0 +1,2 @@+import Distribution.Simple+main = defaultMain
+ linear-code.cabal view
@@ -0,0 +1,77 @@+-- This file has been generated from package.yaml by hpack version 0.28.2.+--+-- see: https://github.com/sol/hpack+--+-- hash: 515c75757e8c9b5fe6719710a1cfb652f7f850ab85098dc5d664b0b0aaf02230++name: linear-code+version: 0.1.0+synopsis: A simple library for linear codes (coding theory, error correction)+description: Please see the README on GitHub at <https://github.com/wchresta/linear-code#readme>+category: Math+homepage: https://github.com/wchresta/linear-code#readme+bug-reports: https://github.com/wchresta/linear-code/issues+author: Wanja Chresta+maintainer: wanja.hs@chrummibei.ch+copyright: 2018, Wanja Chresta+license: GPL-3+license-file: LICENSE+build-type: Simple+cabal-version: >= 1.10+extra-source-files:+ ChangeLog.md+ README.md++source-repository head+ type: git+ location: https://github.com/wchresta/linear-code++library+ exposed-modules:+ Math.Algebra.Code.Linear+ Math.Algebra.Field.Instances+ Math.Algebra.Field.Static+ Math.Algebra.Matrix+ other-modules:+ Paths_linear_code+ hs-source-dirs:+ src+ ghc-options: -Wall+ build-depends:+ HaskellForMaths+ , base >=4.7 && <5+ , combinat+ , containers+ , data-default+ , ghc-typelits-knownnat+ , ghc-typelits-natnormalise+ , matrix+ , random+ default-language: Haskell2010++test-suite linear-code-test+ type: exitcode-stdio-1.0+ main-is: Main.hs+ other-modules:+ Paths_linear_code+ hs-source-dirs:+ test+ ghc-options: -threaded -rtsopts -with-rtsopts=-N+ build-depends:+ HaskellForMaths+ , QuickCheck+ , base >=4.7 && <5+ , combinat+ , containers+ , data-default+ , ghc-typelits-knownnat+ , ghc-typelits-natnormalise+ , linear-code+ , matrix+ , random+ , smallcheck+ , tasty+ , tasty-hunit+ , tasty-quickcheck+ , tasty-smallcheck+ default-language: Haskell2010
+ src/Math/Algebra/Code/Linear.hs view
@@ -0,0 +1,529 @@+{-# LANGUAGE DataKinds #-}+{-# LANGUAGE KindSignatures #-}+{-# LANGUAGE ScopedTypeVariables #-}+{-# LANGUAGE FlexibleContexts #-}+{-# LANGUAGE TypeApplications #-}+{-# LANGUAGE TypeOperators #-}+{-# LANGUAGE TypeFamilies #-}+{-# OPTIONS_GHC -fplugin GHC.TypeLits.Normalise #-}+{-# OPTIONS_GHC -fplugin GHC.TypeLits.KnownNat.Solver #-}+{-+ This file is part of linear-codes.++ Linear-Codes is free software: you can redistribute it and/or modify+ it under the terms of the GNU General Public License as published by+ the Free Software Foundation, either version 3 of the License, or+ (at your option) any later version.++ Foobar is distributed in the hope that it will be useful,+ but WITHOUT ANY WARRANTY; without even the implied warranty of+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the+ GNU General Public License for more details.++ You should have received a copy of the GNU General Public License+ along with Foobar. If not, see <https://www.gnu.org/licenses/>.+-}+{-|+Module : Math.Algebra.Code.Linear+Description : Linear codes over arbitrary fields+Copyright : (c) Wanja Chresta, 2018+License : GPL-3+Maintainer : wanja.hs@chrummibei.ch+Stability : experimental+Portability : POSIX++Naive implementation of coding theory linear codes and error correcting codes+over arbitrary fields, including finite fields. Goes well with the+@HaskellForMath@ library and its finite field implementations in+@Math.Algebra.Field@. To use extension fields (fields of prime power, i.e.+ \( F_{p^k} \) with \(k>1\), use one of the exported finite fields in+"Math.Algebra.Field.Extension" like 'F16' and its generator 'a16'.++As theoretical basis, Introduction to Coding Theory by Yehuda Lindell is used.+It can be found at+<http://u.cs.biu.ac.il/~lindell/89-662/coding_theory-lecture-notes.pdf>++= Usage++@+>>> :set -XDataKinds+>>> c <- randomIO :: IO (LinearCode 7 4 F5)+>>> c+[7,4]_5-Code+>>> generatorMatrix c+( 1 0 1 0 0 2 0 )+( 0 2 0 0 1 2 0 )+( 0 1 0 1 0 1 0 )+( 1 0 0 0 0 1 1 )+>>> e1 :: Vector 4 F5+( 1 0 0 0 )+>>> v = encode c e1+>>> v+( 1 0 1 0 0 2 0 )+>>> 2 ^* e4 :: Vector 7 F3+( 0 0 0 2 0 0 0 )+>>> vWithError = v + 2 ^* e4+>>> vWithError+( 1 0 1 2 0 2 0 )+>>> isCodeword c v+True+>>> isCodeword c vWithError+False+>>> decode c vWithError+Just ( 1 0 2 2 2 2 0 )+@++Notice, the returned vector is NOT the one without error. The reason for this+is that a random code most likely does not have a distance >2 which would be+needed to correct one error. Let's try with a hamming code++@+>>> c = hamming :: BinaryCode 7 4+>>> generatorMatrix c+( 1 1 0 1 0 0 0 )+( 1 0 1 0 1 0 0 )+( 0 1 1 0 0 1 0 )+( 1 1 1 0 0 0 1 )+>>> v = encode c e2+>>> vWithError = v + e3+>>> Just v' = decode c vWithError+>>> v' == v+True+@++-}+module Math.Algebra.Code.Linear+ ( LinearCode (..)