diff --git a/ChangeLog.md b/ChangeLog.md new file mode 100644 --- /dev/null +++ b/ChangeLog.md @@ -0,0 +1,7 @@ +0.1.0 +----- + +* Initial release + - Includes trivial, hamming and random codes + - Implements syndrome decoding + diff --git a/LICENSE b/LICENSE new file mode 100644 --- /dev/null +++ b/LICENSE @@ -0,0 +1,674 @@ + GNU GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The GNU General Public License is a free, copyleft license for +software and other kinds of works. + + The licenses for most software and other practical works are designed +to take away your freedom to share and change the works. By contrast, +the GNU General Public License is intended to guarantee your freedom to +share and change all versions of a program--to make sure it remains free +software for all its users. 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But first, please read +. diff --git a/README.md b/README.md new file mode 100644 --- /dev/null +++ b/README.md @@ -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 +``` + diff --git a/Setup.hs b/Setup.hs new file mode 100644 --- /dev/null +++ b/Setup.hs @@ -0,0 +1,2 @@ +import Distribution.Simple +main = defaultMain diff --git a/linear-code.cabal b/linear-code.cabal new file mode 100644 --- /dev/null +++ b/linear-code.cabal @@ -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 +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 diff --git a/src/Math/Algebra/Code/Linear.hs b/src/Math/Algebra/Code/Linear.hs new file mode 100644 --- /dev/null +++ b/src/Math/Algebra/Code/Linear.hs @@ -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 . +-} +{-| +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 + + += 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 diff --git a/src/Math/Algebra/Field/Instances.hs b/src/Math/Algebra/Field/Instances.hs new file mode 100644 --- /dev/null +++ b/src/Math/Algebra/Field/Instances.hs @@ -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 . +-} +{-| +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 + + diff --git a/src/Math/Algebra/Field/Static.hs b/src/Math/Algebra/Field/Static.hs new file mode 100644 --- /dev/null +++ b/src/Math/Algebra/Field/Static.hs @@ -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 . +-} +{-| +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 + + diff --git a/src/Math/Algebra/Matrix.hs b/src/Math/Algebra/Matrix.hs new file mode 100644 --- /dev/null +++ b/src/Math/Algebra/Matrix.hs @@ -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 . +-} +{-| +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 diff --git a/test/Main.hs b/test/Main.hs new file mode 100644 --- /dev/null +++ b/test/Main.hs @@ -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