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Prepared byKHO LEE CHIN

KNL 3503Information Theory and Error Coding

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P art II Topic

1. Cyclic Block Codes (WK8-10)2. Convolution Codes (WK11 -12)3. Coded Modulation (WK13)

4. Trellis Coded Modulation (WK14)

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Cyclic Block Codey Cyclic codes form an important subclass of linear block

codes.y Attractive due to:

y

Encoding and syndrome computation can be implementedeasily by employing shift registers with feedback connections.y Considerable inherent algebraic structure, it is possible to

diverse various practical methods for decoding them.y

Widely used in communication system error control.

y Efficient for error detection

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Cyclic Block Codey Cyclic codes have many advantages.

Elegant algebraic descriptions:c(x) = m(x)g(x) , where g(x) is generator polynomial

c(x)h(x) = 0 mod (xn 1) , where h(x) is parity-checkpolynomialc(1) = 0, . . . , c(t) = 0 , where i2 GF(qm)Shift register encoders and syndrome unitsSimple burst error correction (error trapping)Random error correction by solving polynomial equations

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Cyclic Block Code

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Error Detection Techniques

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Linear (N,K) Block Codes

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1. Definitiony The (7,4) linear code given in Table 5.1 is a cyclic codey To develop the algebraic properties of cyclic code, we treat

the components of a codeword as thecoefficients of a polynomial as follows:

y Thus, each codeword corresponds to a polynomial of degreen-1 or less. If the degree of v(X)is less than n-1.

y The correspondence between the codeword v and thepolynomial v(X) is one to one.

y Call v(X) the code polynomial of v.

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Table 5 .1. A (7,4) cyclic code generated by g(X)= 1+ X+ X^3

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y The term codeword and code polynomial is interchangeably.y The code polynomial that corresponds to codeword is

y There exist an interesting algebraic between v(X) and (X)y Multiplying v(X) by , we obtain

y The preceding equation can manipulated into following form

y The code polynomial (X) is simply the reminder resultingfrom dividing the polynomial

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Theorem1.

The nonzero code polynomial of minimum degree in a cyclic code C isunique.2. Let be the nonzero code polynomial of

minimum degree in an (n,k) cyclic code C. Then, the constant term must be equal to1.

3. Let be the nonzero code polynomial of minimum degree in an (n,k) cyclic code C. A binary polynomial of degree n-1 or less is a code polynomial if and only if it is a multiple of g(X).

4. In an (n,k) cyclic code, there exists one and only one code polynomial of degree n-k.

Every code polynomial is a multiple of g(X), and every binary polynomial of degree n-1 or less that is a multiple of g(X) is a code polynomial.

5. The generator polynomial g(X) of an (n,k) cyclic code is a factor of 6. If g(X) is a polynomial of degree n-k and is a factor of ,then g(X)

generates an (n,k) cyclic code.

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Example 1The polynomial can be factored as follows:

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y In summary, encoding in systematic form consists of threesteps:1. Premultiply the message2. Obtain the remainder b(X) (the parity check digits) from

dividing by generator polynomial g(X)3. Combine b(X) and to obtain the code polynomial

b(X)+

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Example 2Consider the (7,4) cyclic code generated by g(X)= 1+ X+ X 3.

Let u(X)= 1+ X 3 be the message to be encoded. Dividingx^3 u(x)=x^3+ X^6 by g(x)

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Exercise1. Cyclic code of blocklength 8. Find the generator polynomial

and the minimum distance for each binary cyclic code of blocklength 8.

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