+ , Generator, CheckMatrix+ , codeFromA++ , standardForm, standardFormGenerator++ -- * Code-Vectors and codewords+ , Vector, encode, isCodeword, hasError, weight, codewords+ , allVectors, fullVectors, hammingWords, lighterWords++ -- * Decoding+ , syndrome, decode, syndromeDecode, calcSyndromeTable, recalcSyndromeTable+ , SyndromeTable++ -- * Code transformers+ , dualCode, permuteCode++ -- * Special codes and their generators+ , trivialCode, simplex, hamming+ , BinaryCode++ -- * Helper functions+ , randomPermMatrix+ , codeLength+ , rank++ , eVec, e1, e2, e3, e4, e5, e6, e7, e8, e9, e10+ , char++ -- * Reexported matrix functions from "Math.Algebra.Matrix"+ , matrix, zero, transpose, fromList, fromLists++ -- * Reexported finite fields from @Math.Algebra.Field@+ , F2, F3, F5, F7, F11+ , F4, F8, F16, F9+ ) where++-- Linear codes from mathematical coding theory, including error correcting+-- codes+import GHC.TypeLits+ ( Nat, KnownNat, natVal+ , type (<=), type (+), type (-), type (^)+ )++import Data.Bifunctor (first)+import Data.Monoid ((<>))+import Data.Maybe (fromMaybe)+import Data.List (permutations)+import qualified Data.Map.Strict as M+import Data.Proxy (Proxy (..))+import System.Random (Random, RandomGen, random, randomR)++import Math.Core.Utils (FinSet, elts)+import Math.Combinat.Permutations (_randomPermutation)+import Math.Common.IntegerAsType (IntegerAsType)+import Math.Algebra.Field.Base (Fp, F2, F3, F5, F7, F11)+import Math.Algebra.Field.Static (Size, Characteristic, char)+import Math.Algebra.Field.Extension (F4, F8, F16, F9)+import Math.Algebra.Field.Instances () -- import Random instances for Fields+import Math.Algebra.Matrix+ ( Matrix, matrix, transpose, (<|>), (.*)+ , identity, zero, fromList, fromLists, Vector, rref, submatrix+ )+++-- | A 'Generator' is the generator matrix of a linear code, not necessarily+-- in standard form.+type Generator (n :: Nat) (k :: Nat) = Matrix k n++-- | A 'CheckMatrix' or parity matrix is the dual of a 'Generator'. It can+-- be used to check if a word is a valid code word for the code. Also,+-- \[ \forall v \in f^k: cG \cdot H^\top = 0 \]+-- i.e. the code is generated by the kernel of a check matrix.+type CheckMatrix (n :: Nat) (k :: Nat) = Matrix (n-k) n++-- | A \([n,k]\)-Linear code over the field @f@. The code parameters @f@,@n@ and+-- @k@ are carried on the type level.+-- A linear code is a subspace @C@ of \(f^n\) generated by the generator matrix.+data LinearCode (n :: Nat) (k :: Nat) (f :: *)+ = LinearCode { generatorMatrix :: Generator n k f+ -- ^ Generator matrix, used for most of the operations+ , checkMatrix :: CheckMatrix n k f+ -- ^ Check matrix which can be automatically calculated+ -- from the standard form generator.+ , distance :: Maybe Int+ -- ^ The minimal distance of the code. This is the parameter+ -- \(d\) in \([n,k,d]_q\) notation of code parameters. The+ -- problem of finding the minimal distance is NP-Hard, thus+ -- might not be available.+ , syndromeTable :: SyndromeTable n k f+ -- ^ A map of all possible syndromes to their error vector.+ -- It is used to use syndrome decoding, a very slow decoding+ -- algorithm.+ }++-- | Extract an Int from a type level 'KnownNat'.+natToInt :: forall k. KnownNat k => Proxy k -> Int+natToInt = fromInteger . natVal++instance forall n k f. (Eq f, Fractional f, KnownNat n, KnownNat k, k <= n)+ => Eq (LinearCode n k f) where+ c == d = standardFormGenerator c == standardFormGenerator d++-- We do not show d since it might be expensive to calculate+instance forall n k f.+ (KnownNat n, KnownNat k, KnownNat (Characteristic f))+ => Show (LinearCode n k f) where+ show LinearCode{distance=md} =+ '[':show n<>","<>show k<>dist<>"]_"<>show c<>"-Code"+ where c = char (Proxy :: Proxy f)+ n = natToInt @n Proxy+ k = natToInt @k Proxy+ dist = fromMaybe "" $ fmap (\d -> ',':show d) md++instance forall n k f.+ (KnownNat n, KnownNat k, k <= n, Eq f, FinSet f, Num f, Ord f)+ => Bounded (LinearCode n k f) where+ minBound = trivialCode+ maxBound = codeFromA $ matrix (const $ last elts)+++-- | A random permutation matrix+randomPermMatrix :: forall g n r. (KnownNat n, Num r, RandomGen g)+ => g -> (Matrix n n r, g)+randomPermMatrix g =+ let n = natToInt @n Proxy+ delta i j = if i == j then 1 else 0+ (perm,g') = _randomPermutation n g+ in (fromLists [ [ delta i (perm !! (j-1))+ | j <- [1..n] ]+ | i <- [1..n] ],g')++-- | A random code with a generator in standard form. This does not generate+-- all possible codes but only one representant of the equivalence class+-- modulo similar codes.+randomStandardFormCode :: forall n k f g.+ ( KnownNat n, KnownNat k, k <= n+ , Eq f, FinSet f, Num f, Ord f, Random f, RandomGen g)+ => g -> (LinearCode n k f, g)+randomStandardFormCode = first codeFromA . randomAMatrix+ where+ randomAMatrix :: RandomGen g => g -> (Matrix k (n-k) f,g)+ randomAMatrix = random+++instance forall n k f.