The Math Behind the Compact Disc

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The Math Behind the Compact Disc

• Linear Algebra and Error-Correcting Codes

william j. martin. mathematical sciences.
wpi wednesday december 3. 2008 fairfield
university
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How the device works
The compact disc is a complex system
incorporating interesting ideas from engineering,
physics, CS and math. We will focus only on the
mathematics of the error- correction
strategy. For more info on the CD, see Kelin
Kuhns book Laser Engineering
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Borrowed from K J Kuhns book Laser Engineering
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The Pits

• Each pit is 0.5 microns wide
• and 0.83 to 3.56 microns long.
• Tracks are separated by 1.6 microns of land
• Wavelength of green light is about 0.5 micron
• 40 tracks under one strand of human hair

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Modelling a CommunicationsChannel
Linear algebra model r me (vector add.)
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Channel with Error Correction
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Turn it into an algebra problem!

• A number system that the computer can understand
• F 0, 1
• Ordinary multiplication
• Addition 110

• Now music is turned into binary vectors!

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A bit (or a nibble?) of graph theory

• The n-cube is a type of Hamming graph
• Vertices are all binary n-tuples
• n-tuples are adjacent if they differ in only one
  coordinate
• Nice eigenvalues!

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Binary Vector Spaces

• The vectors are all possible binary n-tuples
  

0 0 1 0 1 1 1 0 1 0 1 1 0 0 0

0 0 1 1 1 1 0 0 0 0 0 0 0 0 1

0 0 0 1 0 0 1 0 1 0 1 1 0 0 1
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Hamming Distance

• The distance between two binary n-tuples x and y
  is the number of coordinates in which they differ

dist( 001100, 001011 ) 3

• This is a metric
• dist( x, y ) ? 0 with dist( x, y ) 0 iff
  xy
• dist( x, y ) dist( y, x )
• Triangle inequality
• dist( x, z ) ? dist( x, y ) dist( y, z )

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Theorem
n

• Let C (the code) be a subset of F with
  minimum distance between any two codewords equal
  to d.
• Then there exists an algorithm which corrects up
  to t errors per transmitted codeword if and only
  if d ? 2t 1.

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Proof

• If x and y are distinct codewords, then the
  balls of radius t around them are disjoint. So if
  the received vector is within distance t of x, it
  must be at distance gt t from any other codeword.
  So decoding is unique.

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A Useful Extension of the Theorem

• The above (computationally infeasible)
  decoding algorithm also correctly recovers from
  any t symbol errors and any s symbol erasures
  provided d gt 2ts.

transmit 0 1 1 2 2 3 0 receive 0 1 3 3 ? ?
? (here, t2 errors and s3 erasures)
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Small Example

• Let C denote the rowspace of the matrix
• Then C 000000, 110100, 011010, 101110,
• 001101, 111001, 010111, 100011
  
• and C has minimum distance 3 so C allows
  correction of any single-bit error in any
  transmitted codeword.

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The binary Hamming code

• Codewords 0 0 0 0 0 0 0 1 1 1 1 1 1 1
  
• 1 1 0 1 0 0 0 0 0
  1 0 1 1 1
• 0 1 1 0 1 0 0 1 0
  0 1 0 1 1
• 0 0 1 1 0 1 0 1 1
  0 0 1 0 1
• 0 0 0 1 1 0 1 1 1
  1 0 0 1 0
• 1 0 0 0 1 1 0 0 1
  1 1 0 0 1
• 0 1 0 0 0 1 1 1 0
  1 1 1 0 0
• 1 0 1 0 0 0 1 0 1
  0 1 1 1 0

• Quadratic Residues!
• In we have
• 1 6 1
• 4 5 4
• 3 2 4 2

Z
Z
7
2
2
2
2
2
2
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The Fano projective plane
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Vector Space F Poynts 1-dim.
subspaces Lynes 2-dim. subspaces
2
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C nullsp(H) where
All codewords 0 0 0 0 0 0 0
1 1 1 1 1 1 1 0 0 0 1 1 1 1
1 1 1 0 0 0 0 0 1 1 0 0 1 1
1 0 0 1 1 0 0 0 1 1 1
1 0 0 1 0 0 0 0 1
1 1 0 1 0 1 0 1 0 1
0 1 0 1 0 1 0 1 1 0 1 0
0 1 0 0 1 0 1 1 1 0 0 1 1 0
0 0 1 1 0 0 1 1 1 0 1 0 0 1
0 0 1 0 1 1 0
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Codes from polynomials

