Cyclic codes

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332437 Lecture 21 Cyclic and Hamming Codes

• Cyclic codes
• Berger codes
• Hamming codes
• Summary

Material from Design and Analysis of Fault
Tolerant Digital Systems, by Barry Johnson,
Addison Wesley
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Cyclic Codes

• Used on current disk systems
• Designed so that any end-around shift of code
  word produces another code word.
• Has a Boolean generator polynomial G (x) of
  degree n-k or greater
• n -- number bits in completed code word
• k -- number bits in original information to be
  encoded
• G (x) coefficients are 0 or 1

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Cyclic Codes (contd.)

• Called an (n, k) cyclic code
• Able to detect all single errors and all
  multiple, adjacent errors
• Very important for communications applications -
  have burst errors
• Transient fault that causes several adjacent
  errors in a data item
• Represent data as a Boolean polynomial

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Example

• 10112 is
• r2, is modulo 2 addition
• Is decimal multiplication
• Code word is the coefficients of a polynomial

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Cyclic Code

• Represents n-bit code word by polynomial of
  degree (n-1) or less
• V (x) called the code polynomial of the code word
  v
• Generate code polynomials by multiplying
  polynomial (representing data to be encoded) by
  generator polynomial

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Example

• Want to encode binary word
• Code polynomial
• Code is the coefficients (1010001)

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Another Example

• V(1001011)

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Cyclic Code Words for 4-bit Information Words
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Polynomial Interpretation
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Distance 3 Code

• Any 2-bit errors can be detected
• Find Hamming distance
• Compare all code words
• See that all possible pairs differ in at least 3
  bit positions

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Circuit Example for Generating a Cyclic Code Word
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Circuit Example

• For generating
• Performs polynomial multiplication
• Implement in digital logic using storage gates
  (flip-flops) and XORs
• Storage elements provide a time delay
• Data appears serially on D (x)
• Bits of code word appear serially on V (x)
• Registers initialized to 0

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Circuit Example for Generating a Cyclic Code Word
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Encoding Process for the Circuit
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Decoding Procedure

• Non-separable code
• Determine if code word (r0, r1,
  r2, , rn-1) is valid
• Represents polynomial r0 r1 x
  r2 x2 rn-1 xn-1
• Obtained supposedly as
• R (x) D (x) G (x) S (x)

Should be 0 --If not 0, invalid code
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Decoding Procedure (contd.)

• Divide R (x) by G (x) - see if remainder is 0
• S (x) is syndrome polynomial

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Division Circuit
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Circuit Analysis

• B (x) (x3 x) D (x)
• V (x) -- coded data
• D (x) decoded data
• Modulo 2 arithmetic -- and - same

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Proof that Circuit Divides
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Polynomial Division

• Initialize register to 0
• Circuit implements polynomial division
• After division, registers have value of syndrome
  or remainder of division
• 0 syndrome - V (x) was a valid code, otherwise,
  V (x) is invalid

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Decoding
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Example

• Code word should be 1010001
• Error caused it to be 1011001

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Decoding Circuit With Generator Polynomial G(x)
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Decoding Process With Erroneous Information
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Main Disadvantage of Cyclic Codes

• Not separable
• Possible to create a separable cyclic code
  (Nelson Carrole)
• n -- bits in complete code
• k -- original data bits
• Generate an (n,k) code
• Multiply D (x) by xn-k, divide result by G (x)
  to give R (x)

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Berger Codes

• Append check bits to each information word -
  separable
• Berger code of length n has
• I information bits
• k check bits

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Berger Code Creation

• Create binary having ones in I bits
• Complement and append to I bits

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Number of Required Check Bits in a Berger Code
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Berger Code Words for 4-bit Information Words
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Horizontal Vertical Parity

• 0001 0
• 1001 1
• 1010 1
• 1111 1
• 0010 0 Extra Vertical Parity
• Extra Horizontal Parity Bit

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Detects Any Single Error

• Can correct it too, because you know the location
  of the error
• Cannot help for multiple errors - can detect
  them but not correct them. If rows (i j) and
  columns (k m) have incorrect parity, error
  could be in bit positions (i,k), (i,m), (j,k), or
  (j,m)

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Hamming Error Correcting Codes (ECCs)

• Most common extension to Parity
• Inexpensive
• Only 10-40 redundancy
• Efficient - correction is fast
• Small time delays for encoding and decoding
• ECC circuit available on cheap chips
• Memory constitutes 60-70 of digital system
  faults transient faults more common as memories
  get denser

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Overlapping Parity

• Use c parity check bits to protect k bits of
  information
• 2c gt ck1
• Code word length nck
• Partition information into parity groups - set a
  parity bit for each group

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Example

• Information bits (d3,d2,d1,d0)
• Check bits (c1,c2,c3)
• Parity Groups (d3,d1,d0,c1),
• (d3,d2,d0,c2), (d3,d2,d1,c3)
• Each check bit set for parity (even or odd) of
  its group

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Check Bits Affected by Single Data Bit Errors
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Check Bits Are Syndrome

• Check bits affected for each possible erroneous
  bit
• Each bit error effect is unique -
• Erroneous bit can be located

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Process

• Encode original data - generate Cg parity check
  bits
• Transmit information and Cg bits
• Regenerate encoding Cc of received information
  bits
• If CcCg, information is correct

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Process (contd.)

• Otherwise
• Obtain syndrome S Cg Cc
• Number bits right to left and place information
  /check bits so that syndrome gives erroneous bit
  position number
• Feed syndrome to controlled complementation unit
  - complements erroneous bit to correct word

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Syndromes for Each Possible Single-Bit Error
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Syndrome Generator Hardware
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Controlled Complementer (Corrects Wrong Bits)

• Syndrome determines which bit (if an error
  occurs) is complemented

Corrected bits
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Hamming Single-error Correcting Code Memory
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Practical Hamming Code Systems

• Keep a log of correction quantities so that
  defective memory chips can be replaced
• Hamming code - erroneously corrects double-bit
  errors
• If both d0 and c2 are wrong, syndrome is 100
  which says that c1 should be corrected (not right)

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Modified Hamming Code

• Need to modify code to both correct single-bit
  errors and detect double-bit errors
• Add another parity bit over the entire Hamming
  Code word (including check bits)

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Modified Code Double-error Detection
Correction
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Single-bit and Double-bit Errors

• Single-bit error - syndrome says how to correct
  it, overall parity is wrong
• Double-bit error - overall parity is correct but
  non-zero syndrome occurs. Treat as an error but
  avoid erroneous correction

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Commercial ECC IC
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Time Overheads of Hamming Codes

• Extra write time - time to generate check bits
• Extra error detection time - 52 ns
• Correction detection time - 67 ns

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Code Selection Issues

• Must it be separable?
• How much extra time, storage, circuitry can be
  tolerated?
• How many bit errors can be detected/corrected?
• Can encoding/decoding occur in parallel with
  transmission/reception (yes for cyclic codes)

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Code Selection Issues (contd.)

• Do we need error detection, error correction, or
  both?
• bit errors that need to be detected/corrected?
• Determines Hamming distance Hd

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Summary

• Hamming code error detection/correction now
  necessary on present-day semiconductor memory
• Memory contributes 60-70 of digital system
  faults
• Transient faults more common as memories get
  denser
• Lack of EEC on memory in cheap PCs is a major
  source of their unreliability