+ ( KnownNat n, KnownNat k, k <= n+ , Eq f, FinSet f, Num f, Ord f, Random f)+ => Random (LinearCode n k f) where+ random g = uncurry shuffleCode $ randomStandardFormCode g++ randomR (hc,lc) g =+ let k = natToInt @k Proxy+ extractA = submatrix 1 k . generatorMatrix+ (rmat,g2) = randomR (extractA hc, extractA lc) g+ rcode = codeFromA rmat+ in shuffleCode rcode g2+++-- | Uses Gaussian eleminiation via 'rref' from 'Data.Matrix.Safe' to+-- find the standard form of generators. This might fail since not all+-- codes can be converted to standard form without permutation of columns.+standardForm :: forall n k f.+ (Eq f, Fractional f, KnownNat n, KnownNat k, k <= n)+ => Generator n k f -> Generator n k f+standardForm = rref+++-- | The standard from generator of a linear code. Uses 'standardForm' to+-- try to create a standard form generator which might fail.+standardFormGenerator :: forall n k f.+ (Eq f, Fractional f, KnownNat n, KnownNat k, k <= n)+ => LinearCode n k f -> Generator n k f+standardFormGenerator = standardForm . generatorMatrix+++-- | Convenience function to extract the length @n@ from the type level+codeLength :: forall n k f. KnownNat n => LinearCode n k f -> Int+codeLength _ = natToInt @n Proxy++-- | Convenience function to extract the rank @k@ from the type level.+rank :: forall n k f. KnownNat k => LinearCode n k f -> Int+rank _ = natToInt @k Proxy++-- | The hamming weight of a Vector is an 'Int' between 0 and n+weight :: forall f m. (Eq f, Num f, Functor m, Foldable m) => m f -> Int+weight = sum . fmap (\x -> if x==0 then 0 else 1)++-- | Generate a linear [n,k]_q-Code over the field a with the generator in+-- standard form (I|A), where the given function generates the k×(n-k)-matrix+-- A.+codeFromA :: forall k n f.+ (KnownNat n, KnownNat k, k <= n, Eq f, FinSet f, Num f, Ord f)+ => Matrix k (n-k) f+ -- ^ Elements of A where top-left is (1,1) and bottom right (k,n-k)+ -> LinearCode n k f+codeFromA a = recalcSyndromeTable LinearCode+ { generatorMatrix = identity <|> a+ , checkMatrix = (-transpose a) <|> identity -- () are important for f/=F2+ , distance = Nothing+ , syndromeTable = undefined+ }+++-- * Codewords and their properties++-- | Get the codeword generated by the given k-sized vector.+encode :: forall n k f. Num f => LinearCode n k f -> Vector k f -> Vector n f+encode code vs = vs .* generatorMatrix code+++-- | List all vectors of length n over field f+allVectors :: forall n f. (KnownNat n, FinSet f, Num f, Eq f) => [Vector n f]+allVectors = fromList <$> allVectorsI (natToInt @n Proxy)++-- | List all lists given length over field f+allVectorsI :: forall f. (FinSet f, Num f, Eq f) => Int -> [[f]]+allVectorsI n = iterate addDim [[]] !! n+ where addDim vs = [ x:v | v <- vs, x <- elts ]++-- | List all vectors of length n with non-zero elements over field f+fullVectors :: forall n f. (KnownNat n, FinSet f, Num f, Eq f) => [Vector n f]+fullVectors = fromList <$> fullVectorsI (natToInt @n Proxy)++-- | List all vectors of given length with non-zero elements over field f+fullVectorsI :: forall f. (FinSet f, Num f, Eq f) => Int -> [[f]]+fullVectorsI n = iterate addDim [[]] !! n+ where addDim vs = [ x:v | v <- vs, x <- elts, x /= 0 ]++-- | List of all words with given hamming weight+hammingWords :: forall n f. (KnownNat n, FinSet f, Num f, Eq f)+ => Int -> [Vector n f]+hammingWords w = fromList <$> shuffledVecs+ where+ n = natToInt @n Proxy+ orderedVecs :: [[f]] -- [1,x,1,1,0..0]+ orderedVecs = (++) (replicate (n-w) 0) <$> fullVectorsI w+ shuffledVecs :: [[f]]+ shuffledVecs = orderedVecs >>= permutations++-- | List of all words with (non-zero) hamming weight smaller than a given +-- boundary+lighterWords :: forall n f. (KnownNat n, FinSet f, Num f, Eq f)+ => Int -> [Vector n f]+lighterWords w = concat [ hammingWords l | l <- [1..w] ]++-- | A list of all codewords+codewords :: forall n k f.+ (KnownNat n, KnownNat k, k <= n, Num f, Eq f, FinSet f)+ => LinearCode n k f -> [Vector n f]+codewords c = map (encode c) allVectors++-- | Give the syndrome of a word for the given code. This is 0 if the word+-- is a valid code word.+syndrome :: forall n k f. Num f+ => LinearCode n k f -> Vector n f -> Syndrome n k f+syndrome c w = w .* transpose (checkMatrix c)++-- | Uses the exponential-time syndrome decoding algorithm for general codes.+-- c.f: https://en.wikipedia.org/wiki/Decoding_methods#Syndrome_decoding+syndromeDecode :: forall n k f.+ (KnownNat n, KnownNat k, k <= n, Ord f, Num f)+ => LinearCode n k f -> Vector n f -> Maybe (Vector n f)+syndromeDecode c w =+ let syn = syndrome c w+ e = M.lookup syn $ syndromeTable c+ in (w+) <$> e++-- | Synonym for syndromeDecoding, an inefficient decoding algorithm that works+-- for all linear codes.+decode :: forall n k f.+ (KnownNat n, KnownNat k, k <= n, Ord f, Num f)+ => LinearCode n k f -> Vector n f -> Maybe (Vector n f)+decode = syndromeDecode++-- | Pairs of (e,S(e)) where e is an error vector and S(e) is its syndrome.