• Lets replace F0,1 with F0,1,,6 (with
  modular arithmetic). Now consider the vector
  space Fz of all polynomials in z with
  coefficients in F. For any subset N of F, we
  have a linear transformation
• L Fz ? F
• via f(z) ? f(0), f(1), f(2), f(3), f(4), f(5)
  (Here, we use, N0,1,2,3,4,5.)
• This is a Reed-Solomon code.

N
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Polynomials to Codewords

• Example
• Let the message be 1, 2, 2 (working mod 7)
• Polynomial is f(z) z 2 z 2
• Codeword is
• f(0), f(1), f(2), f(3), f(4), f(5) 2,
  5, 3, 3, 5, 2

2
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Reed-Solomon Codes

• FACT Two polynomials of degree less than k
  having k points of intersection must be equal.
• SO Reed-Solomon code of length nltq and dim k has
  min. dist. n-k1

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Compact Disc Parameters

• SONY/Philips design (1980)
• Music is sampled 44,100 times per second
• Each sample consists of 32 bits, representing
• left and right channel signal magnitude
  065535 (Pulse Code Modulation PCM)
• So chip must process 1,411,200 raw data bits per
  second
• But it gets much worse!

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Cross-Interleaved RS Codes

• Inner code is a 28-dimensional subspace of a
• 32-dimensional vector space over a finite field
  of size 256.
• Outer code is a 24-dimensional subspace of a
  28-dimensional vector space.
• Six 32-bit samples make up a 192-bit frame which
  is encoded as a 224-bit codeword. (Eventually,
  codewords have length 588 bits!)

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Encoding The numbers

• The codewords from the first code are interleaved
  into a virtually infinite array of 28 rows of
  symbols over GF(256).
• We pull out 8 binary columns (one symbol) to
  obtain a 28x8224-bit frame which is then encoded
  using another Reed-Solomon code to obtain a
  codeword of length 256 bits.

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Interleaving to disperse errors

• Codewords of first code are stacked like bricks
• 28 rows of vectors over GF(256)
• Extract columns and re-encode using second
  Reed-Solomon code

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Splitting Odd and Even Bits
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Back to the Pits

• Each pit is 0.5 microns wide
• and 0.83 to 3.56 microns long.
• Tracks are separated by 1.6 microns of land
• Not all 01-sequences can be recorded

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EFM Eight-to-Fourteen Modulation

• This encoding scheme can only store sequences
  where each consecutive pair of ones is separated
  by at least 2 and at most 10 zeros
• This is achieved by a mapping F ? F
• which is given by a lookup table.

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8
2
2
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Further Processing

• Three more merge bits are added to each of
  these 14
• So 256826433x8 bits, carrying six samples, or
  192 information bits, gets encoded as 588 channel
  bits on the disk
• This represents 0.000136 seconds of music

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What actually goes on the disc?

• We must do this 7,350 times per second
• So CD player reads 4,321,800 bits per second of
  music produced
• To get 74 minutes of music, we must store
• 74x60x4321800 19,188,792,000
• bits of data on the compact disc!

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When in doubt, erase

• Inner code has minimum distance 5 (over GF(256))
• Rather than correct two-symbol errors, the CD
  just erases the entire received vector.

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Sohow good is it?

• The two Reed-Solomon codes team up to correct
  burst errors of up to 4000 consecutive data
  bits (2.5 mm scratch on disc)
• If signal at time t cannot be recovered,
  interpolate
• With smart data distribution, this allows for
  recovery from burst errors of up to 12,000 data
  bits (7.5 mm track length on disc)
• If all else fails, mute, giving 0.00028 sec of
  silence.

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Other Applications

• Space communications (Mariner,Voyager,etc.)
• DVD, CD-R, CD-ROM
• Cell phones, internet packets
• Memory chips, hard drives, USB sticks
• RAID disk arrays
• Quantum computing

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The Last Slide
Thank You All!