ecc1 - General Model message (m) 15-853:Algorithms in the...

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1 15-853 Page1 15-853:Algorithms in the Real World Error Correcting Codes I –Overv iew – Hamming Codes – Linear Codes 15-853 Page2 General Model codeword (c) coder noisy channel decoder message (m) message or error codeword’ (c’) Errors introduced by the noisy channel: • changed fields in the codeword (e.g. a flipped bit) • missing fields in the codeword (e.g. a lost byte). Called erasures How the decoder deals with errors. error detection vs. error correction 15-853 Page3 Applications Storage : CDs, DVDs, “hard drives”, Wireless : Cell phones, wireless links Satellite and Space : TV, Mars rover, … Digital Television : DVD, MPEG2 layover High Speed Modems : ADSL, DSL, . . Reed-Solomon codes are by far the most used in practice, including pretty much all the examples mentioned above. Algorithms for decoding are quite sophisticated. 15-853 Page4 Block Codes Each message and codeword is of fixed size = codeword alphabet k =|m| n = |c| q = | | C ⊆Σ n (codewords) Δ (x,y) = number of positions s.t. x i y i d = min{ Δ (x,y) : x,y C, x y} s = max{ Δ (c,c’)} that the code can correct Code described as: (n,k,d) q codeword (c) coder noisy channel decoder message (m) message or error
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15-853 Page5 Hierarchy of Codes cyclic linear BCH Hamming Reed-Solomon These are all block codes. Bose-Chaudhuri-Hochquenghem C forms a linear subspace of n of dimension k C is linear and c 0 c 1 c 2 …c n-1 is a codeword implies c 1 c 2 …c n-1 c 0 is a codeword 15-853 Page6 Binary Codes Today we will mostly be considering = {0,1} and will sometimes use (n,k,d) as shorthand for (n,k,d) 2 In binary Δ( x,y) is often called the Hamming distance 15-853 Page7 Hypercube Interpretation Consider codewords as vertices on a hypercube. 000 001 111 100 101 011 110 010 codeword The distance between nodes on the hypercube is the Hamming distance Δ . The minimum distance is d. 001 is equidistance from 000, 011 and 101. For s-bit error detection d s + 1 For s-bit error correction d 2s + 1 d = 2 = min distance n = 3 = dimensionality 2 n = 8 = number of nodes 15-853 Page8 Error Detection with Parity Bit A (k+1,k,2) 2 systematic code Encoding : m 1 m 2 …m k m 1 m 2 …m k p k+1 where p k+1 = m 1 m 2 m k d = 2 since the parity is always even (it takes two bit changes to go from one codeword to another). Detects one-bit error
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ecc1 - General Model message (m) 15-853:Algorithms in the...

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