+type Syndrome n k f = Vector (n-k) f++-- | A syndrome table is a map from syndromes to their minimal weight+-- representative. Every vector @v@ has a syndrome \( S(v) \). This table+-- reverses the syndrome function @S@ and chooses the vector with the smallest+-- hamming weight from it's image. This is a lookup table for syndrome+-- decoding.+type SyndromeTable n k f = M.Map (Syndrome n k f) (Vector n f)++-- | Return a syndrome table for the given linear code. If the distance is not+-- known (i.e. 'minDist' @c@ = Nothing) this is very inefficient.+calcSyndromeTable :: forall n k f.+ (KnownNat n, KnownNat k, k <= n, Eq f, FinSet f, Num f, Ord f)+ => LinearCode n k f -> SyndromeTable n k f+-- We need to build a syndrome table for all codewords of wgt < floor $ (d-1)/2+-- If we do not know the weight (because distance code = Nothing), we assume+-- the worst case with a maximum distance of n-k+1+calcSyndromeTable c = M.fromListWith minWt allSyndromes+ where minWt x y = if weight x < weight y then x else y+ n = natToInt $ Proxy @n+ k = natToInt $ Proxy @k+ w = fromMaybe (n-k+1) $ distance c++ allSyndromes :: [(Syndrome n k f, Vector n f)]+ allSyndromes = [(syndrome c e,e) | e <- lighterWords w]++-- | Replace the 'syndromeTable' of a code with a newly calculated syndrome+-- table for the (current) generator. Useful to get a syndrome table for+-- transformed codes when the table cannot be transformed, too.+recalcSyndromeTable :: forall n k f.+ (KnownNat n, KnownNat k, k <= n, Eq f, FinSet f, Num f, Ord f)+ => LinearCode n k f -> LinearCode n k f+recalcSyndromeTable c = c { syndromeTable = calcSyndromeTable c }+++-- | Check if the given candidate code word is a valid code word for the+-- given linear code. If not, the party check failed.+isCodeword :: forall n k f. (Eq f, Num f, KnownNat n, KnownNat k, k <= n)+ => LinearCode n k f -> Vector n f -> Bool+isCodeword c w = syndrome c w == zero+++-- | Check if the given candidate code word has errors, i.e. if some element+-- changed during transmission. This is equivalent with @not@ 'isCodeword'+hasError :: forall n k f. (Eq f, Num f, KnownNat n, KnownNat k, k <= n)+ => LinearCode n k f -> Vector n f -> Bool+hasError g = not . isCodeword g+++-- * Code transformers++-- |The dual code is the code generated by the check matrix+dualCode :: forall n k f.+ (KnownNat n, KnownNat k, k <= n, Eq f, FinSet f, Num f, Ord f)+ => LinearCode n k f -> LinearCode n (n-k) f+dualCode c = recalcSyndromeTable+ LinearCode { generatorMatrix = checkMatrix c+ , checkMatrix = generatorMatrix c+ , distance = distance c+ , syndromeTable = undefined+ }+++-- | Permute the rows of a code with a permutation matrix. The given permutation+-- matrix must be a valid permutation matrix; this is not checked.+-- This effectively multiplies the generator and check matrix from the right+permuteCode :: forall n k f.+ (KnownNat n, KnownNat k, k <= n, Eq f, FinSet f, Num f, Ord f)+ => LinearCode n k f -> Matrix n n f -> LinearCode n k f+permuteCode c p = recalcSyndromeTable+ LinearCode { generatorMatrix = generatorMatrix c .* p+ , checkMatrix = checkMatrix c .* p+ , distance = distance c+ , syndromeTable = undefined+ -- TODO: Permute syndrome table+ }+++-- | Randomly permute the elements of the code. This is a shuffle of the+-- positions of elements of all codewords+shuffleCode :: forall n k f g.+ (KnownNat n, KnownNat k, k <= n, RandomGen g, Eq f, FinSet f, Num f, Ord f)+ => LinearCode n k f -> g -> (LinearCode n k f, g)+shuffleCode c g =+ let (p,g') = randomPermMatrix g+ in (permuteCode c p, g')+++-- * Special codes and their generators++-- | A binary code is a linear code over the field GF(2)+type BinaryCode n k = LinearCode n k F2++-- | The trivial code is the identity code where the parity bits are all zero.+trivialCode :: forall n k f.+ (KnownNat n, KnownNat k, k <= n, Eq f, FinSet f, Num f, Ord f)+ => LinearCode n k f+trivialCode = codeFromA (zero :: Matrix k (n-k) f)+++-- | A simplex code is a code generated by all possible codewords consisting+-- of 0's and 1's except the zero vector.+simplex :: forall k p s.+ ( KnownNat s, KnownNat k , IntegerAsType p+ , 1 <= s^k, k <= s^k, 1+k <= s^k, Size (Fp p) ~ s)+ => LinearCode (s^k-1) k (Fp p)+simplex = codeFromA . transpose $ fromLists nonUnit+ where+ k = natToInt @k Proxy+ nonUnit = filter ((>1) . weight) $ allVectorsI k++-- | The /Hamming(7,4)/-code. It is a [7,4,3]_2 code+hamming :: (KnownNat m, 2 <= m, m <= 2^m, 1+m <= 2^m)+ => LinearCode (2^m-1) (2^m-m-1) F2+hamming = dualCode simplex { distance = Just 3 }+++-- * Helper functions++-- | Standard base vector [0..0,1,0..0] for any field. Parameter must be >=1+eVec :: forall n f. (KnownNat n, Num f) => Int -> Vector n f+eVec i = fromList $ replicate (i-1) 0 ++ 1 : replicate (n-i) 0+ where+ n = natToInt @n Proxy++-- | First base vector [1,0..0]+e1 :: forall n f. (KnownNat n, Num f) => Vector n f+e1 = eVec 1++-- | Second base vector [0,1,0..0]+e2 :: forall n f. (KnownNat n, Num f) => Vector n f+e2 = eVec 2++e3 :: forall n f. (KnownNat n, Num f) => Vector n f+e3 = eVec 3++e4 :: forall n f. (KnownNat n, Num f) => Vector n f+e4 = eVec 4++e5 :: forall n f. (KnownNat n, Num f) => Vector n f+e5 = eVec 5++e6 :: forall n f. (KnownNat n, Num f) => Vector n f+e6 = eVec 6++e7 :: forall n f. (KnownNat n, Num f) => Vector n f+e7 = eVec 7++e8 :: forall n f. (KnownNat n, Num f) => Vector n f+e8 = eVec 8++e9 :: forall n f. (KnownNat n, Num f) => Vector n f+e9 = eVec 9++e10 :: forall n f. (KnownNat n, Num f) => Vector n f+e10 = eVec 10++-- vim : set colorcolumn=80
+ src/Math/Algebra/Field/Instances.hs view
@@ -0,0 +1,73 @@+{-# LANGUAGE ScopedTypeVariables #-}+{-# OPTIONS_GHC -fno-warn-orphans #-}+{-+ This file is part of linear-codes.++ Linear-Codes is free software: you can redistribute it and/or modify+ it under the terms of the GNU General Public License as published by+ the Free Software Foundation, either version 3 of the License, or+ (at your option) any later version.++ Foobar is distributed in the hope that it will be useful,+ but WITHOUT ANY WARRANTY; without even the implied warranty of+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the+ GNU General Public License for more details.++ You should have received a copy of the GNU General Public License+ along with Foobar. If not, see <https://www.gnu.org/licenses/>.+-}+{-|+Module : Math.Algebra.Field.Instances+Description : Missing instnaces for @HaskellForMaths@'s 'Math.Algebra.Field'+Copyright : (c) Wanja Chresta, 2018+License : GPL-3+Maintainer : wanja.hs@chrummibei.ch+Stability : experimental+Portability : POSIX++Some important instances like 'Random' and 'Bounded' are missing from+@HaskellForMath@'s implementation of finite fiels. Here, orphan instances+for these classes are added.+-}++module Math.Algebra.Field.Instances() where++import System.Random+import Data.Bifunctor (first)+import qualified Math.Algebra.Field.Base as F+import qualified Math.Algebra.Field.Extension as F+import qualified Math.Common.IntegerAsType as F+import qualified Math.Core.Utils as F++choose :: RandomGen g => [a] -> g -> (a,g)+choose [] = error "Cannot choose from empty list"+choose as = first (as !!) . randomR (0,length as-1)++-- Make prime fields Random+instance forall p. F.IntegerAsType p => Random (F.Fp p) where+ randomR (l,h) = choose $ filter (\x -> l <= x && x <= h + || l >= x && x >= h) F.elts+ random = choose F.elts++-- Make extension fields Random+instance forall fp poly.+ (F.FinSet fp, Ord fp, Num fp, F.PolynomialAsType fp poly)+ => Random (F.ExtensionField fp poly) where+ randomR (l,h) = choose $ filter (\x -> l <= x && x <= h + || l >= x && x >= h) F.elts+ random = choose F.elts++-- Make prime fields bounded+instance forall p. F.IntegerAsType p => Bounded (F.Fp p) where+ minBound = head F.elts+ maxBound = last F.elts+++-- Make extension fields bounded+instance forall fp poly. + (F.FinSet fp, Eq fp, Num fp, F.PolynomialAsType fp poly) + => Bounded (F.ExtensionField fp poly) where+ minBound = head F.elts+ maxBound = last F.elts++
+ src/Math/Algebra/Field/Static.hs view
@@ -0,0 +1,103 @@+{-# LANGUAGE DataKinds #-}+{-# LANGUAGE TypeApplications #-}+{-# LANGUAGE TypeFamilies #-}+{-# LANGUAGE TypeOperators #-}+{-# LANGUAGE ScopedTypeVariables #-}+{-# LANGUAGE UndecidableInstances #-}+{-+ This file is part of linear-codes.++ Linear-Codes is free software: you can redistribute it and/or modify+ it under the terms of the GNU General Public License as published by+ the Free Software Foundation, either version 3 of the License, or+ (at your option) any later version.++ Foobar is distributed in the hope that it will be useful,+ but WITHOUT ANY WARRANTY; without even the implied warranty of+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the+ GNU General Public License for more details.++ You should have received a copy of the GNU General Public License+ along with Foobar. If not, see <https://www.gnu.org/licenses/>.+-}+{-|+Module : Math.Algebra.Field.Static+Description : Some type families extracting finite field parameters+Copyright : (c) Wanja Chresta, 2018+License : GPL-3+Maintainer : wanja.hs@chrummibei.ch+Stability : experimental+Portability : POSIX++Some finite field parameters are missing from @HaskellForMaths@ implementation.+Here, we add type classes to add these parameters to the type level.