Cyclic Code

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Cyclic Code
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Linear Block Code

• Hamming Code is a Linear Block Code. Linear Block
  Code means that the codeword is generated by
  multiplying the message vector with the generator
  matrix.
• Minimum weight as large as possible. If minimum
  weight is 2t1, capable of detecting 2t error
  bits and correcting t error bits.

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Cyclic Codes

• Hamming code is useful but there exist codes that
  offers same (if not larger) error control
  capabilities while can be implemented much
  simpler.
• Cyclic code is a linear code that any cyclic
  shift of a codeword is still a codeword.
• Makes encoding/decoding much simpler, no need of
  matrix multiplication.

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Cyclic code

• Polynomial representation of cyclic codes. C(x)
  C_n-1xn-1 C_n-2xn-2 C_1x1
  C_0x0, where, in this course, the
  coefficients belong to the binary field 0,1.
• That is, if your code is (1010011) (c6 first, c0
  last), you write it as x6x4x1.
• Addition and subtraction of polynomials --- Done
  by doing binary addition or subtraction on each
  bit individually, no carry and no borrow.
• Division and multiplication of polynomials. Try
  divide x3x2x1 by x1.

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Cyclic Code

• A (n,k) cyclic code can be generated by a
  polynomial g(x) which has degree n-k and is a
  factor of xn-1. Call it the generator
  polynomial.
• Given message bits, (m_k-1, , m_1, m_0), the
  code is generated simply as

• In other words, C(x) can be considered as the
  product of m(x) and g(x).

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Example

• A (7,4) cyclic code g(x) x3x1.
• If m(x) x31, C(x) x6x4x1.

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Cyclic Code

• One way of thinking it is to write it out as the
  generator matrix

• So, clearly, it is a linear code. Each row of the
  generator matrix is just a shifted version of the
  first row. Unlike Hamming Code.
• Why is it a cyclic code?

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Example

• The cyclic shift of C(x) x6x4x1 is C1(x)
  x5x2x1.
• It is still a code polynomial, because it the
  code polynomial if m(x) x21.

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Cyclic Code

• Given a code polynomial

• We have

• C1(x) is the cyclic shift of C(x) and (1) has a
  degree of no more than n-1 and (2) divides g(x)
  (why?) hence is a code polynomial.

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Cyclic Code

• So, to generate a cyclic code is to find a
  polynomial that (1) has degree n-k and (2) is a
  factor of xn-1.

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Generating Systematic Cyclic Code

• A systematic code means that the first k bits are
  the data bits and the rest n-k bits are parity
  checking bits.
• To generate it, you let

where

• The claim is that C(x) must divide g(x) hence is
  a code polynomial. 33 mod 7 6. Hence 33-628
  can be divided by 7.

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Division Circuit

• Division of polynomials can be done efficiently
  by the division circuit. (just to know there
  exists such a thing, no need to understand it)

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Remaining Questions for Those Really Interested

• Decoding. Divide the received polynomial by g(x).
  If there is no error you should get a 0 (why?).
  Make sure that the error polynomial you have in
  mind does not divide g(x).
• How to make sure to choose a good g(x) to make
  the minimum degree larger? Turns out to learn
  this you have to study more its the BCH code.

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Cyclic code used in IEEE 802

• g(x) x32 x26 x23 x22 x16 x12 x11
  x10 x8 x7 x5 x4 x2 x 1
• all single and double bit errors
• all errors with an odd number of bits
• all burst errors of length 32 or less

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Other codes

• RS code. Block code. Used in CD, DVD, HDTV
  transmission.
• LDPC code. Also block code. Reinvented after
  first proposed 40 some years ago. Proposed to be
  used in 802.11n. Achieve close-to-Shannon bound
• Trellis code. Not block code. More closely
  coupled with modulation.
• Turbo code. Achieve close-to-Shannon bound.