+-}+module Math.Algebra.Field.Static where++import Data.Proxy (Proxy(Proxy))+import GHC.TypeLits (Nat, KnownNat, type (^), natVal)+import qualified Math.Algebra.Field.Base as F+import qualified Math.Algebra.Field.Extension as F+++-- | The characteristic of a finite field on the type level. The characteristic+-- is: For any element @x@ in the field @f@ with characteristic @c@, we have:+-- @c * x = x + x + .. + x (c times) = 0@+type family Characteristic (f :: *) :: Nat+type instance Characteristic F.F2 = 2+type instance Characteristic F.F3 = 3+type instance Characteristic F.F5 = 5+type instance Characteristic F.F7 = 7+type instance Characteristic F.F11 = 11+type instance Characteristic F.F13 = 13+type instance Characteristic F.F17 = 17+type instance Characteristic F.F19 = 19+type instance Characteristic F.F23 = 23+type instance Characteristic F.F29 = 29+type instance Characteristic F.F31 = 31+type instance Characteristic F.F37 = 37+type instance Characteristic F.F41 = 41+type instance Characteristic F.F43 = 43+type instance Characteristic F.F47 = 47+type instance Characteristic F.F53 = 53+type instance Characteristic F.F59 = 59+type instance Characteristic F.F61 = 61+type instance Characteristic F.F67 = 67+type instance Characteristic F.F71 = 71+type instance Characteristic F.F73 = 73+type instance Characteristic F.F79 = 79+type instance Characteristic F.F83 = 83+type instance Characteristic F.F89 = 89+type instance Characteristic F.F97 = 97+type instance Characteristic (F.ExtensionField k poly)+ = Characteristic k -- Extension fields have their base fields char+++-- | Characteristic of a field. It takes a finite field type in the proxy+-- value and gives the characteristic. This is done using type families+-- To support new finite field types, you need to add a type instance+-- for the type family 'Characteristic'.+char :: forall c f. (KnownNat c, c ~ Characteristic f) => Proxy f -> Int+char Proxy = fromInteger . natVal $ Proxy @c+++-- | Type family which gives the degree of a polynomial type. This is used to+-- extract type level information from 'Math.Algebra.Field.Extension'+type family PolyDegree (f :: *) :: Nat+type instance PolyDegree F.ConwayF4 = 2+type instance PolyDegree F.ConwayF8 = 3+type instance PolyDegree F.ConwayF9 = 2+type instance PolyDegree F.ConwayF16 = 4+type instance PolyDegree F.ConwayF25 = 2+type instance PolyDegree F.ConwayF27 = 3+type instance PolyDegree F.ConwayF32 = 5+++-- | Type family which gives the size of a field, i.e. the number of elements+-- of a finite field.+type family Size (f :: *) :: Nat+type instance Size (F.Fp p) = Characteristic (F.Fp p)+type instance Size (F.ExtensionField fp poly) =+ Characteristic fp ^ PolyDegree poly++
+ src/Math/Algebra/Matrix.hs view
@@ -0,0 +1,237 @@+{-# LANGUAGE DataKinds #-}+{-# LANGUAGE DeriveTraversable #-}+{-# LANGUAGE GeneralizedNewtypeDeriving #-}+{-# LANGUAGE KindSignatures #-}+{-# LANGUAGE MultiParamTypeClasses #-}+{-# LANGUAGE ScopedTypeVariables #-}+{-# LANGUAGE TypeApplications #-}+{-# LANGUAGE TypeFamilies #-}+{-# LANGUAGE TypeOperators #-}+{-+ This file is part of linear-codes.++ Linear-Codes is free software: you can redistribute it and/or modify+ it under the terms of the GNU General Public License as published by+ the Free Software Foundation, either version 3 of the License, or+ (at your option) any later version.++ Foobar is distributed in the hope that it will be useful,+ but WITHOUT ANY WARRANTY; without even the implied warranty of+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the+ GNU General Public License for more details.++ You should have received a copy of the GNU General Public License+ along with Foobar. If not, see <https://www.gnu.org/licenses/>.+-}+{-|+Module : Math.Algebra.Matrix+Description : Type safe matrix wrapper over the matrix library+Copyright : (c) Wanja Chresta, 2018+License : GPL-3+Maintainer : wanja.hs@chrummibei.ch+Stability : experimental+Portability : POSIX++Math.Algebra.Matrix wraps @matrix@'s Data.Matrix functions and adds size+information on the type level. Additionally, it fixes some issues that makes+the library work well with finite fields. The name of most functions is the+same as in Data.Matrix+-}++module Math.Algebra.Matrix+ ( Matrix(..)+ , matrix+ , Vector+ , transpose+ , (<|>)+ , identity+ , zero+ , fromList+ , fromLists+ , toList+ , toLists+ , (.*)+ , (^*)+ , rref+ , submatrix+ ) where++import GHC.TypeLits (Nat, KnownNat, natVal, type (+), type (<=))+import Data.List (find)+import Data.Proxy (Proxy(..))+import Data.Semigroup ((<>))+import Data.Maybe (isNothing)++import qualified Data.Matrix as M+import qualified System.Random as R+++-- | A matrix over the type @f@ with @m@ rows and @n@ columns. This just wraps+-- the 'Data.Matrix.Matrix' constructor and adds size information to the type+newtype Matrix (m :: Nat) (n :: Nat) (f :: *) = Matrix (M.Matrix f)+ deriving (Eq, Functor, Applicative, Foldable, Traversable, Monoid)++instance forall m n f. Show f => Show (Matrix m n f) where+ show (Matrix mat) = M.prettyMatrix mat++instance forall m n f. Ord f => Ord (Matrix m n f) where+ compare x y = toList x `compare` toList y -- TODO: Do not use `toList`?++instance forall f m n. Num f => Num (Matrix m n f) where+ (Matrix x) + (Matrix y) = Matrix $ x + y+ (Matrix x) - (Matrix y) = Matrix $ x - y+ (*) = error "Data.Matrix.Safe: (*) not allowed. Use (.*) instead"+ negate = fmap negate+ abs = fmap abs+ signum = fmap signum+ fromInteger = Matrix . fromInteger++instance forall m n a. (KnownNat m, KnownNat n, R.Random a)+ => R.Random (Matrix m n a) where+ random g =+ let m = fromInteger . natVal $ Proxy @m+ n = fromInteger . natVal $ Proxy @n+ (g1,g2) = R.split g+ rmat = fromList . take (m*n) . R.randoms $ g1+ in (rmat, g2)+ randomR (lm,hm) g =+ -- lm and hm are matrices. We zip the elements and use these as+ -- hi/lo bounds for the random generator+ let zipEls :: [(a,a)]+ zipEls = zip (toList lm) (toList hm)+ rmatStep :: R.RandomGen g => (a,a) -> ([a],g) -> ([a],g)+ rmatStep hilo (as,g1) = let (a,g2) = R.randomR hilo g1+ in (a:as,g2)+ (rElList,g') = foldr rmatStep ([],g) zipEls+ in (fromList rElList,g')+++-- | Type safe matrix multiplication+(.*) :: forall m k n a. Num a => Matrix m k a -> Matrix k n a -> Matrix m n a+(Matrix m) .* (Matrix n) = Matrix $ m * n++-- | Type safe scalar multiplication+(^*) :: forall m n a. Num a => a -> Matrix m n a -> Matrix m n a+x ^* (Matrix n) = Matrix $ M.scaleMatrix x n++-- | A row vector (a matrix with one row).+type Vector = Matrix 1++-- | /O(rows*cols)/. Generate a matrix from a generator function.+-- | The elements are 1-indexed, i.e. top-left element is @(1,1)@.+matrix :: forall m n a. (KnownNat m, KnownNat n)+ => ((Int, Int) -> a) -> Matrix (m :: Nat) (n :: Nat) a+matrix = Matrix . M.matrix m' n'+ where m' = fromInteger $ natVal @m Proxy+ n' = fromInteger $ natVal @n Proxy++-- | /O(rows*cols)/. The transpose of a matrix.+transpose :: forall m n a. Matrix m n a -> Matrix n m a+transpose (Matrix m) = Matrix . M.transpose $ m++-- | Horizontally join two matrices. Visually:+--+-- > ( A ) <|> ( B ) = ( A | B )+(<|>) :: forall m n k a. (KnownNat n, KnownNat k)+ => Matrix m n a -> Matrix m k a -> Matrix m (k+n) a+(Matrix x) <|> (Matrix y) = Matrix $ x M.<|> y++-- | /O(rows*cols)/. Identity matrix+identity :: forall n a. (Num a, KnownNat n) => Matrix n n a+identity = Matrix $ M.identity n'+ where n' = fromInteger $ natVal @n Proxy++-- | /O(rows*cols)/. The zero matrix+zero :: forall m n a. (Num a, KnownNat n, KnownNat m) => Matrix m n a+zero = Matrix $ M.zero m' n'+ where n' = fromInteger $ natVal @n Proxy+ m' = fromInteger $ natVal @m Proxy++-- | Create a matrix from a list of elements.+-- The list must have exactly length @n*m@. This is checked or else an +-- exception is thrown.+fromList :: forall m n a. (KnownNat m, KnownNat n) => [a] -> Matrix m n a+fromList as = if length as == n*m+ then Matrix $ M.fromList m n as+ else error $ "List has wrong dimension: "+ <>show (length as)+ <>" instead of "+ <>show (n*m)+ where n = fromInteger $ natVal @n Proxy+ m = fromInteger $ natVal @m Proxy++-- | Create a matrix from a list of rows. The list must have exactly @m@+-- lists of length @n@. An exception is thrown otherwise.+fromLists :: forall m n a. (KnownNat m, KnownNat n) => [[a]] -> Matrix m n a+fromLists as = if length as == m && all (\row -> length row == n) as+ then Matrix $ M.fromLists as+ else error $ "List has wrong dimension: "+ <>show (length as)<>":"+ <>show (length $ head as)+ <>" instead of "+ <>show m <>":"<> show n+ where n = fromInteger $ natVal @n Proxy+ m = fromInteger $ natVal @m Proxy++-- | Get the elements of a matrix stored in a list.+toList :: forall m n a. Matrix m n a -> [a]+toList (Matrix m) = M.toList m++-- | Get the elements of a matrix stored in a list of lists,+-- where each list contains the elements of a single row.+toLists :: forall m n a. Matrix m n a -> [[a]]+toLists (Matrix m) = M.toLists m+++-- | /O(1)/. Extract a submatrix from the given position. The size of the+-- extract is determined by the types, i.e. the parameters define which+-- element is the top-left element of the extract.+-- CAUTION: It is not checked if an extract is possible. Wrong parameters+-- will cause an exception.+submatrix :: forall m n m' n' a.+ (KnownNat m, KnownNat n, KnownNat m', KnownNat n'+ , m' <= m, n' <= n)+ => Int -> Int -> Matrix m n a -> Matrix m' n' a+submatrix i j (Matrix mat) = Matrix $ M.submatrix i (i+m'-1) j (j+n'-1) mat+ where n' = fromInteger $ natVal @n' Proxy+ m' = fromInteger $ natVal @m' Proxy++++-- | Reduced row echelon form. Taken from rosettacode. This is not the+-- implementation provided by the 'matrix' package.+-- https://rosettacode.org/wiki/Reduced_row_echelon_form#Haskell+rref :: forall m n a. (KnownNat m, KnownNat n, m <= n, Fractional a, Eq a)+ => Matrix m n a -> Matrix m n a+rref mat = fromLists $ f matM 0 [0 .. rows - 1]+ where + matM = toLists mat+ rows = length matM+ cols = length $ head matM++ f m _ [] = m+ f m lead (r : rs)+ | isNothing indices = m+ | otherwise = f m' (lead' + 1) rs+ where + indices = find p l+ p (col, row) = m !! row !! col /= 0+ l = [(col, row) |+ col <- [lead .. cols - 1],+ row <- [r .. rows - 1]]++ Just (lead', i) = indices+ newRow = map (/ m !! i !! lead') $ m !! i++ m' = zipWith g [0..] $+ replace r newRow $+ replace i (m !! r) m+ g n row+ | n == r = row+ | otherwise = zipWith h newRow row+ where h = subtract . (* row !! lead')++ replace :: Int -> b -> [b] -> [b]+ {- Replaces the element at the given index. -}+ replace n e t = a ++ e : b+ where (a, _ : b) = splitAt n t
+ test/Main.hs view
@@ -0,0 +1,121 @@+{-# LANGUAGE ScopedTypeVariables, DataKinds, TypeOperators, TypeFamilies #-}+{-# LANGUAGE FlexibleContexts, FlexibleInstances, MultiParamTypeClasses #-}+module Main where++import GHC.TypeLits (KnownNat, natVal, type (<=))+import Data.Maybe (fromJust)+import Data.Proxy (Proxy(..))+import Control.Applicative (empty)++import qualified Math.Algebra.Matrix as M+import Math.Algebra.Field.Instances -- Import random instances+import qualified Math.Core.Utils as F+import qualified Math.Algebra.Field.Base as F+import qualified Math.Algebra.Field.Extension as F+import qualified Math.Common.IntegerAsType as F+import Math.Algebra.Code.Linear+import System.Random (Random)++import Test.Tasty+import Test.Tasty.HUnit+import qualified Test.Tasty.SmallCheck as S+import qualified Test.Tasty.QuickCheck as Q+import qualified Test.SmallCheck.Series as S+import qualified Test.QuickCheck.Arbitrary as Q++main :: IO ()+main = defaultMain tests++tests = testGroup "linear-code" [ fieldTests, codeTests ]++fieldTests :: TestTree+fieldTests = testGroup "Associativity"+ [ S.testProperty "Associativity for (F2,+)" $+ prop_associativity ((+) :: F2 -> F2 -> F2)+ , S.testProperty "Associativity for (F2,*)" $+ prop_associativity ((*) :: F2 -> F2 -> F2)+ ]++codeTests :: TestTree+codeTests =+ let tc = trivialCode :: BinaryCode 5 3+ hamming74 = hamming :: BinaryCode 7 4+ in testGroup "Codes"+ [ testGroup "Instances"+ [ testCase "Show works for unknown distance" $+ show (trivialCode {distance=Nothing} :: LinearCode 7 4 F.F3)+ @?= "[7,4]_3-Code"+ , testCase "Show works for known distance" $+ show (trivialCode {distance=Just 3} :: LinearCode 7 4 F.F3)+ @?= "[7,4,3]_3-Code"+ ]+ , testGroup "Trivial code"+ [ testCase "Trivial binary code == codeFromA zero, [5,3]" $+ tc @?= codeFromA zero+ , testCase "Trivial binary code == codeFromA zero, [3,3]" $+ (trivialCode :: BinaryCode 3 3) @?= codeFromA zero+ , testCase "Trivial binary code == codeFromA zero, [7,1]" $+ (trivialCode :: BinaryCode 7 1) @?= codeFromA zero+ , testCase "zero vector is a code word" $+ assertBool ("H*c' = "++show (syndrome tc zero)) $+ isCodeword tc zero+ , testCase "ones-vector is not a code word" $+ let ones = fromList [1,1,1,1,1]+ in assertBool ("H*c' = "++show (syndrome tc ones)) $+ not $ isCodeword tc ones+ ]+ , testGroup "Random Code"+ [ Q.testProperty "Random code generation works" $+ \(c :: LinearCode 7 4 F.F3) -> seq c True+ , Q.testProperty "All generated codewords are codewords" $+ \c x y z w -> isCodeword (c :: LinearCode 7 4 F.F5) $+ encode c $ fromList ([x,y,z,w] :: [F.F5])+ ]+ , testGroup "Hamming(7,4)"+ [ S.testProperty "All encoded words are codewords" $+ \((x,y,z,w)::(F2,F2,F2,F2)) -> isCodeword hamming74+ (encode hamming74 (fromList [x,y,z,w]))+ , Q.testProperty "List all codewords" $+ \(c :: LinearCode 7 4 F.F5) ->+ length (codewords c) == 5^4+ , Q.testProperty "Simple decode of single error" $+ \(v :: Vector 4 F2) ->+ let c = encode hamming74 v :: Vector 7 F2+ in decode hamming74 (c + e2) == Just c+ ]+ , testGroup "Standard form"+ [ Q.testProperty "Standard form of standard form is equal" $+ \(c :: LinearCode 7 4 F.F3) ->+ let sc = standardFormGenerator c+ in sc == standardForm sc+ ]+ --, testGroup "Code transformers"+ -- [ testProperty "Dual of dual is identitiy" $+ -- \(c :: LinearCode 7 4 F2) -> (dualCode . dualCode) c == c+ -- ]+ ]++-- SmallCheck Series for GF+instance forall m f. (Monad m, F.FiniteField f) => S.Serial m f where+ series = S.generate $ \d -> take (d+1) (F.eltsFq 1 :: [f])++instance forall m n f. (KnownNat m, KnownNat n, Q.Arbitrary f)+ => Q.Arbitrary (M.Matrix m n f) where+ arbitrary = fromList <$> Q.vectorOf (n*m) Q.arbitrary+ where+ n = fromInteger . natVal $ (Proxy :: Proxy n)+ m = fromInteger . natVal $ (Proxy :: Proxy m)++instance forall p. F.IntegerAsType p => Q.Arbitrary (F.Fp p) where+ arbitrary = Q.arbitraryBoundedRandom++instance forall n k f.+ (KnownNat n, KnownNat k, k <= n, Num f, Ord f, Eq f, F.FinSet f, Random f)+ => Q.Arbitrary (LinearCode n k f) where+ arbitrary = Q.arbitraryBoundedRandom+++prop_associativity :: Eq m => (m -> m -> m) -> m -> m -> m -> Bool+prop_associativity (%) x y z = (x % y) % z == x % (y % z)++-- vim : set colorcolumn=80